Abstract
We consider the space of ordered pairs of distinct \({\mathbb{C}{\mathrm{P}}}^{1}\)structures on Riemann surfaces (of any orientations) which have identical holonomy, so that the quasiFuchsian space is identified with a connected component of this space. This space holomorphically maps to the product of the Teichmüller spaces minus its diagonal.
In this paper, we prove that this mapping is a complete local branched covering map. As a corollary, we reprove Bers’ simultaneous uniformization theorem without any quasiconformal deformation theory. Our main theorem is that the intersection of arbitrary two Poincaré holonomy varieties (\(\operatorname{SL}_{2}\mathbb{C}\)opers) is a nonempty discrete set, which is closely related to the mapping.
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1 Introduction
In 1960, Bers established a bijection between pairs of Riemann surface structures of opposite orientations and typical discrete and faithful representations of a surface group into \({\mathrm{PSL}}(2, \mathbb{C})\) up to conjugacy ([Ber60]). It is called Bers’ simultaneous uniformization theorem, and it gave a foundation for the later evolutional development of the hyperbolic threemanifold theory by Thurston ([Thu78]) and many others. In this paper, we partially generalize Bers’ theorem, in a certain sense, to generic surface representations into \({\mathrm{PSL}}(2, \mathbb{C})\), which are not necessarily discrete.
Throughout this paper, let S be a closed orientable surface of genus g>1. Given a quasiFuchsian representation \(\rho \colon \pi _{1}(S) \to {\mathrm{PSL}}(2, \mathbb{C})\), the domain of discontinuity is the union of disjoint topological open disks Ω^{+}, Ω^{−} in \({\mathbb{C}{\mathrm{P}}}^{1}\). Then, their quotients Ω^{+}/Imρ, Ω^{−}/Imρ have marked Riemann surface structures with opposite orientations.
Let S^{+}, S^{−} be S with opposite orientations. Then Bers’ simultaneous uniformization theorem asserts that this correspondence gives a biholomorphism
where QF is space of the quasiFuchsian representations \(\rho \colon \pi _{1}(S) \to {\mathrm{PSL}}(2, \mathbb{C})\) up to conjugation, T is the Teichmüller space of S^{+} and T^{∗} is the Teichmüller space of S^{−}; see [Hub06] [EK06] for the analyticity. (Note that T^{∗} is indeed antiholomorphic to T; see [Wol10].)
The \({\mathrm{PSL}}(2, \mathbb{C})\)character variety of S is the space of homomorphisms \(\pi _{1}(S) \to {\mathrm{PSL}}(2, \mathbb{C})\), roughly, up to conjugation, and it has two connected components ([Gol88]). Let denote the component consisting of representations \(\pi _{1}(S) \to {\mathrm{PSL}}(2,\mathbb{C})\) which lift to \(\pi _{1}(S) \to \mathrm{SL}(2, \mathbb{C})\); then strictly contains the (Euclidean) closure of QF.
A \({\mathbb{C}{\mathrm{P}}}^{1}\)structure on S is a locally homogeneous structure modeled on \({\mathbb{C}{\mathrm{P}}}^{1}\), and its holonomy is in . The quotients Ω^{+}/Imρ and Ω^{−}/Imρ discussed above have not only Riemann surfaces structures but also \({\mathbb{C}{\mathrm{P}}}^{1}\)structures on S^{+} and S^{−}, respectively. In fact, almost every representation in is the holonomy of some \({\mathbb{C}{\mathrm{P}}}^{1}\)structure on S [GKM00]; see §2.1 for details.
In fact, each \({\mathbb{C}{\mathrm{P}}}^{1}\)structure on S corresponds to a holomorphic quadratic differential on a Riemann surface structure on S (§2.1.2). Let P be the space all (marked) \({\mathbb{C}{\mathrm{P}}}^{1}\)structures on S^{+} with the fixed orientation, which is identified with the cotangent bundle of T. Similarly, let P^{∗} be the space of all marked \({\mathbb{C}{\mathrm{P}}}^{1}\) on S^{−}, identified with the cotangent bundle of T^{∗}.
By sending each quasiFuchsian representation \(\rho \colon \pi _{1}(S) \to {\mathrm{PSL}}(2, \mathbb{C})\) to the \({\mathbb{C}{\mathrm{P}}}^{1}\)structures Ω^{+}/Imρ and Ω^{−}/Imρ, the quasiFuchsian space QF holomorphically embeds into P×P^{∗} as a closed halfdimensional submanifold. The holonomy map
takes each \({\mathbb{C}{\mathrm{P}}}^{1}\)structure to its holonomy representation. Now we introduce the space of all ordered pairs of distinct \({\mathbb{C}{\mathrm{P}}}^{1}\)structures sharing holonomy
Let us denote this space by B for appreciation of the work of Bers. Since \(\operatorname{Hol}\) is locally biholomorphic, B is also a halfdimensional closed holomorphic submanifold. The map switching the order of C and D is a fixedpointfree biholomorphic involution of B. Then, the quasiFuchsian space QF is biholomorphically identified with two connected components of B, which are identified by this involution (Lemma 13.1). Every connected component of (P⊔P^{∗})^{2} contains at least one component of B which does not correspond to QF (see Lemma 2.5).
Let ψ:P⊔P^{∗}→T⊔T^{∗} be the projection from the space of all \({\mathbb{C}{\mathrm{P}}}^{1}\)structures on S^{+} and S^{−} to the space of all Riemann surface structures on S^{+} and S^{−}. Define Ψ:B→(T⊔T^{∗})^{2}∖Δ by Ψ(C,D)=(ψ(C),ψ(D)), where Δ is the diagonal {(X,X)∣X∈T⊔T^{∗}} (which can not intersect Ψ(B)).
It is a natural question to ask to what extent connected components of B resemble the quasiFuchsian space QF. In this paper, we prove a local and a global property of the holomorphic map Ψ:
Theorem A
The map Ψ is a complete local branched covering map.
(For the definition of complete local branched covering maps, see §2.5.) In particular, Ψ is open, and its fibers are discrete subsets of B. Thus its ramification locus is a nowheredense analytic subset, which may possibly be the empty set. (The completeness of Theorem A is given by Theorem 12.2, and the local property by Theorem B below.)
Note that, by the completeness in Theorem A, for every connected component Q of B, the restriction ΨQ is surjective onto its corresponding component of (T⊔T^{∗})^{2}∖Δ. We also show that, towards the diagonal Δ, the holonomy of \({\mathbb{C}{\mathrm{P}}}^{1}\)structures leaves every compact set in (see Proposition 12.6).
The deformation theory of hyperbolic cone manifolds is developed, especially, by Hodgson, Kerckhoff and Bromberg [HK98, HK05, HK08, Bro041, Bro042]). If cone angles exceed 2π, their deformation theory is established only under the assumption that the cone singularity is short and, thus, the tube radius is large. More generally, a conjecture of McMullen ([McM98, Conjecture 8.1]) asserts that the deformation space of geometricallyfinite hyperbolic conemanifolds is parametrized by using the cone angles and the conformal structures on the ideal boundary. Theorem A provides some additional evidence for the conjecture, when the cone angles are 2πmultiples (c.f. [Bro07]).
Bers’ simultaneous uniformization theorem is a consequence of the measurable Riemann mapping theorem. It thus is important that the domain Ω^{+}⊔Ω^{−} is a (full measure) subset of \({\mathbb{C}{\mathrm{P}}}^{1}\). However, in general, developing maps of \({\mathbb{C}{\mathrm{P}}}^{1}\)structures are not embeddings, and Bers’ proof does not apply to the other components of B. In fact, Theorem A implies the simultaneous uniformization theorem (§13). Thus we reprove Bers’ theorem genuinely from a complex analytic viewpoint, without any quasiconformal deformation theory.
Next we describe the local property in Theorem A. Since \(\operatorname{Hol}\) is locally biholomorphic, for every (C,D)∈B, if an open neighborhood V of (C,D) in B is sufficiently small, then \(\operatorname{Hol}\) embeds V onto a neighborhood U of \(\operatorname{Hol}(C) = \operatorname{Hol}(D)\) in . Let T_{C} and T_{D} be T or T^{∗} so that ψ(C)∈T_{C} and ψ(D)∈T_{D}, and define a holomorphic map Ψ_{C,D}:U→T_{C}×T_{D} by the restriction of Ψ to V and the identification V≅U. The following gives a finitetoone “parametrization” of U by pairs of Riemann surface structures associated with V.
Theorem B
Let (C,D)∈B. Then, there is a neighborhood V of (C,D) in B, such that \(\operatorname{Hol}\) embeds V into , and the restriction of Ψ to V is a branched covering map onto its image in T_{C}×T_{D} (Theorem 10.3.)
By the simultaneous uniformization theorem, for every X∈T^{∗} and Y∈T, the slices T×{Y} and {X}×T^{∗}, called the Bers’ slices, intersect transversally in the point in QF corresponding to (X,Y) by (1). The Teichmüller spaces T and T^{∗} are, as complex manifolds, open bounded pseudoconvex domains in \(\mathbb{C}^{3g3}\), where g is the genus of S. In order to prove Theorem A and Theorem B, we consider the analytic extensions of T×{Y} and {X}×T^{∗} in the character variety and analyze their intersection.
For each X∈T⊔T^{∗}, let P_{X} be the space of all \({\mathbb{C}{\mathrm{P}}}^{1}\)structures on X. Then P_{X} is an affine space of holomorphic quadratic differentials on X, and thus \(\mathsf{P}_{X} \cong \mathbb{C}^{3g3}\). Although the restrictions of the holonomy map \(\operatorname{Hol}\) to P and P^{∗} are nonproper and noninjective, the restriction of \(\operatorname{Hol}\) to P_{X} is a proper embedding ([Poi84, GKM00], see also [Tan99, Kap95, Dum17]). Let , which we shall call the Poincaré holonomy variety of X as its injectivity is due to Poincaré. Note that, if X∈T, then contains {X}×T^{∗} as a bounded pseudoconvex subset, and similarly, if Y∈T^{∗}, then contains T×{Y} as a bounded open subset.
The intersection theory of subvarieties and submanifolds in the character variety has been important ([Dum15, DW08] [Fal83, Theorem 12]). Since is half of , it is a basic question to ask what the intersection of such smooth subvarieties looks like.
Theorem C
For all distinct X, Y in T⊔T^{∗}, the intersection of and is a nonempty discrete set.
More precisely, we will show that contains at least one point if the orientations of X and Y are the same, and at least two points if the orientations are opposite (Corollary 12.7). Such a global understanding of in Theorem C is completely new. In fact, much of this paper is devoted to proving the discreteness of .
The deformation spaces, P and P^{∗}, of \({\mathbb{C}{\mathrm{P}}}^{1}\)structures have two distinguished parametrizations: namely, Schwarzian parametrization (§2.1.2) and Thurston parametrization (§2.1.6). In order to understand points in , we give a comparison theorem between those two parametrizations.
Let C be a \({\mathbb{C}{\mathrm{P}}}^{1}\)structure on a Riemann surface X. Then the quadratic differential of its Schwarzian parameters gives a vertical measured (singular) foliation V on X. The Thurston parametrization of C gives the measured geodesic lamination L on the hyperbolic surface. Dumas showed that V and L projectively coincide in the limit as C leaves every compact set in P_{X} ([Dum06, Dum07]), see also [O+].)
The measured geodesic lamination L of the Thurston parameter is also realized as a circular measured lamination \(\mathcal{L}\) on C, so that \(\mathcal{L}\) and L are the same measured lamination on S (§2.1.7). In this paper, we prove a more explicit asymptotic relation between the Thurston lamination \(\mathcal{L}\) and the vertical foliation V, without projectivization. For a quadratic differential q=ϕ dz^{2} on a Riemann surface X, let ∥q∥=∫_{X}ϕ dx dy, the L^{1}norm. Then we have the following.
Theorem D
Let X∈T⊔T^{∗}. For every ϵ>0, there is r>0, such that, if the holomorphic quadratic differential q on X satisfies ∥q∥>r, then, letting C be the \({\mathbb{C}{\mathrm{P}}}^{1}\)structure on X given by q, the vertical foliation V of q is (1+ϵ,ϵ)quasiisometric to \(\sqrt{2}\) times the Thurston lamination \(\mathcal{L}\) on C, up to an isotopy of X supported on the ϵneighborhood of the zero set of q in the uniformizing hyperbolic metric on X. (Theorem 4.1.)
(See 4 for the definition of being quasiisometric, and see §2.1.7 for the Thurston lamination on a \({\mathbb{C}{\mathrm{P}}}^{1}\)surface.) Theorem D is reminiscent of the (refined) estimates of high energy harmonic maps between hyperbolic Riemann surfaces by Wolf ([Wol91]).
Last we address that, in our setting, a variation of McMullen’s conjecture can be stated in a global manner:
Question 1.1
For every (or even some) nonquasiFuchsian component Q of B, is the restriction of Φ to Q a biholomorphic mapping onto its corresponding component of (T⊔T^{∗})^{2}?
1.1 Outline of this paper
In §3, we analyze the geometry of EpsteinSchwarz surfaces corresponding to \({\mathbb{C}{\mathrm{P}}}^{1}\)structures, using [Dum17] and [Bab]. In §4, we analyze the horizontal foliations of \({\mathbb{C}{\mathrm{P}}}^{1}\)structures on X and Y corresponding to the intersection points of in Theorem C. In fact, we show that such horizontal projectivized measured foliations projectively coincide towards infinity of (Theorem 4.5).
A (fat) traintrack is a surface obtained by identifying edges of rectangles in a certain manner. In §5, we introduce more general traintracks whose branches are not necessarily rectangles but more general polygons, cylinders, and even surfaces with staircase boundary (surface train tracks). In §6, given a certain pair of flat surfaces, we decompose them into the surface train tracks in a compatible manner.
In §7, we prove Theorem D. In §8, for every holonomy ρ in outside a large compact subset K of , we construct certain surface traintrack decompositions of \({\mathbb{C}{\mathrm{P}}}^{1}\)structures on X and Y corresponding to ρ in a compatible manner, using the decomposition of flat surfaces. In §9, from the compatible decompositions of the \({\mathbb{C}{\mathrm{P}}}^{1}\)structures, we construct an integervalued cocycle which changes continuously in . In §10, by this cocycle and some complex geometry, we prove the discreteness in Theorem C. In §12, the completeness of Theorem C is proven. In §11, we discuss the case when the orientations of X and Y are opposite. In §13, we give a new proof of Bers’ theorem.
2 Preliminaries
2.1 \({\mathbb{C}{\mathrm{P}}}^{1}\)Structures
(General references are [Dum09], [Kap01, §7].) Let F be a connected orientable surface. A \({\mathbb{C}{\mathrm{P}}}^{1}\)structure on F is a \(({\mathbb{C}{\mathrm{P}}}^{1}, {\mathrm{PSL}}(2, \mathbb{C}))\)structure. That is, a maximal atlas of charts embedding open sets of F into \({\mathbb{C}{\mathrm{P}}}^{1}\) with transition maps in \({\mathrm{PSL}}(2, \mathbb{C})\). Let \(\tilde{F}\) be the universal cover of F. Then, equivalently, a \({\mathbb{C}{\mathrm{P}}}^{1}\)structure is a pair of

a local homeomorphism \(f\colon \tilde{F} \to {\mathbb{C}{\mathrm{P}}}^{1}\) and

a homomorphism \(\rho \colon \pi _{1}(S) \to {\mathrm{PSL}}(2,\mathbb{C})\)
such that f is ρequivariant ([Thu97]). It is defined up to an isotopy of the surface and an element α of \({\mathrm{PSL}}(2, \mathbb{C})\), i.e. (f,ρ)∼(αf,α^{−1}ρα). The local homeomorphism f is called the developing map and the homomorphism ρ is called the holonomy representation of a \({\mathbb{C}{\mathrm{P}}}^{1}\)structure. We also write the developing map of C by \(\operatorname{dev}C\).
2.1.1 The holonomy map
The \({\mathrm{PSL}}(2, \mathbb{C})\)character variety of S is the space of the equivalence classes homomorphisms
where the quotient is the GITquotient (see [New06] for example). For the holonomy representations of \({\mathbb{C}{\mathrm{P}}}^{1}\)structures on S, the quotient is exactly given by the conjugation by \({\mathrm{PSL}}(2, \mathbb{C})\). Then, the character variety has exactly two connected components, distinguished by the lifting property to \(\mathrm{SL}(2, \mathbb{C})\); see [Gol88]. Let be the component consisting of representations which lift to \(\pi _{1}(S) \to \mathrm{SL}(2, \mathbb{C})\), and let P be the space of marked \({\mathbb{C}{\mathrm{P}}}^{1}\)structures on S. Then the holonomy map
takes each \({\mathbb{C}{\mathrm{P}}}^{1}\)structure to its holonomy representation. Then \(\operatorname{Hol}\) is a locally biholomorphic map, but not a covering map onto its image ([Hej75, Hub81, Ear81]). By Gallo, Kapovich, and Marden ([GKM00]), \(\rho \in {\mathrm{Im}}\operatorname{Hol}\) if and only if ρ is nonelementary and ρ has a lift to \(\pi _{1}(S) \to \mathrm{SL}(2,\mathbb{C})\). In particular, \(\operatorname{Hol}\) is almost onto .
2.1.2 The Schwarzian parametrization
(See [Dum09] [Leh87].) Let X be a Riemann surface structure on S. Then, the hyperbolic structure τ_{X} uniformizing X is, in particular, a \({\mathbb{C}{\mathrm{P}}}^{1}\)structure on X. For an arbitrary \({\mathbb{C}{\mathrm{P}}}^{1}\)structure C on X, the Schwarzian derivative gives a holomorphic quadratic differential on X by comparing with τ_{X}, so that τ_{X} corresponds to the zero differential. Then (X,q) is the Schwarzian parameters of C. Let QD(X) be the space of the holomorphic quadratic differentials on X, which is a complex vector space of dimension 3g−3. Thus, the space P_{X} of all \({\mathbb{C}{\mathrm{P}}}^{1}\) structures on X is identified with QD(X).
Theorem 2.1
[Poi84, Kap95], see also [Tan99, Dum17]
For every Riemann surface structure X on S, the set P_{X} of projective structures on X is property embedded in by \(\operatorname{Hol}\).
For X∈T⊔T^{∗}, let denote the smooth analytic subvariety . Pick any metric d on T and T^{∗} compatible with their topology (for example, the Teichmüller metric or the WeilPeterson metric).
Lemma 2.2
Let B be an arbitrary bounded subset of either T or T^{∗}. For every compact subset K in , there is ϵ>0, such that, if distinct X,Y∈B satisfy d(X,Y)<ϵ, then .
Proof
For each X∈T⊔T^{∗}, by Theorem 2.1, P_{X} is properly embedded in . For a neighborhood U of X, let D_{r}(U) denote the set of all holomorphic quadratic differentials q on Riemann surfaces Y in U such that the L^{1}norm ∥q∥ is less than r. Since \(\operatorname{Hol}\) is a local biholomorphism, for every X∈T⊔T^{∗} and \(r \in \mathbb{R}_{>0}\), there is a neighborhood U of X, \(\operatorname{Hol}\) embeds D_{r}(U) into . Let P_{U} be the space of all \({\mathbb{C}{\mathrm{P}}}^{1}\)structures whose complex structures are in U. Then, if r>0 is sufficiently large, we can, in addition, assume that \(K \cap \operatorname{Hol}(\mathsf{P}_{U}) = K \cap \operatorname{Hol}(D_{r}(U))\). Therefore, for all Y,W∈U, we have . □
2.1.3 Singular Euclidean structures
(See [Str84], [FD12].) Let q=ϕ dz^{2} be a quadratic differential on a Riemann surface X. Then q induces a singular Euclidean structure E on S from the Euclidean structure on \(\mathbb{C}\): Namely, for each nonsingular point z∈X, we can identify a neighborhood U_{z} of z with an open subset of \(\mathbb{C}\cong \mathbb{E}^{2}\) by the integral
along a path connecting z and w, where w∈U_{z} is a fixed base point (for details, see [Str84]). Then the zeros of q correspond to the singular points of E. Note that, for r>0, if the differential q is scaled by r, then the Euclidean metric E is scaled by \(\sqrt{r}\). Let E^{1} denote the normalization \(\frac{E}{ {\mathrm{Area}}E}\) of E by the area.
The complex plane \(\mathbb{C}\) is foliated by horizontal lines and, by the identification \(\mathbb{C}= \mathbb{E}^{2}\), the vertical length dy gives a canonical transversal measure to the foliation. Similarly, \(\mathbb{C}\) is also foliated by the vertical lines, and the horizontal length dx gives a canonical transversal measure to the foliation. Then, those vertical and horizontal foliations on \(\mathbb{C}\) induce vertical and horizontal singular foliations on E which meet orthogonally.
In this paper, a flat surface is the singular Euclidean structure obtained by a quadratic differential on a Riemann surface, which has vertical and horizontal foliations.
2.1.4 Measured laminations
(See [Thu78, EM87] for details) Let σ be a hyperbolic structure on the closed surface S. A geodesic lamination on σ is a set of disjoint geodesics whose union is a closed subset of S. A measured (geodesic) lamination L on σ is a pair of a geodesic lamination and its transversal measure. In this paper, for an arc α on σ transversal to L, we denote, by L(α), the transversal measure of α given by L. If we take a different hyperbolic structure σ′ on S, there is a unique geodesic representative on L on σ′. We thus can define measured laminations without fixing a specific hyperbolic structure on S.
2.1.5 Bending a geodesic in the hyperbolic threespace
The following wellknown lemma describes a closeness of a geodesic and a piecewise geodesic in \(\mathbb{H}^{3}\) with a small amount of bending.
Lemma 2.3
[CEG87, Theorem I.4.2.10]
Let \(c\colon [0,\ell ] \to \mathbb{H}^{3}\) be a piecewise geodesic parametrized by arc length. Let s(t) be the geodesic segment in \(\mathbb{H}^{3}\) connecting c(0) to c(t). Let θ(t) be the angle between the forward tangent vector of c at t and the forward tangent vector of s(t) at c(t).
For every ϵ>0 and r>0, there is δ>0 such that, if each smooth geodesic segment of c has length at least r and the exterior angle of c at every singular point of c is less than δ, then θ(t)<ϵ for all t∈[0,ℓ].
2.1.6 Thurston’s parameterization
By the uniformization theorem of Riemann surfaces, the space of all marked hyperbolic structures on S is identified with the space T of all marked Riemann surface structures. Let ML be the space of measured laminations on S. Note that \({\mathbb{C}{\mathrm{P}}}^{1}\) is the ideal boundary of \(\mathbb{H}^{3}\), so that \(\operatorname{Aut} {\mathbb{C}{\mathrm{P}}}^{1} = {\mathrm{Isom}}^{+} \mathbb{H}^{3}\). In fact, Thurston gave a parameterization of P using the threedimensional hyperbolic geometry.
Theorem 2.4
There is a natural (tangential) homeomorphism
Suppose that, by this homeomorphism, C=(f,ρ)∈P corresponds to a pair (σ,L)∈T×ML. Let \(\tilde{L}\) be the π_{1}(S)invariant measured lamination on \(\mathbb{H}^{2}\) obtained by lifting L. Then (σ,L) yields a ρequivariant pleated surface \(\beta \colon \mathbb{H}^{2} \to \mathbb{H}^{3}\), obtained by bending \(\mathbb{H}^{2}\) along \(\tilde{L}\) by the angles given by its transversal measure. The map β is called a bending map, and it is unique up to postcomposing with \({\mathrm{PSL}}(2, \mathbb{C})\).
2.1.7 Collapsing maps
([KP94]; see also [Bab20].) Let C≅(τ,L) be a \({\mathbb{C}{\mathrm{P}}}^{1}\)structure expressed in Thurston parameters. Let \(\tilde{C}\) be the universal cover of C. Then \(\tilde{C}\) can be regarded as the domain of f, so that \(\tilde{C}\) is holomorphically immersed in \({\mathbb{C}{\mathrm{P}}}^{1}\). A round disk is a topological open disk whose development is a round disk in \({\mathbb{C}{\mathrm{P}}}^{1}\), and a maximal disk is a round disk which is not contained in a strictly bigger round disk. In fact, for all \(z \in \tilde{C}\), there is a unique maximal disk D_{z} whose core contains z. Then there is a measured lamination \(\mathcal{L}\) on C obtained from the cores of maximal disks in the universal cover \(\tilde{C}\), such that \(\mathcal{L}\) is equivalent to L in ML. This lamination is the Thurston lamination on C. In addition, there is an associated continuous map κ:C→τ which takes \(\mathcal{L}\) to L, called the collapsing map.
Then, the bending map and the developing of C are related by the collapsing map κ and appropriate nearest point projections in \(\mathbb{H}^{3}\): Let \(\tilde {\kappa }\colon \tilde{C} \to \mathbb{H}^{2}\) be the lift of κ to a map between the universal covers. Let H_{z} be the hyperbolic plane in \(\mathbb{H}^{3}\) bounded by the boundary circle of D_{z}. There is a unique nearest point projection from D_{z} to H_{z}. Then \(\beta \circ \tilde {\kappa }(z)\) is the nearest point projection of f(z) to H_{z}.
2.2 Bers’ space
Recall, from §1, that B is the space of ordered pairs of \({\mathbb{C}{\mathrm{P}}}^{1}\)structures on S with identical holonomy, which may have different orientations.
Lemma 2.5
Every component of (P⊔P^{∗})^{2} contains, at least, one connected component of B which is not identified with the quasiFuchsian space.
Proof
By [GKM00], every nonelementary representation \(\rho \colon \pi _{1}(S) \to \mathrm{SL}(2, \mathbb{C})\) is the holonomy representation of infinitely many \({\mathbb{C}{\mathrm{P}}}^{1}\)structures on S^{+} whose developing maps are not embedding, and also of infinitely many \({\mathbb{C}{\mathrm{P}}}^{1}\)structures of S^{−} whose developing maps are not embedding. Therefore, since a quasiFuchsian component of B consists of pairs of \({\mathbb{C}{\mathrm{P}}}^{1}\)structures whose developing maps are embedding, every component of (P⊔P^{∗})^{2} contains at least one connected component of B, which is not a quasiFuchsian component. □
Lemma 2.6
B is a closed analytic submanifold of P⊔P^{∗} of complex dimension 6g−6.
Proof
It is a holomorphic submanifold, since is a local biholomorphism. As , the complex dimension of B is also 6g−6. Let (C_{i},D_{i}) be a sequence in B converging to (C,D) in (P⊔P^{∗})^{2}. Then, since \(\operatorname{Hol}C_{i} = \operatorname{Hol}D_{i}\), by the continuity of \(\operatorname{Hol}\), \(\operatorname{Hol}(C) = \operatorname{Hol}(D)\). Therefore B is closed. □
2.3 Angles between laminations
Let F be a surface with a hyperbolic or singular Euclidean metric. Let ℓ_{1}, ℓ_{2} be (nonoriented) geodesics on F with nonempty intersection. Then, for p∈ℓ_{1}∩ℓ_{2}, let ∠_{p}(ℓ_{1},ℓ_{2})∈[0,π/2] denote the angle between ℓ_{1} and ℓ_{2} at p.
Let L_{1} L_{2} be geodesic laminations or foliations on F. Then ∠(L_{1},L_{2}) be the infimum of ∠_{p}(ℓ_{1},ℓ_{2})∈[0,π/2] over all p∈L_{1}∩L_{2} where ℓ_{1} and ℓ_{2} are leaves of L_{1} and L_{2}, respectively, containing p. By convention, if L_{1}∩L_{2}=∅, then ∠(L_{1},L_{2})=0. We say that L_{1} and L_{2} are ϵparallel, if ∠(L_{1},L_{2})<ϵ.
2.4 The MorganShalen compactification
(See [CS83, MS84], see also [Kap01, §10.3].) The MorganShalen compactification is a compactification of \({\mathrm{PSL}}(2, \mathbb{C})\)character variety, introduced in [CS83, MS84]. For our , each boundary point corresponds to a minimal action of π_{1}(S) on a \(\mathbb{R}\)tree, π_{1}(S)↷T.
Every holonomy \(\rho \colon \pi _{1}(S) \to {\mathrm{PSL}}(2, \mathbb{C})\) induces a translation length function \(\rho ^{\ast}\colon \pi _{1}(S) \to \mathbb{R}_{\geq 0}\), and a minimal action π_{1}(S) on a \(\mathbb{R}\)tree also induces a translation length function. Then converges to a boundary point π_{1}(S)↷T if the length function \(\rho ^{\ast}_{i}\) projectively converges to the projective class of the translation function of π_{1}(S)↷T as i→∞.
2.5 Complex geometry
We recall some basic complex geometry used in this paper. Let U, W be complex manifolds of the same dimension. A holomorphic map ϕ:U→W is a (finite) branched covering map if

there are closed analytic subsets U′, W′ of dimensions strictly smaller than dimU=dimW, such that the restriction of ϕ to U∖U′ is a covering map onto W∖W′, and

its covering degree is finite. (See [FG02, p227].)
A holomorphic map ϕ:U→W is a local branched covering map if, for every z∈U, there is a neighborhood V of z in U such that the restriction ϕV is a branched covering map onto its image. A holomorphic map U→W is complete if it has the (not necessarily unique) path lifting property ([AS60]).
Let U be an open subset of \(\mathbb{C}^{n}\). Then a subset V of U is analytic if it is locally an intersection of zeros of finitely many holomorphic functions.
Proposition 2.7
Proposition 6.1 in [FG02]
Every connected bounded analytic set in \(\mathbb{C}^{n}\) is a discrete set.
Theorem 2.8
p107 in [GR84], Theorem 7.9 in [HY99]
Let \(U \subset \mathbb{C}^{n}\) be a region. Suppose that \(f\colon U \to \mathbb{C}^{n}\) is a holomorphic map with discrete fibers. Then it is an open map.
3 Approximations of EpsteinSchwarz surfaces
3.1 Epstein surfaces
(See Epstein [Eps], and also Dumas [Dum17].) Let C be a \({\mathbb{C}{\mathrm{P}}}^{1}\)structure on S. Fix a developing pair (f,ρ) of C, where \(f \colon \tilde{C} \to {\mathbb{C}{\mathrm{P}}}^{1}\) is the developing map and \(\rho \colon \pi _{1}(S) \to {\mathrm{PSL}}(2,\mathbb{C})\) is the holonomy representation, which is unique up to \({\mathrm{PSL}}(2,\mathbb{C})\). For \(z \in \mathbb{H}^{3}\), by normalizing the ball model of \(\mathbb{H}^{3}\) so that z is the center, we obtain a spherical metric \(\nu _{\mathbb{S}^{2}}(z)\) on \(\partial _{\infty }\mathbb{H}^{3} = {\mathbb{C}{\mathrm{P}}}^{1}\).
Given a conformal metric μ on C, there is a unique map \(\operatorname{Ep}\colon \tilde{C} \to \mathbb{H}^{3}\) such that, for each \(x \in \tilde{C}\), the pull back of \(\nu _{\mathbb{S}^{2}} \operatorname{Ep}(z)\) coincides with \(\tilde{\mu}\) at z. This map is ρequivariant, and called the Epstein surface.
3.2 Approximation
Let C=(X,q) be a \({\mathbb{C}{\mathrm{P}}}^{1}\)structure on S expressed in Schwarzian coordinates, where q is a holomorphic quadratic differential on a Riemann surface X. Then q yields a flat surface structure E on S. Moreover q gives a vertical measured foliation V and a horizontal measured foliation H on E.
Let \(\operatorname{Ep}\colon \tilde{S} \to \mathbb{H}^{3}\) be the Epstein surface of C with the conformal metric given by E. Then, let \(\operatorname{Ep}^{\ast}\colon T\tilde{S} \to T \mathbb{H}^{3}\) be the derivative of \(\operatorname{Ep}\), where \(T\tilde{S}\) and \(T \mathbb{H}^{3}\) denote the tangent bundles. Let \(d\colon \tilde{E} \to \mathbb{R}_{\geq 0}\) be the distance function from the singular set \(\tilde{Z}_{q}\) with respect to the singular Euclidean metric of \(\tilde{E}\).
Let v′(z) be the vertical unit tangent vector of \(\tilde{E}\) at a smooth point z. Similarly, let h′(z) be the horizontal unit tangent vector at a smooth point z of \(\tilde{E}\).
Lemma 3.1
[Eps], Lemma 2.6 and Lemma 3.4 in [Dum17]

(1)
\(\ \operatorname{Ep}^{\ast }h'(z)\ < \frac{6}{d(z)^{2}}\);

(2)
\(\sqrt{2} < \ \operatorname{Ep}^{\ast }v'(z)\ < \sqrt{2} + \frac{6}{d(z)^{2}}\);

(3)
h′(z), v′(z) are principal directions of \(\operatorname{Ep}\) at z;

(4)
\(k_{v} < \frac{6}{d(z)^{2}}\), where k_{v} is the principal curvature of \(\operatorname{Ep}\) in the vertical direction.
Consider the Euclidean metric on \(\mathbb{C}\cong \mathbb{E}^{2}\). By the exponential map \(\exp \colon \mathbb{C}\to \mathbb{C}^{\ast}\), we push forward a complete Euclidean metric to \(\mathbb{C}^{\ast}\), which is invariant under the action of \(\mathbb{C}^{\ast}\). If a simply connected region Q in the flat surface E contains no singular points, then Q is immersed into \(\mathbb{C}\) locally isometrically preserving horizontal and vertical directions. Using Lemma 3.1 and the definition of Epstein surfaces, one obtains the following.
Lemma 3.2
[Bab, Lemma 12.15]
For every ϵ>0, there is r>0, such that if Q is a region in E satisfying

Q has Ediameter less than r, and

the distance from the singular set of E is more than r.
then \(\exp \colon \mathbb{C}\to \mathbb{C}^{\ast}\) and the developing map are ϵclose pointwise with respect to the complete Euclidean metrics.
We shall further analyze vertical curves on Epstein surfaces. Let \(v\colon [0,\ell ] \to \tilde{E}\) be a path in a vertical leaf, such that v contains no singular point and has a constant speed \(\frac{1}{\sqrt{2}}\) in the Euclidean metric. Let \(\operatorname{Ep}^{\perp}(z)\) be the unit normal vector of the Epstein surface \(\operatorname{Ep}\) at each smooth point \(z \in \tilde{E}\). Let s_{t} be the geodesic segment in \(\mathbb{H}^{3}\) connecting \(\operatorname{Ep}v(0)\) to \(\operatorname{Ep}v(t)\); see Fig. 1.
The following lemma is an analogue of Lemma 2.3 for smooth curves.
Lemma 3.3
For every ϵ>0, there is (large) ω>0 only depending on ϵ, such that, w.r.t. the Emetric, if the distance of the vertical segment v from the zeros Z_{q} of q is more than ω, then the angle between \(\operatorname{Ep}^{\ast }v'(t)\) and the geodesic containing s_{t} is less than ϵ for all t. (Fig. 1.)
Proof
In fact, the proof of this lemma is essentially reduced to the analogous lemma (Lemma 2.3) for piecewise geodesic curves as follows.
Fix a Riemannian metric on the tangent bundle of \(\mathbb{H}^{3}\) which is invariant under the isometries of \(\mathbb{H}^{3}\). Then, by Lemma 3.1 (2) and (4), for every ϵ_{1}>0, there is sufficiently large ω>0 such that, if a vertical segment \(v\colon [0, \ell ] \to \tilde{E}\) of unit speed has length less than \(\frac{1}{\epsilon }\) and distance from Z_{p} at least ω, then the smooth curve \(\operatorname{Ep}\circ \,v\) is ϵ_{1}close to the geodesic segment connecting the endpoints of \(\operatorname{Ep}\circ \,v\) in the C^{1}topology with respect to the invariant metric. Therefore, the lemma holds true under the additional assumption that the length of v is uniformly bounded from above.
Now, without any upper bound on the length, let \(v\colon [0, \ell ] \to \tilde{E}\) be a vertical segment of unit speed which has distance at least ω from Z_{p}. Let ϵ_{1}>0 be a constant. Then we decompose v into n segments v_{1},v_{2},…,v_{n} so that the first n−1 segments v_{1},v_{2},…,v_{n−1} have length exactly \(\frac{1}{\epsilon _{1}}\) and the last segment v_{n} has length at most \(\frac{1}{\epsilon _{1}}\). For all i=1,2,…,n, let u_{i} be the geodesic segment connecting the endpoints of \(\operatorname{Ep}\circ v_{i}\). Then, by the argument above, for every ϵ_{2}>0, if ϵ_{1}>0 is sufficiently small, then the piecewise geodesic curve \(\cup _{i = 1}^{n} u_{i}\) is ϵ_{2}close to \(\operatorname{Ep}\circ v\) in C^{1}topology. We can, in addition, assume that the exterior angle at the common endpoint of u_{i} and u_{i+1} is less than ϵ_{2} for all i=1,2,…,n_{1}. Therefore, by Lemma 2.3, for every ϵ_{2}>0, if ϵ_{1}>0 is sufficiently small, then the piecewise geodesic curve \(\cup _{i = 1}^{n} u_{i}\) is ϵ_{2}close to the geodesic segment connecting the endpoints of \(\operatorname{Ep}\circ v\) in C^{1}topology. (See Fig. 2.)
Therefore, for every ϵ>0, if ϵ_{1}>0 is sufficiently small, then \(\operatorname{Ep}\circ v\) is ϵclose to the geodesic segment connecting its endpoints. Then the lemma immediately follows. □
Define \(\theta \colon [0,\ell ] \to T_{\operatorname{Ep}v(0)}\) by the parallel transport of \(\operatorname{Ep}^{\perp }(t)\) along s_{t} to the starting point \(\operatorname{Ep}(v(0))\); see Fig. 3. Let H be the (totally geodesic) hyperbolic plane in \(\mathbb{H}^{3}\) orthogonal to the tangent vector \(\operatorname{Ep}^{\ast }v'(0)\), so that H contains \(\operatorname{Ep}^{\perp }v(0)\). Then, Lemma 3.3, implies:
Corollary 3.4
For every ϵ>0, there is (large) ω>0 only depending on ϵ such that, if the Hausdorff distance between v and the zeros Z_{q} of q is more than ω w.r.t. the Emetric, then ∠_{v(0)}(θ(t),H)<ϵ for all t∈[0,ℓ].
Recall that the \({\mathrm{PSL}}_{2}\mathbb{C}\)character variety χ of the surface S is an affine algebraic variety. Then we say a compact subset K in the character variety χ or the holonomy variety χ_{X} for \(X \in \overline {\mathsf{T}}\) is sufficiently large, if K contains a sufficiently large ball in the ambient affine space centered at the origin.
Proposition 3.5
Total curvature bound in the vertical direction
For all X∈T∪T^{∗} and all ϵ>0, there is a bounded subset K=K(X,ϵ) in , such that, for , if a vertical segment v has normalized length less than \(\frac{1}{\epsilon }\) and has normalized Euclidean distance from the zeros of q_{X,ρ} at least ϵ, then the total curvature along v is less than ϵ.
Proof
For every r>0, if K is sufficiently large, then, if a \({\mathbb{C}{\mathrm{P}}}^{1}\)structure C=(X,q) on X has holonomy outside K, then the distance from Z_{q} to v is at least r. Then the proposition immediately follows from Dumas’ estimate in Lemma 3.1 (4). □
Consider the projection \(\hat{\theta}(t)\) of \(\theta (t) \in T^{1}_{v(0)} \mathbb{H}^{3}\) to the unit tangent vector in H at v_{0}. Let \(\eta \colon [0, \ell ] \to \mathbb{R}\) be the continuous function of the total increase of \(\hat{\theta}(t) \colon [0, \ell ] \to \mathbb{R}\), so that η(0)=0 and \(\eta '(t) = \hat{\theta}'(t)\).
Proposition 3.6
Let X∈T⊔T^{∗}. For every ϵ>0, there is a bounded subset K=K(X,ϵ)>0 in , such that, if

C∈P_{X} has holonomy in ;

a vertical segment v of the normalized flat surface \(E_{C}^{1}\) has the length less than \(\frac{1}{\epsilon }\);

the normalized distance of v from the singular set Z_{C} of \(E_{C}^{1}\) is more than ϵ,
then, η′(t)<ϵ for t∈[0,ℓ] and \(\int _{0}^{\ell }\eta '(t)  < \epsilon \). In particular, η(t)<ϵ for all t∈[0,ℓ].
Proof
The absolute value of θ′(t) is bounded from above by the curvature of \(\operatorname{Ep}\circ v\colon [0, \ell ] \to \mathbb{H}^{3}\) at t. Therefore η′(t) is bounded from above the curvature. Thus, for every ϵ>0, if K is sufficiently large, then by Lemma 3.1 (4), then η′(t)<ϵ for all t∈[0,ℓ], regardless of the choice of the vertical segment v. Therefore, by Proposition 3.5, if K is sufficiently large, \(\int _{0}^{\ell }\eta '(t)  < \epsilon \) holds. □
Let α be the biinfinite geodesic in \(\mathbb{H}^{3}\) through \(\operatorname{Ep}(v(0))\) and \(\operatorname{Ep}(v(\ell ))\). Let p_{1}, p_{2} denote the endpoints of α in \({\mathbb{C}{\mathrm{P}}}^{1}\). If a hyperbolic plane in \(\mathbb{H}^{3}\) is orthogonal to α, then its ideal boundary is a round circle in \({\mathbb{C}{\mathrm{P}}}^{1} \setminus \{p_{1}, p_{2}\}\). Moreover \({\mathbb{C}{\mathrm{P}}}^{1} \setminus \{p_{1}, p_{2}\}\) is foliated by round circles which bound hyperbolic planes orthogonal to α.
If a hyperbolic plane in \(\mathbb{H}^{3}\) contains the geodesic α, then its ideal boundary is a round circle containing p_{1} and p_{2}. Then, by considering all such hyperbolic planes, we obtain another foliation \(\mathcal{V}\) of \({\mathbb{C}{\mathrm{P}}}^{1} \setminus \{p_{1}, p_{2}\}\) by circular arcs connecting p_{1} and p_{2}. Then \(\mathcal{V}\) is orthogonal to the foliation by round circles. Note that \(\mathcal{V}\) has a natural transversal measure given by the angles between the circular arcs at p_{1} (and p_{2}). Then the transversal measure is invariant under the rotations of \(\mathbb{H}^{3}\) about α, and its total measure is 2π. Given a smooth curve c on \({\mathbb{C}{\mathrm{P}}}^{1} \setminus \{p_{1}, p_{2}\}\) such that c decomposes into finitely many segments c_{1},c_{2},…c_{n} which are transversal to \(\mathcal{V}\), possibly, except at their endpoints. Let \(\mathcal{V} (c)\) denote the “total” transversal measure of c given by \(\mathcal{V}\), the sum of the transversal measures of c_{1},c_{2},…,c_{n}. Then, Proposition 3.6 implies the following.
Corollary 3.7
For every ϵ>0, there is a bounded subset , such that, if

C∈P_{X} has holonomy in ,

a vertical segment v of E_{C} has the normalized length less than \(\frac{1}{\epsilon }\), and

the normalized distance of v from the zeros Z_{C} is more than ϵ,
then, the curve \(f  v\colon [0, \ell ] \to {\mathbb{C}{\mathrm{P}}}^{1}\) intersects \(\mathcal{V}\) at angles less than ϵ, and the total \(\mathcal{V}\)transversal measure of the curve is less than ϵ.
Definition 3.8
Let v be a unit tangent vector of \(\mathbb{H}^{3}\) at \(p \in \mathbb{H}^{3}\). Let H be a totally geodesic hyperbolic plane in \(\mathbb{H}^{3}\). For ϵ>0, v is ϵalmost orthogonal to H if \(dist_{\mathbb{H}^{3}}(H, p) < \epsilon \) and the angle between the geodesic g tangent to v at p and H is ϵclose to π/2.
Fix a metric on the unit tangent space \(T^{1} \mathbb{H}^{3}\) invariant under \({\mathrm{PSL}}(2, \mathbb{C})\). For ϵ>0, let \(N_{\epsilon }Z_{X, \rho}^{1}\) denote the ϵneighborhood of the singular set \(Z_{X, \rho}^{1}\) of the normalized flat surface \(E^{1}_{X, \rho}\).
Theorem 3.9
Fix arbitrary X∈T⊔T^{∗}. For every ϵ>0, if a bounded subset is sufficiently large, then, for all ,

(1)
if a vertical segment v in \(E_{X, \rho}^{1} \setminus N_{\epsilon }Z_{X, \rho}^{1}\) has length less than \(\frac{1}{\epsilon }\), then the total curvature of \(\operatorname{Ep}_{X, \rho}  v\) is less than ϵ, and

(2)
if a horizontal segment h in \(E_{X, \rho}^{1} \setminus N_{\epsilon }Z_{X, \rho}^{1}\) has length less than \(\frac{1}{\epsilon }\), then for the vertical tangent vectors w along the horizontal segment h, their images \(\operatorname{Ep}_{X, \rho}^{\ast}(w)\) are ϵclose in the unit tangent bundle of \(\mathbb{H}^{3}\).
Proof
(1) is already by Proposition 3.5. By [Bab, Proposition 4.7], we have (2).
4 Comparing measured foliations
4.1 Thurston laminations and vertical foliations
Let L_{1}, L_{2} be measured laminations or foliations on a surface F. Then L_{1} and L_{2} each define a pseudometric almost everywhere on F: for all x,y∈F not contained in a leaf of L_{i} with atomic measure, consider the minimal transversal measure of all arcs connecting x to y. We say that, for ϵ>0, L_{1} is (1+ϵ,ϵ)quasiisometric to L_{2}, if for almost all x,y∈F,
We shall compare a measured lamination of the Thurston parametrization and a measured foliation from the Schwarzian parametrization of a \({\mathbb{C}{\mathrm{P}}}^{1}\)surface.
Theorem 4.1
For every ϵ>0, there is r>0 with the following property: For every C∈P⊔P^{∗}, then, letting (E,V) be its associated flat surface, if disk D in E has radius less than \(\frac{1}{\epsilon }\) and the distance between D and the singular set Z of E is more than r, then the vertical foliation V of C is (1+ϵ,ϵ)quasi isometric to \(\sqrt{2}\) times the Thurston lamination L of C on D.
Proof of Theorem 4.1
It suffices to show the assertion when D is a unit disk. Since D contains no singular point, we can regard D as a disk in \(\mathbb{C}\) by the natural coordinates given by the quadratic differential. The scaled exponential map
is a good approximation of the developing map sufficiently away from zero (Lemma 3.2), which was proved using Dumas’ work [Dum17]). Let C_{0} be the \({\mathbb{C}{\mathrm{P}}}^{1}\)structure on \(\mathbb{C}\) whose developing map is \(\exp (\sqrt{2}\, \ast )\). The next lemma immediately follows from the construction of Thurston coordinates.
Lemma 4.2
The Thurston lamination on C_{0} is the vertical foliation of \(\mathbb{C}\) with a transversal measure given by the horizontal Euclidean distance.
Let D_{x} be the maximal disk in \(\tilde{C}\) centered at x. Let D_{0,x} be the maximal disk in C_{0} centered at x by the inclusion \(D \subset \mathbb{C}\). When \({\mathbb{C}{\mathrm{P}}}^{1}\) is identified with \(\mathbb{S}^{2}\) so that the center O of the disk D map to the north pole and the maximal disk in \(\tilde{C}\) centered at O maps to the upper hemisphere. If r>0 is sufficiently large, then the \(\operatorname{dev} D\) is close to \(\exp (\sqrt{2}\, \ast )\). Then, for every x∈D, its maximal disk D_{x} in \(\tilde{C}\) is ϵclose to the maximal disk D_{0,x} in C_{0}, and the ideal point ∂_{∞}D_{x} is ϵHausdorff close to the idea boundary ∂_{∞}D_{0,x} on \(\mathbb{S}^{2}\).
Therefore, by [Bab17, Theorem 11.1, Proposition 3.6], the convergence of canonical neighborhoods implies the assertion.
A staircase polygon is a polygon in a flat surface whose edges are horizontal or vertical (see Definition 5.1).
Theorem 4.3
For every X∈T⊔T^{∗} and every ϵ>0, there is a constant r>0 with the following property: Suppose that C is a \({\mathbb{C}{\mathrm{P}}}^{1}\)structure on X and C contains a staircase polygon P w.r.t. its flat surface structure (E,V), such that the (E)distance from ∂P to the singular set Z of E is more than r. Then, letting \(\mathcal{L}\) denote the Thurston lamination of C, the restriction of \(\mathcal{L}\) of C to P with its transversal measure scaled by \(\sqrt{2}\) is (1+ϵ,ϵ)quasiisometric to the vertical foliation V on P up to a diffeomorphism supported on the r/2neighborhood of the singular set in P.
Proof
Let N_{r/2}Z denote the r/2neighborhood of Z. If r>0 is sufficiently large, then, for each disk D of radius \(\frac{r}{4}\) centered at a point on E∖N_{r/2}(Z), the assertion holds by Theorem 4.1.
Since ∂P∩N_{r/2}Z=∅, there is an upper bound for lengths of edges of such staircase polygons P with respect to the normalized Euclidean metric E^{1}.
Lemma 4.4
For every ϵ>0, if r>0 is sufficiently large, then for every vertical segment v of VP whose distance from the singular set Z is more than r/2, we have \(\mathcal{L}(v) < \epsilon \).
Proof
This follows from Corollary 3.7. □
By Theorem 4.1 and Lemma 4.4, V and \(\mathcal{L}\) are (1+ϵ,ϵ)quasiisometric on P minus N_{r/2}Z. Note that V and \(\mathcal{L}\) in P∩N_{r/2} are determined by V and \(\mathcal{L}\) in P∖N_{r/2} up to an isotopy, respectively. Therefore, as desired, V and \(\mathcal{L}\) are (1+ϵ,ϵ)quasiisometric on P, up to a diffeomorphism supported on N_{r/2}Z.
4.2 Horizontal foliations asymptotically coincide
Let X,Y∈T⊔T^{∗} with X≠Y. Let and let , the holonomy varieties of X and Y, respectively. Suppose that ρ_{i} is a sequence in which leaves every compact set in . Then, let C_{X,i} and C_{Y,i} be the \({\mathbb{C}{\mathrm{P}}}^{1}\)structures on X and Y, respectively, with holonomy ρ_{i}. Similarly, let H_{X,i} and H_{Y,i} denote the horizontal measured foliations of C_{X,i} and C_{Y,i}. Then, up to a subsequence, we may assume that ρ_{i} converges to a π_{1}(S)tree T in the MorganShalen boundary of , and that the projective horizontal foliations [H_{X,i}] and [H_{Y,i}] converge to [H_{X}] and [H_{Y}]∈PML(S), respectively, as i→∞. Let ζ:π_{1}(S)→Isom T denote the representation given by the isometric action in the limit, where Isom T is the group of isometries of T.
Let \(\tilde{H}_{X}\) be the total lift of the horizontal foliation H_{X} to the universal cover of X, which is a π_{1}(S)invariant measured foliation. Then, collapsing each leaf of \(\tilde{H}_{X}\) to a point, we obtain a \(\mathbb{R}\)tree T_{X}, where the metric is induced by the transversal measure (dual tree of \(\tilde{H}\)). Let \(\phi _{X}\colon \tilde{S} \to T_{X}\) be the quotient collapsing map, which commutes with the π_{1}(S)action. By Dumas ([Dum17, Theorem A, §6]), there is a unique straight map ψ_{X}:T_{X}→T such that ψ_{X} is also π_{1}(S)equivariant, and that every nonsingular vertical leaf of \(\tilde{V}_{X}\) maps to a geodesic in T.
Similarly, let \(\phi _{Y}\colon \tilde{S} \to T_{Y}\) be the map which collapses each leaf of \(\tilde{H}_{Y}\) to a point. (See Fig. 4.) Let ψ_{Y}:T_{Y}→T be the straight map. Let d_{T} be the induced metric on T.
Next we show the horizontal foliations coincide in the limit as projective laminations.
Theorem 4.5
[H_{X}]=[H_{Y}] in PML.
Proof
Pick a diffeomorphism X→Y preserving the marking. Let ξ:X→Y be a piecewise linear homeomorphism which is a good approximation of ξ with respect to the limit singular Euclidean structures E_{X}, E_{Y} on X and Y; let \(E_{X} = \cup _{j = 1}^{p} \sigma _{j}\) be the piecewise linear decomposition of E_{X} for ξ, where σ_{1},…,σ_{p} are convex polygons in E_{X} with disjoint interiors. We diffeomorphically identify X, Y with the base surface S, so that the identifications induce ξ. Let \(\tilde{\xi}\colon \tilde{X} \to \tilde{Y}\) be the lift of ξ:X→Y to a π_{1}(S)equivariant map between the universal covers \(\tilde{X}\), \(\tilde{Y}\).
Recall that the dual tree T is a geodesic metric space. Therefore, the ζequivariant maps \(\psi _{X} \circ \phi _{X}\colon \tilde{X} \to T\) and \(\psi _{Y} \circ \phi _{X}\colon \tilde{Y} \to T\) are ζequivariantly homotopic when identifying their domains by \(\tilde{\xi}\). Namely, for each \(x \in \tilde{S}\), for t∈[0,1], let η_{t}(p) be the point dividing the geodesic segment from ψ_{X}∘ϕ_{X}(p) to ψ_{Y}∘ϕ_{Y}(p) in the ratio t:1−t. By subdividing the piecewise linear decomposition \(E_{X} = \cup _{j = 1}^{p} \sigma _{j}\) if necessary, we may assume that for each j=1,…,p, \(\psi _{X}\circ \phi _{X} (\tilde{\sigma}_{j})\) and \(\psi _{Y}\circ \phi _{Y} (\tilde{\sigma}_{j})\) are the geodesic segments in T contained in a common geodesic in T for all lifts \(\tilde{\sigma}_{j}\) of linear pieces σ_{j} (j=1,…,p), where \(\tilde{\sigma}_{j}\) is a lift of σ_{j} to the universal cover \(\tilde{E}_{X}\). Note that \(\eta _{t}(\tilde{\sigma}_{j})\) may be a single point in T for t∈(0,1); however this degeneration may happen only at most a single time point t∈[0,1] for each j. Let 0<t_{1}<t_{2}<⋯t_{m}<0 be the time points such that \(\eta _{t_{i}}\) takes some piece \(\tilde{\sigma}_{j}\) to a single point in T.
Suppose \(\eta _{t} (\tilde{\sigma}_{j})\) is a segment in T for t∈[0,1]. Then the fibers of η_{t} yield a foliation on σ_{j}. Moreover the pullback of the distance in T gives the transversal measure on the foliation. That is, if an arc in σ is transversal to the foliation, its transversal measure is the distance in T between the images of the endpoints of the arc. Therefore, if t≠t_{1},t_{2},…,t_{m}, η_{t} gives a singular measured foliation H_{t} on S, where singular points are contained in the boundary of the linear pieces. Then, recalling that we have fixed a metric on T in its projective class, we have H_{0}=H_{X} and H_{1}=H_{Y} as ψ_{X} and ψ_{Y} are straight maps, up to scaling of H_{X} and H_{Y}.
At time t_{i}, the \(\eta _{t_{i}}\)image of \(\tilde{\sigma}_{j}\) is a single point in T for some j. Then, since all points on \(\tilde{\sigma}_{j}\) map to the same point on T, the pullback of the distance on T by \(\eta _{t_{i}}\) can be regarded as the empty lamination on σ_{j}. Thus, we obtain a measured lamination \(H_{t_{i}}\) on S, pulling back the distance by \(\eta _{t_{i}}\). Therefore, we obtain a measured lamination H_{t} on S for all t∈[0,1]. Moreover, as the ζequivariant homotopy \(\eta _{t}\colon \tilde{S} \to T\) changes continuously in t, H_{t} changes continuously on t∈(0,1).
For each j=1,…,p, let U_{j} be a small piecewise linear neighborhood of σ_{j} homeomorphic to a disk in E_{X}. Then, for every ϵ>0, we can approximate the homotopy η_{t} (0≤t≤1) by ξ_{t} such that

η_{0}=ξ_{0} and η_{1}=ξ_{1};

η_{t} is piecewise linear;

η_{t} is ϵclose to ξ_{t} in C^{0}topology;

there is a sequence 0=u_{0}<u_{1}<u_{2}<⋯<u_{m}=1, such that, for each i=0,1,…,m−1, the homotopy ξ_{t} is supported on the neighborhood U_{j} of some σ_{j} for u_{i}≤t≤u_{i+1}.
For each t∈[0,1], similarly ξ_{t} induces a measured lamination W_{t} on S so that, in each linear piece, the fibers of ξ_{t} yield strata of the lamination and the distance T the transversal measure. Then, when ϵ>0 is small, W_{t} is a good approximation of W_{t}. By the continuity of ξ, the measured lamination W_{t} changes continuously in t∈[0,1].
We shall modify the measured lamination W_{t} by certain homotopy, removing the “loose part” of W_{t} in order to make η_{t} “tight”. By tightening, with respect to the pullback of the metric of T, the minimal measure of the homotopy class of every closed curve does not increase. Thus this tightening operation removes an obviously unnecessary part of the pullback measure. See Fig. 5 for some examples.
Let \(\tilde{W}_{t}\) be the π_{1}(S)invariant measured lamination on \(\tilde{S}\) obtained by lifting W_{t}. Let T_{t} be the dual tree of \(\tilde{W}_{t}\). Then let \(\phi _{t} \colon \tilde{S} \to T_{t}\) denote the collapsing map. Let ψ_{t}:T_{t}→T denote the folding map so that η_{t}=ψ_{t}∘ϕ_{t}. Suppose that there is a bounded connected subtree γ of T_{t} such that

γ is a closed subset of T_{t};

The boundary of γ maps to a single point z_{γ} on T by ψ_{t}, and the interior of γ maps into a single component of z_{γ}∖z_{γ};

for every α∈π_{1}(S), int (αγ) is disjoint from int γ.
We call such a subtree loose. For a technical reason, we allow γ to be a single point on T_{t}. However, we will later identify a single point subtree of T with the empty set when we consider deformations of such subtrees.
For t∈(0,1), fix a loose subtree γ of T_{t}. Then let \(\psi _{t}' \colon T_{t} \to T\) be the ζequivariant continuous “collapsing” map, such that \(\psi _{t}' (\gamma )\) is the point ψ_{t}(∂γ), \(\psi _{t}' (\alpha \gamma )\) is the point ψ_{t}(αγ) for each γ∈π_{1}(S), and \(\psi _{t}'(x) = \psi _{t}(x)\) for all x∈T not contained in the union of π_{1}(S)orbits of γ. Notice that ψ_{t}(γ) is a subtree of T, and ψ_{t}(∂γ) is an endpoint of the subtree. Therefore, there is a ζequivariant homotopy from ψ_{t} to \(\psi _{t}'\). Thus we call \(\psi _{t}'\) a tightening of ψ_{t} w.r.t. γ. Notice that \(\phi _{t}^{1}(\gamma )\) is a closed simply connected region in \(\tilde{S}\) bounded by some strata of \(\tilde{W}_{t}\) which all map to the same point z_{γ} on T by ψ_{t}∘ϕ_{t}.
More generally, suppose that there are finitely many loose subtrees γ_{1},γ_{2},…,γ_{n} of T_{t}, such that π_{1}(S)orbits of their interiors intγ_{1},intγ_{2},…,intγ_{n} are all disjoint. Then we can homotopy the holding map ψ_{t}:T_{t}→T, simultaneously tightening all loose subtrees γ_{1},γ_{2},…,γ_{n}.
Pick a maximal collection of such loose subtrees γ_{1},γ_{2},…,γ_{n} of T_{t}, so that we can not enlarge any of those loose subtrees or add another one. Then let \(\psi '_{t}\colon T_{t} \to T\) be the tightening of ψ_{t} w.r.t. γ_{1},γ_{2},…,γ_{n} (maximal tightening). Let \(W_{t}'\) be the (singular) measured lamination on S given by the tightened holding map \(\psi _{t}'\colon T_{t} \to T\), where strata are connected components of fibers and the transversal measure is given by the pullback metric. In addition, let R_{t} be the collection \(\{\phi _{t}^{1}(\gamma _{i})\}_{i = 1}^{n_{t}}\) of the closed simply connected regions \(\phi _{t}^{1}(\gamma _{i})\) in \(\tilde{S}\).
As the homotopy \(\xi _{t}\colon \tilde{S} \to T\) changes continuously in t∈[0,1], we can show that the collection of maximal loose subtrees \(\gamma _{t, 1}, \gamma _{t, 2}, \dots , \gamma _{t, n_{t}}\) of T_{t} continuously in t∈[0,1], so that the collection R_{t} changes continuously in t. To be precise, by continuity, we mean that the subsets \(\phi _{t}^{1} (\gamma _{1}) \cup \cdots \cup \phi _{t}^{1} (\gamma _{n_{t}})\) and \(\phi _{t}^{1} (\partial \gamma _{1}) \cup \cdots \cup \phi _{t}^{1} ( \partial \gamma _{n_{t}})\) of \(\tilde{S}\) change continuously in the GromovHausdorff topology on the subsets of \(\tilde{S}\), except that, if γ_{t,i} maps to a single point in T for some t∈[0,1], we identify the collection \(\gamma _{t, 1}, \gamma _{t, 2}, \dots , \gamma _{t, n_{t}}\) with the collection minus γ_{t,i}.
Therefore, by the continuity of the maximal loose subtrees, ϕ_{t} changes continuously in t∈[0,1], and thus \(W'_{t}\) changes continuously in t. Since, in each interval [u_{i},u_{i+1}], the homotopy ξ_{t} is supported on the topological disk U_{j}, the change of W_{t} is also supported in U_{j}. Therefore one can show moreover, for all t∈[u_{i},u_{i+1}]:

for every arc c in U_{j} with endpoints on the boundary of U_{j}, the minimum \(W'_{t}\)measure of the homotopy class of c remains the same when the endpoints of c are fixed;

for every arc c in S∖U_{j} with endpoints on the boundary of U_{j}, the \(W_{t}'\)measure of c remains the same;

for every loop ℓ in S∖U_{j} with endpoints on the boundary of U_{j}, the \(W_{t}'\)measure of ℓ remains the same.
Therefore, for every loop ℓ on S, the tightened measure \(W_{t}'\) gives a constant measure to the homology class of ℓ, for all t∈[u_{i},u_{i+1}]. Hence, \(W_{t}'\) on homotopy classes of loops stays constant for all t∈[0,1].
Since ψ_{X} is a straight map, T_{X} contains no loose subtree. Therefore H_{X}=W_{0}. Similarly, H_{Y}=W_{1}. Hence H_{X}=H_{Y} with respect to the normalization above, and thus [H_{X}]=[H_{Y}] in PML. (As ϵ>0 is arbitrary, we can also show that H_{t} is a constant foliation after collapsing.)
Recall that the translation lengths of loops given by ζ:π_{1}(S)→Isom T is the scaled limit on the translation lengths of \(\rho _{i}\colon \pi _{1}(S) \to {\mathrm{PSL}}(2, \mathbb{C})\) as i→∞. Since \(\psi _{X}\circ \phi _{X}\colon \tilde{S} \to T\) and \(\psi _{Y}\circ \phi _{Y}\colon \tilde{S} \to T\) both ζequivariant and the translation lengths of ρ_{i} in the (asymptotically) same scale when i is very large, Theorem 4.5 implies the following.
Corollary 4.6
There are sequences of positive real numbers k_{i} and \(k_{i}'\), such that \(\lim _{i \to \infty }\frac{k_{i}}{k_{i}'} = 1\) and \(\lim _{i \to \infty } k_{i} H_{X, \rho _{i}} = \lim _{i \to \infty } k_{i}'H_{Y, \rho _{i}}\) in ML.
5 Train tracks
5.1 Traintrack graphs
A train track graph is a C^{1}smooth graph Γ embedded in a smooth surface in the following sense:

Each edge of Γ is C^{1}smoothly embedded in the surface.

At every vertex v of Γ, the unit vectors at v tangent to the edges starting from v are unique up to a sign, and the opposite unit tangent vectors are both realized by the edges.
A weight system w of a traintrack graph is an assignment of a nonnegative real number w(e) to each edge e of Γ, such that at every vertex v of Γ, letting e_{1},…,e_{n} be the edges from one direction and \(e_{1}', e_{2}', \dots , e_{m}'\) the opposite direction, the equation \(w(e_{1}) + w(e_{2}) + \cdots + w(e_{n}) = w(e_{1}') + w(e_{2}') + \cdots + w(e_{n}')\) holds.
5.2 Singular Euclidean surfaces
A singular Euclidean structure on a surface is given by a Euclidean metric with a discrete set of cone points. In this paper, all cone angles of singular Euclidean structures are πmultiples, as we consider singular Euclidean structures induced by holomorphic quadratic differentials. In addition, by a singular Euclidean polygon, we mean a polygon with geodesic edges and a discrete set of singular points whose cone angles are πmultiples. A polygon is rightangled if the interior angles are π/2 or π/3 at all vertices. A Euclidean cylinder is a nonsingular Euclidean structure on a cylinder with geodesic boundary. By a flat surface, we mean a singular Euclidean surface with (singular) vertical and horizontal foliations, which intersect orthogonally.
Definition 5.1
Let E be a flat surface. A curve ℓ on E is a staircase, if ℓ contains no singular point and ℓ is piecewise vertical or horizontal. Then, a staircase curve is monotone if the angles at the vertices alternate between π/2 and 3π/2 along the curve, so that it is a geodesic in the L^{∞}metric (the infinitesimal length is the maximum of the infinitesimal horizontal length and the infinitesimal vertical length). A staircase curve is vertically geodesic, if for every horizontal segment, the angle at one endpoint is π/2 and the angle at the other endpoint is 3π/2; see Fig. 6.
A staircase surface is a flat surface whose boundary components are staircase curves. In particular, a staircase polygon is a flat surface homeomorphic to a disk bounded by a staircase curve. A (L^{∞})convex staircase polygon P is a staircase polygon, such that, if p_{1}, p_{2} are adjacent vertices of P, then at least, one of the interior angles at p_{1} and p_{2} is π/2. A staircase cylinder A embedded in a flat surface E is a spiral cylinder, if A contains no singular point and each boundary component is a monotone staircase loop (see Fig. 7).
Clearly, we have the following decomposition.
Lemma 5.2
Every spiral cylinder decomposes into finitely many rectangles when cut along some horizontal segments each starting from a vertex of a boundary component. (Fig. 7.)
5.3 Surface train tracks
Let F be a compact surface with boundary, such that each boundary component of F is either a smooth loop or a loop with an even number of corner points. Then a (boundary)marking of F is an assignment of “horizontal” or “vertical” to every smooth boundary segment, such that every smooth boundary component is horizontal and, along every nonsmooth boundary component, horizontal edges and vertical edges alternate. From the second condition, every boundary component with at least one corner point has an even number of corner points.
For example, a marking of a rectangle is an assignment of horizontal edges to one pair of opposite edges and vertical edges to the other pair, and a marking of a 2ngon is an assignment of horizontal and vertical edges, such that the horizontal and vertical edges alternate along the boundary. Clearly, there are exactly two ways to give a 2ngon a marking. A marking of a flat cylinder is the unique assignment of horizontal components to both boundary components.
Recall that a (fat) train track T is a surface with boundary and corners obtained by gluing marked rectangles R_{i} along their horizontal edges, in such a way that the identification is given by subdividing every horizontal edge into finitely many segments, pairing up all edge segments, and identifying the paired segments by a diffeomorphism; see for example [Kap01, §11].
In this paper, we may allow any marked surfaces as branches.
Definition 5.3
A surface train track T is a surface having boundary with corners, obtained by gluing marked surfaces F_{i} in such a way that the identification is given by (possibly) subdividing each horizontal edge and horizontal boundary circle of F_{i} into finite segments, pairing up all segments, and identifying each pair of segments by a diffeomorphism.
Given a surface train track T=∪F_{i}, if all branches F_{i} are cylinders with smooth boundary and polygons, then we call T a polygonal train track.
Suppose that a surface F is decomposed into marked surfaces with disjoint interiors so that the horizontal edges of marked surfaces overlap only with other horizontal edges, and vertical edges overlap with other vertical edges (except at corner points); we call this a surface traintrack decomposition of F. Given a traintrack decomposition of a surface F, the union of the boundaries of its branches is a finite graph on F, and we call it the edge graph.
Let F=∪F_{i} be a traintrack decomposition of a surface F. Clearly the interior of a branch is embedded in F, but the boundary of a branch may intersect itself. The closure of a branch F_{i} in F is called the support of the branch, and denoted by F_{i}, which may not be homotopy equivalent to F_{i} on F.
Next, we consider geometric traintrack decompositions of flat surfaces. Let E be a flat surface, and let V and H be it vertical and horizontal foliations, respectively. Then, when we say that a staircase surface F is on E, we always assume that horizontal edges of F are contained in leaves of H and vertical edges in leaves of V. Note that a marked rectangle R on E may selfintersect in its horizontal edges, so that it forms a spiral cylinder. Then a staircase traintrack decomposition of a flat surface E is a decomposition of E into finitely many staircase surfaces on E, such that we obtain a surface traintrack by gluing those staircase surfaces back only along horizontal edges. (Note that, in the context of \({\mathbb{C}{\mathrm{P}}}^{1}\)structures, the vertical direction is regarded as the stable or stretching direction (see Lemma 3.1) and the vertical foliation is carried by this surface traintrack.)
More generally, a trapezoidal traintrack decomposition of E is a surface traintrack decomposition, such that each vertical edge is contained in a vertical leaf and each horizontal edge is a nonvertical line segment disjoint from the singular set of E.
Given a flat surface, we shall construct a canonical staircase traintrack decomposition. Let q be a holomorphic quadratic differential on a Riemann surface X homeomorphic to S. Let E be the flat surface given by q, which is homeomorphic to S. As above, let V, H be the vertical and horizontal foliations of E. Let E^{1} be the unitarea normalization of E, so that \(E^{1} = \frac{E}{{\mathrm{Area}}\, E}\).
Let z_{1},z_{2}…z_{p} be the zeros of q, which are the singular points of E. For each i=1,…,p, let ℓ_{i} be the singular leaf of V containing z_{i}. For r>0, let n_{i} be the closed rneighborhood of z_{i} in ℓ_{i} with respect to the path metric of ℓ_{i} induced by E^{1} (vertical rneighborhood). Let N_{r} be their union n_{1}∪⋯∪n_{p} in E, which may not be a disjoint union as a singular leaf may contain multiple singular points. If r>0 is sufficiently small, then each (connected) component of N_{r} is contractible. Let QD^{1}(X) denote the set of all unit area quadratic differentials on X. Since the set of unit area differentials on X is a sphere, by its compactness, we have the following.
Lemma 5.4
For every X in T^{+}∪T^{−}, if r>0 is sufficiently small, then, for all q∈QD^{1}(X), each component of N_{r} is a simplicial tree (i.e. contractible).
Fix X in T^{+}∪T^{−}, and let r>0 be the small value given by Lemma 5.4. Let p be an endpoint of a component of N_{r}. Then p is contained in horizontal geodesic segments, in E, of finite length, such that their interiors intersect N only in p. Let h_{p} be a maximal horizontal geodesic segment or a horizontal geodesic loop, such that the interior of h_{p} intersects N_{r} only in p. If h_{p} is a geodesic segment, then the endpoints of h_{p} are also on N_{r}. If h_{p} is a geodesic loop, h_{p} intersects N_{r} only in p.
Consider the union ∪_{p}h_{p} over all endpoints p of N_{r}. Then N_{r}∪(∪_{p}h_{p}) decomposes E into staircase rectangles and, possibly, flat cylinders. Thus we obtain a staircase train track decomposition whose branches are all rectangles.
Next, we construct a polygonal traintrack structure of E so that the singular points are contained in the interior of the branches. Let \(b_{i} \in \mathbb{Z}_{\geq 3}\) be the balance of the singular leaf ℓ_{i} at the zero z_{i}, i.e. the number of the segments in ℓ_{i} meeting at the singular point z_{i}.
We constructed the vertical rneighborhood n_{i} of the zero z_{i}. Let \(P_{i}^{r}\) be the set of points on E whose horizontal distance from n_{i} is at most \(\sqrt[4]{r}\) (horizontal neighborhood). Then, as E is fixed, if r>0 is sufficiently small, then \(P_{i}^{r}\) is a convex staircase 2b_{i}gon whose interior contains z_{i}. We say that \(P_{i}^{r}\) is the \((r, \sqrt[4]{r})\)neighborhood of z_{i}. Such \((r, \sqrt[4]{r})\)neighborhoods will be used in the proof of Lemma 6.7.
When we vary q∈QD^{1}(X), fixing r, the convex polygons for different zeros may intersect. Nonetheless, by compactness, we have the following.
Lemma 5.5
Let X∈T^{+}∪T^{−}. If r>0 is sufficiently small, then, for every q∈QD^{1}(X), each connected component of \(P_{1}^{r} \cup P_{2}^{r} \cup \cdots \cup P_{n_{q}}^{r}\) is a staircase polygon.
Then, let r>0 and \(P^{r} (= P^{r}_{q})\) be \(P_{1}^{r} \cup P_{2}^{r} \cup \cdots \cup P_{p_{q}}^{r}\) as in Lemma 5.5. Then, similarly, for each horizontal edge h of P^{r}, let \(\hat{h}\) be a maximal horizontal geodesic segment or a horizontal geodesic loop on E, such that the interior point of \(\hat{h}\) intersects P^{r} exactly in h. Then, either

\(\hat{h}\) is a horizontal geodesic segment whose endpoints are on the boundary of P^{r}, or

\(\hat{h}\) is a horizontal geodesic loop intersecting P^{r} exactly in h.
Consider the union \(\cup _{h} \hat{h}\) over all horizontal edges h of P^{r}. Then the union decomposes E∖P^{r} into finitely many staircase rectangles and, possibly, flat cylinders. Thus we have a staircase traintrack structure, whose branches are polygons and flat cylinders. Note that the singular points are all contained in the interiors of polygonal branches.
For later use, we modify the train track to eliminate thin rectangular branches, i.e. they have short horizontal edges. Note that each vertical edge of a rectangle is contained in a vertical edge of a polygonal branch. Thus, if a rectangular branch R has horizontal length less than \(\sqrt[4]{r}\), then naturally glue R with both adjacent polygonal branches along the vertical edges of R. After applying such gluing for all thin rectangles, we obtain a traintrack structure t^{r} of E.
Lemma 5.6
For every X∈T^{+}∪T^{−}, if r>0 is sufficiently small, then, for every q∈QD^{1}(X), the branches of the traintrack structure t^{r} on E are staircase polygons and staircase flat cylinder, and every rectangular branch of t^{r} has width at least \(\sqrt[4]{r}\).
Definition 5.7
Let E be a flat surface. A traintrack structure T_{1} is a refinement of another traintrack structure T_{2} of E, if the T_{1} is a subdivision of T_{2} (which includes the case that T_{1}=T_{2}).
Let E_{i} be a sequence of flat surfaces converging to a flat surface E. Let T be a traintrack structure on a flat surface E, and let T_{i} be a sequence of traintrack structures on a flat surface E_{i} for each i. Then T_{i} converges to T as i→∞ if the edge graph of \(T_{k_{i}}\) converging to the edge graph of T_{∞} on E in the Hausdorff topology. Then T_{i} semiconverges to T as i→∞ if every subsequence \(T_{k_{i}}\) of T_{i} subconverges to a traintrack structure T′ on E, such that T is a refinement of T′.
Lemma 5.8
\(t^{r}_{q}\) is semicontinuous in the Riemann surface X and the quadratic differential q on X, and the (small) traintrack parameter r>0 given by Lemma 5.6. That is, if r_{i}→r and q_{i}→q, then \(t^{r_{i}}_{q_{i}}\) semiconverges to \(t^{r}_{q}\) as i→∞.
Proof
Clearly, the flat surface E changes continuously in q. Accordingly P^{r} changes continuously in the Hausdorff topology in q and r. Then the semicontinuity easily follows from the construction of \(t^{r}_{q}\). (Note that \(t^{r}_{q}\) isnot necessarily continuous since a branch of \(t^{r_{i}}_{q_{i}}\) may, in the limit, be decomposed some branches including a rectangular branch of horizontal length \(\sqrt[4]{r}\).) □
5.4 Straightening foliations on flat surfaces
Let E be the flat surface homeomorphic to S, and let V be its vertical foliation. Let V′ be another measured foliation on S.
For each smooth leaf ℓ of V′, consider its geodesic representative [ℓ] in E. If ℓ is nonperiodic, the geodesic representative is unique. Suppose that ℓ is periodic. Then, if [ℓ] is not unique, then the set of its geodesic representatives foliates a flat cylinder in E.
Consider all geodesic representatives, in E, of smooth leaves ℓ of V′, and let [V′] be the set of such geodesic representatives and the limits of those geodesics. We still call the geodesics of [V′] leaves. We can regard [V′] as a map from a lamination [V′] on S to E which is a leafwise embedding.
6 Compatible surface train track decompositions
Let X,Y∈T⊔T^{∗} with X≠Y. Clearly, for each , there are unique \({\mathbb{C}{\mathrm{P}}}^{1}\)structures C_{X} and C_{Y} on X and Y, respectively, with holonomy ρ. Set C_{X}=(X,q_{X}) and C_{Y}=(Y,q_{Y}), in Schwarzian coordinates, where q_{X}∈QD(X) and q_{Y}∈QD(Y). Then, define to be the map taking to the ordered pair of the projectivized horizontal foliations of q_{X,ρ} and q_{Y,ρ}. Let Λ_{∞}⊂PML×PML be the set of the accumulation points of η towards the infinity of — namely, (H_{X},H_{Y})∈Λ_{∞} if and only if there is a sequence ρ_{i} in which leaves every compact set in such that η(ρ_{i}) converges to (H_{X},H_{Y}) as i→∞.
Let Δ⊂PML×PML be the diagonal set. Then, by Theorem 4.5, Λ_{∞} is contained in Δ. Given a Riemann surface X and a projective measured foliation H, by Hubbard and Masur [HM79], there is a unique holomorphic quadratic differential on X such that its horizontal foliation coincides with the measured foliation. Let \(E_{X, H} = E_{X, H}^{1}\) denote the unitarea flat surface induced by the differential. Given H_{X}∈PML, let V_{X} be the vertical measured foliation realized by (X,H_{X}), and let V_{Y} be the vertical foliation of (Y,H_{Y}).
Noting that a smooth leaf of a (singular) foliation may be contained in a singular leaf of another foliation, we let Δ^{∗} be the set of all (H_{X},H_{Y})∈PML×PML which satisfies either

there is a leaf of H_{X} contained in a leaf of V_{Y};

there is a leaf of V_{Y} contained in a leaf of H_{X};

there is a leaf of H_{Y} contained in a leaf of V_{X}; or

there is a leaf of V_{X} contained in a leaf of H_{X}.
Then Δ^{∗} is a closed measurezero subset of PML×PML, and disjoint from the diagonal Δ. (For the proof of our theorems, we will only consider a sufficiently small neighborhood of Δ, which is disjoint from Δ^{∗}.)
6.1 Straightening maps
Fix a transversal pair (H_{X},H_{Y})∈(PML×PML)∖Δ^{∗}. Let p be a smooth point in \(E_{Y, H_{Y}}\), and let \(\tilde{p}\) be a lift of p to the universal cover \(\tilde{E}_{Y, H_{Y}}\).
Let v be the leaf of the vertical foliation \(\tilde{V}_{Y}\) on \(\tilde{E}_{Y, H_{Y}}\) which contains \(\tilde{p}\), and let h be the leaf of the horizontal foliation \(\tilde{H}_{Y}\) on the universal cover which contains \(\tilde{p}\). Then, let [v]_{X} denote the geodesic representative of v in \(\tilde{E}_{X, H_{X}}\), and let [h]_{X} denote the geodesic representative of h in \(\tilde{E}_{X, H_{X}}\). Since \(\tilde{E}_{X, H_{X}}\) is a nonpositively curved space, [v]_{X}∩[h]_{X} is a point or a segment of a finite length in \(\tilde{E}_{X, H_{X}}\); let \(\operatorname{st}(p)\) be the subset of \(E_{X, H_{X}}\) obtained by projecting the point or a finite segment.
6.2 Nontransversal graphs
Let E be a flat surface with horizontal foliation H. Let \(\ell \colon \mathbb{R}\to E\) be a (nonconstant) geodesic on E parametrized by arc length. A horizontal segment of ℓ is a maximal segment of ℓ which is tangent to the horizontal foliation H. Note that a horizontal segment is, in general, only immersed in E.
Let X,Y∈T⊔T^{∗} with X≠Y, and let (H_{X},H_{Y})∈(PML×PML)∖Δ^{∗}. For a smooth leaf ℓ_{Y} of V_{Y}, let [ℓ_{Y}]_{X} denote the geodesic representative of ℓ_{Y} on the flat surface \(E_{X, H_{X}}\). The geodesic [ℓ_{Y}]_{X} is not necessarily embedding and should be regarded as an immersion \(\mathbb{R}\to E_{X, H_{X}}\).
Lemma 6.1
Every horizontal segment v of [ℓ_{Y}]_{X} is a segment (i.e. finite length) connecting singular points of E.
Proof
If h has infinite length, then ℓ_{Y} must be contained in a leaf of H_{Y}. This contradicts (H_{X},H_{Y})∈(PML×PML)∖Δ^{∗}. □
Let [V_{Y}]_{X} denote the set of all geodesic representatives of smooth leaves of V_{Y} on E_{X,H}. Let \(G_{Y} \subset E_{X, H_{X}}\) be the union of (the images of) all horizontal segments of [V_{Y}]_{X}. Then it follows that G_{Y} is a finite graph, such that

every connected component of G_{Y} is contained in a horizontal leaf of H_{X}, and

every vertex of G_{Y} is a singular point of E_{X,H}.
Proposition 6.2
For all distinct X,Y∈T⊔T^{∗} and all (H_{X},H_{Y})∈PML×PML∖Δ^{∗}, there is B>0, such that, for all leaves ℓ_{Y} of V_{Y}, every horizontal segment of the geodesic representative [ℓ_{Y}]_{X} is bounded by B from above.
Proof
By Lemma 6.1, each horizontal segment has a finite length and its endpoints are at singular points of E_{X,H}. (Although the number of embedded horizontal segments is clearly bounded, a horizontal segment is in general immersed in E_{X,H}.)
Consider all horizontal segments s_{i} (i∈I) of [V_{Y}]_{X}. Each s_{i} is a mapping of a segment of a leaf of V_{Y} into a leaf of H_{X}. Thus, by identifying s_{i} with the segment of V_{Y}, we regard ⊔_{i∈I}s_{i} as a subset of E_{Y,H}. Since s_{i} is immersed into G_{Y} for each i∈I, we have a mapping from ⨆_{i∈I}s_{i} to G_{Y}. Therefore, there are a small regular neighborhood N of G_{Y} and a small homotopy of the mapping ⨆s_{i}→G_{Y}, such that ⨆s_{i} is, after the homotopy, embedded in N and the endpoints of s_{i} are on the boundary of N (close to the vertices of G_{Y} where they map initially). Since G_{Y} is a finite graph and endpoints of s_{i} map to vertices of G_{Y}, there are only finitely many combinatorial types of horizontal segment s_{i}→G_{Y}. In particular, the lengths of s_{i} are bounded from above. □
By continuity and the compactness of PML, the uniformness follows:
Corollary 6.3
The upper bound B can be taken uniformly in H∈PML.
Let \(V'_{Y}\) denote [V_{Y}]_{X}∖G_{Y}, the set of geodesic representatives [ℓ_{Y}]_{X} minus their horizontal segments, for all leaves ℓ_{Y} of V_{Y}. Then \(V_{Y}'\) is transversal to H_{X} at every point, and the angle between them (§2.3) is uniformly bounded away from zero:
Lemma 6.4
For every (H_{X},H_{Y})∈PML×PML∖Δ^{∗}, \(\angle _{E_{X, H}} ( V'_{Y}, H_{X}) > 0\).
Proof
Suppose, to the contrary, that there is a sequence of distinct points x_{i} in \(V_{Y}' \) such that \((0 <) \angle _{x_{i}} (V_{Y}', H_{X}) \to 0\) as i→∞. We may, in addition, assume that x_{i} are smooth points of \(E_{X, H_{X}}\). For each i=1,2,…, let ℓ_{i} be a leaf of [V_{Y}]_{X} containing x_{i}, so that \(\angle _{x_{i}}(\ell _{i}, H_{X}) \to 0\) as i→∞. Let s_{i} be a segment of ℓ_{i} containing x_{i} but disjoint from the singular set of E_{X}. Clearly, the angle \(\angle _{x_{i}}(\ell _{i}, H_{X})\) remains the same when x_{i} moves in s_{i}. By the discreteness of the singular set of E_{X}, we may assume that the length of s_{i} diverges to infinity as i→∞. Then, for sufficiently large i, the segments s_{i} are all disjoint, and thus s_{i} must be all parallel as x_{i} converges to a smooth point. This can not happen as \(\angle _{x_{i}}(\ell _{i}, H_{X}) \to 0\). Therefore \(\angle _{E_{X, H}} ( V'_{Y}, H_{X}) > 0\). □
6.3 Traintrack decompositions for diagonal horizontal foliations
Definition 6.5
Let (E,V) be a flat surface. Let T be a train track decomposition of E. A curve \(\mathbb{R}\to E\) is carried by T, if B is a branch of T, then, for every component s of ℓ∩intB, both endpoints of s are on different horizontal edges of B,
A (topological) lamination on E is carried by T if every leaf is carried by T.
In this paper, a traintrack may have “bigon regions” which correspond to vertical edges of T. Thus a measured lamination may be carried by a train track in essentially different ways. As a lamination is usually defined up to an isotopy on the entire surface, when a measured lamination is carried by a traintrack, we call it a realization of the measured lamination.
Definition 6.6
Let (E,V) be a flat surface. Let T be a train track decomposition of E. A geodesic ℓ on E is essentially carried by T, if, for every rectangular branch B of T and every component s of ℓ∩intB,

both endpoints of s are (different) horizontal edges of B, or

the endpoints of s are on adjacent (horizontal and vertical edges) of B.
The measured foliation V on E is essentially carried by T if every smooth leaf of V is essentially carried by T.
Because of the horizontal segment, [V_{Y}]_{X} is not necessarily carried by \(t^{r}_{X, H}\) even if the traintrack parameter r>0 is very small. Let \({\mathbf{t}}_{X, H}^{r}\) be the traintrack decomposition of E_{X,H} obtained by, for each component of the horizontal graph G_{Y}, taking the union of the branches intersecting the component. A branch of \({\mathbf{t}}_{X, H}^{r}\) is transversal if it is disjoint from G_{Y}, and nontransversal if it contains a component of G_{Y}.
Lemma 6.7
For every (H_{X},H_{Y})∈(PML×PML)∖Δ^{∗}, if r>0 sufficiently small, then

(1)
[V_{Y}]_{X} is essentially carried by \({\mathbf{t}}^{r}_{X, H_{X}} =: {\mathbf{t}}_{X, H_{X}}\), and

(2)
[V_{Y}]_{X} can be homotoped along leaves of H_{X} to a measured lamination W_{Y} carried by \({\mathbf{t}}_{X, H_{X}}\) so that, by the homotopy, every point of [V_{Y}]_{X} either stays in the same branch or moves to the adjacent branch across a vertical edge.
Moreover, these properties hold in a small neighborhood of (H_{X},H_{Y})∈(PML×PML)∖Δ^{∗}.
We call W_{Y} a realization of [V_{Y}]_{X} on \({\mathbf{t}}_{X, H_{X}}\).
Proof
By the construction of the train track, each vertical edge of a rectangular branch has length less than 2r, and each horizontal edge has length at least \(\sqrt[4]{r}\). Then, (1) follows from Lemma 6.4.
We shall show (2). Recall from §5.3, that the construction of \(T_{X, H_{X}}\) started with taking an rneighborhood of the zeros in the vertical direction and then taking points \(\sqrt[4]{r}\)close to the neighborhood in the horizontal direction. Therefore, each vertical edge of \({\mathbf{t}}_{X,H_{X}}\) has length at least \(\sqrt[4]{r}\) and each horizontal edge has length less than r.
Similarly to a Teichmüller mapping, we rescale the Euclidean structure of \(E_{X, H_{X}}\) with area one by scaling the horizontal distance by \(\sqrt[4]{r}\) and the vertical distance by \(\frac{1}{\sqrt[4]{r}}\), its reciprocal. Then, by this mapping, the flat surface E_{X,H} is transformed to another flat surface \(E'_{X, H_{X}}\) and the traintrack structure \({\mathbf{t}}_{X, H_{X}}\) is transformed to \({\mathbf{t}}'_{X, H_{X}}\). Then, the horizontal edges of rectangular branches of \({\mathbf{t}}'_{X, H_{X}}\) have horizontal length at least \(\sqrt{r}\), and the vertical edges have length less than \(2 r^{\frac{3}{4}}\). Thus, as the train track parameter r>0 is sufficiently small, the vertical edge is still much shorter than the horizontal edge. Note that, the foliations V_{X} and H_{X} persist by the map, except the transversal measures are scaled.
As r>0 is sufficiently small, the geodesic representative \([V_{Y}]_{X}'\) of V_{Y} on \(E'_{X, H_{X}}\) is almost parallel to V_{X}. Since N is a compact subset of (PML×PML)∖Δ^{∗}, by Lemma 6.4, \(\angle _{E_{X}}(H_{X}, [V_{Y})]_{X}\) is bounded from below by a positive number uniformly in H=(H_{X},H_{Y})∈N. Then, indeed, for every υ>0, if r>0 is sufficiently small, then \(\angle _{E_{X}'}(V_{X}, [V_{Y}]'_{X}) < \upsilon \).
Then, let ℓ be a leaf of V_{Y}. Let ℓ_{X} be the geodesic representative of ℓ in \(E_{X}'\). Consider the set \(N^{v}_{\sqrt{r}}\) of points on \(E'_{X}\) whose horizontal distance to the set of the vertical edges of \({\mathbf{t}}'_{X, H_{X}}\) is less than \(\sqrt{r}\). Let s be a maximal segment of ℓ_{X}, such that s is contained in \(N^{h}_{\sqrt{r}}\) and that each endpoint of s is connected to a vertex of \({\mathbf{t}}'_{X, H_{X}}\) by a horizontal segment (which may not be contained in a horizontal edge of \({\mathbf{t}}'_{X, H_{X}}\)). Clearly, if r>0 is sufficiently small, s does not intersect the same vertical edge twice nor the same branch twice.
Claim 6.8
There is a staircase curve c on \(E'_{X, H_{X}}\), such that

c is \(r^{\frac{1}{2}}\)close to s in the horizontal direction,

each vertical segment of c is a vertical edge of \({\mathbf{t}}_{X, H_{X}}'\), and

each horizontal segment of c contains no vertex of \({\mathbf{t}}_{X, H_{X}}'\) in its interior.
(See Fig. 8.)
Pick finitely many segments s_{1},…,s_{n} in leaves of \([V_{Y}]_{X}'\) as above, such that if a vertical edge v of \({\mathbf{t}}_{X, H_{X}}'\) intersects \([V_{Y}]_{X}'\), then there is exactly one s_{i} which is \(r^{\frac{3}{4}}\)Hausdorff close to v. Let c_{1},…,c_{n} be their corresponding staircase curves on \(E'_{X}\).
Then, we can homotope [V_{Y}]_{X} in a small neighborhood of the region R_{i} bounded by s_{i} and c_{i}, such that, while homotoping, the leaves do not intersect s_{i}, and that the homotopy moves each point horizontally (Fig. 9).
Each point on \([V_{Y}]_{X}'\) is homotoped at most to an adjacent branch (Fig. 10). Then, after this homotopy, \([V_{Y}]_{X}'\) is carried by \({\mathbf{t}}'_{X, H_{X}}\). This homotopy induces a desired homotopy of [V_{Y}]_{X}.
Let H=(H_{X},H_{Y})∈PML×PML∖Δ^{∗} and W_{Y} denote the realization of [V_{Y}]_{X} on \({\mathbf{t}}_{X, H_{X}}\) given by Lemma 6.7.
A measured lamination in PML is defined up to an isotopy of the surface. The union of the vertical edges of \({\mathbf{t}}_{X, H_{X}}\) consists of disjoint vertical segments. Each vertical segment of the union is called a (vertical) slit. Then, a measured lamination can be carried by a train track in many different ways by homotopy across slits:
Definition 6.9
Shifting
Suppose that T is a traintrack structure of a flat surface E, and let L_{1} be a realization of L∈ML on T. For a vertical slit v of T, consider the branches on T whose boundary intersects v in a segment. A shifting of L_{1} across v is a homotopy of L_{1} on E to another realization L_{2} of L which reduces the weights of the branches on one side of v by some amount and increases the weights of the branches on the other side of v by the same amount (Fig. 11). Two realizations of L on T are related by shifting if they are related by simultaneous shifts across some vertical slits of T.
The homotopy of [V_{Y}]_{X} in Lemma 6.7 moves points at most to adjacent branches in the horizontal direction. Thus we have the following.
Lemma 6.10
In Lemma 6.7, the realizations given by different choices of s_{i} are related by shifting.
Proposition 6.11
Let H_{i}=(H_{X,i},H_{Y,i}) be a sequence in PML×PML∖Δ^{∗} converging to H=(H_{X},H_{Y}) in PML×PML∖Δ^{∗}. Let W_{i} be a realization of \([V_{Y_{i}}]_{X_{i}}\) on \({\mathbf{t}}_{X, H_{X, i}}\), and let W be a realization of [V_{Y}]_{X} on \({\mathbf{t}}_{X, H_{X}}\) given by Lemma 6.7. Then, a limit of the realization W_{i} and the realization W are related by shifting across vertical slits.
Proof
By the semicontinuity of \({\mathbf{t}}_{X, H_{X}}\) in H_{X} (Proposition 6.12), the limit of the train tracks \({\mathbf{t}}_{X, H_{X, i}}\) is a subdivision of \({\mathbf{t}}_{X, H_{X}}\). Let \(s_{i, 1}, \dots , s_{i, k_{i}}\) be the segments from the proof of Lemma 6.7 which determine the realization W_{i}. The segment s_{j,i} converges up to a subsequence. Then, the assertion follows from Lemma 6.10. □
In summary, we have obtained the following.
Proposition 6.12
Staircase train tracks
For all distinct X,Y∈T⊔T^{∗} and a compact neighborhood N_{∞} of Λ_{∞} in (PML×PML)∖Δ^{∗}, if the traintrack parameter r>0 is sufficiently small, then, for every H=(H_{X},H_{Y}) of N_{∞}, the staircase train track \({\mathbf{t}}^{r}_{X, V_{X}}\) satisfies the following:

(1)
\({\mathbf{t}}^{r}_{X, H_{X}}\) changes semicontinuously in H∈N_{∞}.

(2)
V_{Y} is essentially carried by \({\mathbf{t}}^{r}_{X, H_{X}}\), and its realization on \({\mathbf{t}}^{r}_{X, H_{X}}\) changes continuously H∈N_{∞}, up to shifting across vertical slits.
6.4 An induced traintrack structure for diagonal horizontal foliations
We first consider the diagonal case when H_{X}=H_{Y}=:H∈PML. We have constructed a staircase train track decomposition t_{X,H} of E_{X,H}. Moreover, the geodesic representative [V_{Y}]_{X} is essentially carried by t_{X,H}. Thus, we homotope [V_{Y}]_{X} along leaves of H_{X}, so that it is carried by the train track t_{X,H} (Lemma 6.7). Let W_{Y} denote this topological lamination being carried on t_{X,H} which is homotopic to [V_{Y}]_{X}.
From the realization W_{Y} on t_{X,H}, we shall construct a polygonal traintrack structure on \(E_{Y, H_{Y}}\). The flat surfaces E_{X,H} and E_{Y,H} have the same horizontal foliation, and the homotopy of [V_{Y}]_{X} to W_{Y} is along the horizontal foliation. Therefore, for each rectangular branch R_{X} of t_{X,H}, if the weight of W_{Y} is positive, by taking the inverseimage of the straightening map \(\operatorname{st}\colon E_{Y, H} \to E_{X, H}\) in §6.1, we obtain a corresponding rectangle R_{Y} on E_{Y,H} whose vertical length is the same as R_{X} and horizontal length is the weight. Note that an edge of R_{Y} may contain a singular point of \(E_{Y, H_{Y}}\).
Next let P_{X} be a polygonal branch of t_{X,H}. Similarly, let P_{Y} be the inverseimage of P_{X} by the straighten map. Note that P_{Y} is not necessarily homeomorphic to P_{X}. In particular, P_{Y} can be the empty set, a staircase polygon which may have a smaller number of vertices than P_{X}. Moreover, P_{Y} may be disconnected (Fig. 12). Then, we have a (staircase) polygonal traintrack decomposition t_{Y,H} of E_{Y,H}. By convention, nonempty P_{Y}, as above, is called a branch of t_{Y,H} corresponding to P_{X} (which may be disconnected). In comparison to t_{X,H}, the oneskeleton of t_{Y,H} may contain some singular points of \(E_{Y, H_{Y}}\). Since t_{Y,H} changes continuously in the realization W_{Y} of [V_{Y}]_{X} on t_{X,H}, the semicontinuity of t_{X,H} (Proposition 6.12 (1)) gives a semicontinuity of t_{Y,H}.
Lemma 6.13
t_{Y,H} changes semicontinuously in the horizontal foliation H in PML and the realization W_{Y} of [V_{Y}]_{X} on t_{X,H}.
6.5 Filling properties
Lemma 6.14
Let X≠Y∈T⊔T^{∗}. For every diagonal H_{X}=H_{Y}, every component of H_{X,H}∖[V_{Y}]_{X} is contractible, i.e. a tree.
Proof
Recall that H_{Y} and V_{Y} are the horizontal and vertical foliations of the flat surface \(E_{Y, H_{Y}}\). Then, since H_{X}=H_{Y}, the lemma follows. □
A horizontal graph is a connected graph embedded in a horizontal leaf (whose endpoints may not be at singular points). Then, Lemma 6.14 implies the following.
Corollary 6.15
Let X≠Y∈T⊔T^{∗}. For every diagonal pair H_{X}=H_{Y}, let r>0 be the traintrack parameter given by Lemma 6.7. Then, for sufficiently small ϵ>0, if a horizontal graph h of H_{X} has total transversal measure less than ϵ induced by the realization W_{Y}, then h is contractible.
By continuity,
Proposition 6.16
There is a neighborhood N of the diagonal Δ in PML×PML and ϵ>0 such that, if the traintrack parameter r>0 is sufficiently small, then for every (H_{X},H_{Y})∈N, if a horizontal graph h of H_{X} has total transversal measure less than δ induced by W_{Y}, then h is contractible.
6.6 Semidiffeomorphic surface traintrack decompositions
6.6.1 Semidiffeomorphic train tracks for diagonal foliation pairs
Definition 6.17
Let F_{1} and F_{2} be surfaces with staircase boundary. Then F_{1} is semidiffeomorphic to F_{2}, if there is a homotopy equivalence ϕ:F_{1}→F_{2} which collapses some horizontal edges of F_{1} to points: To be more precise,

the restriction of ϕ to the interior intF_{1} is a diffeomorphism onto the interior intF_{2};

ϕ takes ∂F_{1} to ∂F_{2}, and intF_{1} to intF_{2};

for every vertical edge v of F_{1}, the map ϕ takes v diffeomorphically onto a vertical edge or a segment of a vertical edge in F_{2};

for every horizontal edge h of F_{1}, the map ϕ takes h diffeomorphically onto a horizontal edge of F_{2} or collapses h to a single point on a vertical edge of F_{2}.
Let T and T′ be traintrack structures of flat surfaces E and E′, respectively, on S. Then T is semidiffeomorphic to T′, if there is a marking preserving continuous map ϕ:E→E′, such that,

T and T′ are homotopy equivalent by ϕ (i.e. their 1skeletons are homotopy equivalent), and

for each branch B of T, there is a corresponding branch B′ of T′ such that ϕB is a semidiffeomorphism onto B′.
In §6.4, for every H∈PML, we constructed a staircase traintrack structure t_{Y,H} of the flat surface E_{Y,H} with staircase boundary from a realization W_{Y} of [V_{Y}]_{X} on the traintrack structure t_{X,H} of E_{X,H}. However, when a branch B_{X} of t_{X,H} corresponds to a branch B_{Y} of t_{Y,H}, in fact, B_{Y} might not be connected, and in particular not semidiffeomorphic to B_{X} (Fig. 12, Left). In this section, we modify t_{X,H} and t_{Y,H} by gluing some branches in a corresponding manner, so that corresponding branches are semidiffeomorphic after a small perturbation.
Let v be a (minimal) vertical edge of t_{X,H}, i.e. a vertical edge not containing a vertex in its interior. Let B_{X} be a branch of t_{X,H} whose boundary contains v. Suppose that α is an arc in B_{X} connecting different horizontal edges of B_{X}. Then, we say that v and α are vertically parallel in B_{X} if

α is homotopic in B_{X} to an arc α′ transversal to the horizontal foliation HB_{X}, keeping its endpoints on the horizontal edges, and

v diffeomorphically projects into α′ along the horizontal leaves H_{X}B_{X} (see Fig. 13).
The W_{Y}weight of v in B_{X} is the total weight of the leaves of W_{Y}B_{X} which are vertically parallel to v.
Let w be the W_{Y}weight of v in B_{X}. Then, there is a staircase rectangle in B_{Y} such that a vertical edge corresponds to v and the horizontal length is w.
Consider a horizontal arc α_{h} in B connecting a point on v to a point on another vertical edge of B; clearly, the transversal measure of W_{Y} of α_{h} is a nonnegative number. Then, the W_{Y}weight of v in B is the minimum of the W_{Y}transversal measures of all such horizontal arcs α_{h} starting from v.
Fix 0<δ<r to be a sufficiently small positive number. We now consider both branches B_{1}, B_{2} of t_{X,H} whose boundary contains v. Suppose that, the W_{Y}weight of v is less than δ in B_{i} for both i=1,2; then, glue B_{1} and B_{2} along v, so that B_{1} and B_{2} form a single branch. Let \(T_{X, H}^{r, \delta }\), or simply T_{X,H}, denote the traintrack structure of E_{X,H} obtained by applying such gluing, simultaneously, branches of t_{X,H} along all minimal vertical edges satisfying the condition. Then, since t_{X,H} is a refinement of T_{X,H}, the realization W_{Y} of [V_{Y}]_{X} on t_{X,H} is also a realization on T_{X,H}. Similarly, let \(T_{Y, H}^{r ,\delta }\), or simply T_{Y,H}, be the traintrack structure of E_{Y,H} obtained by the realization W_{Y} on T_{X,H}; then t_{Y,H} is a refinement of T_{Y,H}. Lemma 6.14 implies the following.
Lemma 6.18
Every transversal branch of T_{X,H} has a nonnegative Euler characteristic.
Let B be a branch of T_{Y,H}, and let v be a minimal vertical edge of T_{Y,H} contained in the boundary of B. Let B′ be the branch of T_{Y,H} adjacent to B across v. Suppose that the W_{Y}weight of v is less than δ in B. Then, it follows from the construction of T_{X,H}, that there is a staircase rectangle R_{v} in B′, such that the horizontal length of R_{v} is δ/3 and that v is a vertical edge of R_{v}. Let v be a vertical edge of B. Then we enlarge B by gluing the rectangle R_{v} along v, and we remove R_{v} from B′ (Fig. 14)— this cutandpaste operation transforms T_{Y,H} by pushing the vertical edge v by δ/3 into B′ in the horizontal direction. For all minimal vertical edges v of T_{Y,H} whose Wweights are less than δ as above, we apply such modifications simultaneously and obtain a traintrack structure \(T'_{Y, H}\) of E_{Y,H} homotopic to T_{Y,H}. (We push weight only δ/3 across a vertical edge, since, if another δ/3 is pushed out across the opposite vertical edge, at least δ/3weight remains left.)
Lemma 6.19

The edge graph of \(T_{Y, H}'\) is, at least, \(\frac{\delta }{ 3}\) away from the singular set of \(E^{1}_{Y, H}\);

\(T_{Y, H}'\) is δHausdorff close to T_{Y,H} in \(E^{1}_{Y, H}\);

T_{X,H} is semidiffeomorphic to \(T_{Y, H}'\) (Fig. 15);

T_{X,H} changes semicontinuously in H;

T_{Y,H} changes semicontinuously in H, and the realization of W on T_{X,H}.
Proof
The first three assertions follow from the construction of T_{X,H} and T_{Y,H}. The semicontinuity of T_{X,H} is given by its construction and the semicontinuity of t_{X,H} (Proposition 6.12). Similarly, the semicontinuity of T_{Y,H} follows from its construction and the semicontinuity of t_{Y,H}. □
6.6.2 Semidiffeomorphic traintracks for almost diagonal horizontal foliations
In this section, we extend the construction form §6.6.1 to the neighborhood of the diagonal (PML×PML)∖Δ^{∗}. By Lemma 6.4, for every compact neighborhood N of the diagonal Δ in (PML×PML)∖Δ^{∗}, there is δ>0, such that
for all (H_{X},H_{Y})∈N, where V_{Y} is the vertical measured foliation of the flat surface structure on Y with the horizontal foliation H_{Y}. Let \({\mathbf{t}}_{X, H_{X}}^{r} (= {\mathbf{t}}_{X, H_{X}})\) be the traintrack decomposition of \(E_{X, H_{X}}\) obtained in §6.3. Lemma 6.7 clearly implies the following.
Proposition 6.20
For a compact subset N in (PML×PML)∖Δ^{∗}, if the traintrack parameter r>0 is sufficiently small, then for all (H_{X},H_{Y})∈N, \({\mathbf{t}}_{X, H_{X}}^{r}\) essentially carries [V_{Y}]_{X}.
Let W_{Y} be a realization of [V_{Y}]_{X} on \({\mathbf{t}}_{X, H_{X}}\) by a homotopy along horizontal leaf H_{X} (§6.3). For every branch B_{X} of \({\mathbf{t}}_{X, H_{X}}\), consider the subset of \(E_{Y, H_{Y}}\) which maps to W_{Y}B_{X} by the straightening map \(\operatorname{st}\colon E_{Y, H_{Y}} \to E_{X, H_{X}}\) (§6.1) and the horizontal homotopy. Then, the boundary of the subset consists of straight segments in the vertical foliation V_{Y} and curves topologically transversal to V_{Y} (Fig. 16 for the case when B_{X} is a rectangle). We straighten each nonvertical boundary curve of the subset keeping its endpoints (Fig. 16). let B_{Y} be the region in \(E_{Y, H_{Y}}\) after straightening all nonvertical curves, so that the boundary of B_{Y} consists of segments parallel to V_{Y} and segments transversal to V_{Y}. Then, for different branches B_{X} of \({\mathbf{t}}_{X, H_{X}}\), corresponding regions B_{Y} have disjoint interiors; thus the regions B_{Y} yield a trapezoidal surface traintrack decomposition of \(E_{Y, H_{Y}}\).
Let E be a flat surface, and let H be its horizontal foliation. Then, for ϵ>0, a piecewisesmooth curve c on E is ϵalmost horizontal, if ∠_{E}(H,c)<ϵ, i.e. the angles between the tangent vectors along c and the foliation H are less than ϵ. More generally, c is ϵquasi horizontal if c is ϵHausdorff close to a geodesic segment which is ϵalmost horizontal to the horizontal foliation H. (In particular, the length of c is very short, then it is ϵquasi horizontal.)
Definition 6.21
Let E be a flat surface. For ϵ>0, an ϵquasistaircase traintrack structure of E is a trapezoidal traintrack structure of E such that its horizontal edges are all ϵquasi horizontal straight segments.
If H_{X}=H_{Y}, then t_{Y,H} is a staircase traintrack, by continuity, we have the following.
Lemma 6.22
Let r>0 be a traintrack parameter given by Proposition 6.20. Then, for every ϵ>0, if the neighborhood N of the diagonal in PML×PML is sufficiently small, then, for all (H_{X},H_{Y})∈N, the trapezoidal traintrack decomposition \({\mathbf{t}}_{Y, H_{Y}}^{r}\) of \(E_{Y, H_{Y}}\) is ϵquasi staircase.
Next, similarly to §6.6.1, we modify \({\mathbf{t}}_{X, H_{X}}\) and \({\mathbf{t}}_{Y, H_{Y}}\) by gluing some branches, so that corresponding branches have small diffeomorphic neighborhoods. Let W_{Y} be a realization of [V_{Y}]_{X} in \({\mathbf{t}}_{X, H_{X}}\). Fix small δ>0. Let v be a vertical edge v of \({\mathbf{t}}_{X, H_{X}}\), and let B_{1}, B_{2} be the branches of \({\mathbf{t}}_{X, H_{X}}\) whose boundary contains v. We glue B_{1} and B_{2} along v, if the W_{Y}measure of v in B_{i} is less than δ for both i=1,2. By applying such gluing for all vertical edges satisfying the condition, we obtain a staircase traintrack \(T_{X, H_{X}}^{r, \delta } = T_{X, H_{X}}\), so that \({\mathbf{t}}_{X, H_{X}}\) is a refinement of \(T_{X, H_{X}}\).
Then, W_{Y} is still carried by \(T_{X, H_{X}}\). Therefore, let \(T_{Y, H_{X}}\) be the trapezoidal traintrack decomposition of \(E_{Y, H_{Y}}\) obtained by this realization, so that \({\mathbf{t}}_{Y, H_{Y}}\) is its refinement.
Let v be a vertical edge of \(T_{X, H_{X}}\). Let B_{X} be a branch of \(T_{X, H_{X}}\) whose boundary contains v. Let \(B_{X}'\) be the branch of \(T_{X, H_{X}}\) adjacent to B_{X} across v. Let B_{Y} and \(B_{Y}'\) be the branches of \(T_{Y, H_{Y}}\) corresponding to B_{X} and \(B_{X}'\), respectively. Then, there is a vertical edge w of \(T_{Y, H_{Y}}\) corresponding to v, contained in the boundary of both B_{Y} and \(B_{Y}'\).
If the W_{Y}weight of v in B_{X} is less than δ, then the W_{Y}weight of V in \(B_{X}'\) is at least δ, by the construction of \(T_{X, H_{X}}\). Therefore, \(B_{Y}'\) contains an ϵquasistaircase trapezoid R_{Y}, such that w is a vertical edge of R_{Y} and the horizontal length between the vertical edges is δ/3. (c.f. Figure 14.)
Then, we can modify the train track \(T_{Y, H_{Y}}\) by removing R_{Y} from \(B_{Y}'\) and gluing R_{Y} with B_{Y} along w — this modified \(T_{Y, H_{Y}}\) by a homotopy. By simultaneously applying this modification for all vertical edges v of \(T_{X, H_{X}}\) satisfying the condition, we obtain a trapezoidal traintrack decomposition \(T_{Y, H_{Y}}'\).
Proposition 6.23
For an arbitrary compact neighborhood N of the diagonal Δ in (PML×PML)∖Δ^{∗}, fix a sufficiently small traintrack parameter r>0 obtained by Proposition 6.20. Then, if the parameter δ>0 is sufficiently small, then for every (H_{X},H_{Y})∈N,

\(T_{X, H_{X}}\) is semidiffeomorphic to \(T_{Y, H_{Y}}'\);

\(T'_{Y, H_{Y}}\) is δHausdorff close to \(T_{Y, H_{Y}}\) in the normalized metric \(E^{1}_{Y, H_{Y}}\);

the open δ/4neighborhood of the singular set is disjoint from the oneskeleton of \(T'_{Y, H_{Y}}\)
(Fig. 17).
A sliding is an operation of a traintrack moving some vertical edges in the horizontal direction without changing the homotopy type of the traintrack structure. If we change the realization W_{Y} on t_{X,H} by shifting across a vertical slit, the induced traintrack t_{Y,H} changes by sliding its corresponding vertical segment (Fig. 18).
As before, a branch of \(T_{X, H_{X}}\) disjoint from the nontransversal graph G_{Y} is called a transversal branch. A branch of \(T_{X, H_{X}}\) containing a component of G_{Y} is called the nontransversal branch. By the semicontinuity of \(T_{Y, H_{Y}}\) in Lemma 6.19 and the construction of \(T'_{Y, H_{Y}}\), we obtain a semicontinuity of \(T_{Y, H_{Y}}\) up to sliding.
Lemma 6.24
Let H_{i}=(H_{X,i},H_{Y,i}) be a sequence converging to H=(H_{X},H_{Y}). Then, up to a subsequence, \(T_{Y, H_{Y, i}}\) semiconverges to a train track structure \(T_{Y, H_{Y}}''\) of \(E_{Y, H_{Y}}\), such that either \(T_{Y, H_{Y}}'' = T_{Y, H_{Y}}'\) or \(T_{Y, H_{Y}}''\) can be transformed to a refinement of \(T_{Y, H_{Y}}'\) by sliding some vertical edges by δ/3.
6.7 Bounded polygonal train tracks for the Riemann surface X
The train tracks we constructed may so far have rectangular branches with very long horizontal edges. In this section, we further modify the traintrack structures \(T_{X, H_{X}}\) and \(T_{Y, H_{Y}}\) from §6.6 by reshaping those long rectangles into spiral cylinders.
Given a rectangular branch of a train track, although its interior is embedded in a flat surface, its boundary may intersect itself. Let T be a traintrack structure of a flat surface E. The diameter of a branch B of T is the diameter of the interior of B with the path metric in B. The diameter of a train track T is the maximum of the diameters of the branches of T.
Recall that we have fixed a compact neighborhood N⊂(PML×PML)∖Δ^{∗} of the diagonal. Recall that, for (H_{X},H_{Y})∈N, \(E_{X, H_{X}}^{1}\) and \(E_{Y, H_{Y}}^{1}\) are the unitarea flat structures realizing (X,H_{X}) and (Y,H_{Y}), respectively. Pick a small r>0 given by Proposition 6.12, so that, for every (H_{X},H_{Y})∈N, there are traintrack structures \(T_{X, H_{X}}\) of \(E_{X, H_{X}}^{1}\) and \(T_{Y, H_{Y}}\) of \(E_{Y, H_{Y}}^{1}\) from §6.6.2.
Lemma 6.25

(1)
Let H_{i}=(H_{X,i},H_{Y,i})∈N be a sequence converging to H=(H_{X},H_{Y})∈N. Suppose that \(T_{X, H_{X, i}} =: T_{X, i}\) contains a rectangular branch R_{i} for every i, such that the horizontal length of R_{i} diverges to infinity as i→∞. Then, up to a subsequence, the support \(R_{i} \subset E_{X, H_{X, i}}^{1} =: E_{X, i}\) converges to either

a flat cylinder which is a branch of \(T_{X, H_{X}}\) or

a closed leaf of H_{X} which is contained in the union of the horizontal edges of \(T_{X, H_{X}}\).


(2)
Let A be the limit flat cylinder or a loop in (1). For sufficiently large i>0, let \(R_{i, 1}, \dots , R_{i, n_{i}}\) be the set of all rectangular branches of T_{X,i} which converge to A as i→∞ in the Hausdorff metric. Then, the union \(R_{i, 1} \cup \cdots \cup R_{i, n_{i}} \subset E_{X, H_{i}}^{1}\) is a spiral cylinder for all sufficiently large i. (See Fig. 7.)
Proof
(1) Let R_{i} be a rectangular branch of T_{X,i} such that the horizontal length of R_{i} diverges to infinity as i→∞. Then, as Area E_{i}=1, the vertical length of R_{i} must limit to zero. Then, in the universal cover \(\tilde{E}_{i}\) of E_{i}, we can pick a lift \(\tilde{R}_{i}\) of R_{i} which converges, uniformly on compact, to a smooth horizontal leaf of \(\tilde{H}_{X}\) or a copy of \(\mathbb{R}\) contained in a singular leaf of \(\tilde{H}_{X}\). Let \(\tilde{\ell}\) denote the limit, and let ℓ be its projection into a leaf of H_{X}.
Claim 6.26
ℓ is a closed leaf of H_{X}.
Proof
Suppose, to the contrary, that ℓ is not periodic. Then ℓ is either a leaf of an irrational sublamination or a line embedded in a singular leaf of \(H_{X, H_{X}}\). Then, the distance from ℓ to the singular set of \(E_{X, H_{X}}\) is zero.
Recall that the \((r, \sqrt[4]{r})\)neighborhood of the singular set of \(E_{X, H_{X, i}}\) is contained in the (nonrectangular) branches of \(T_{X, H_{X, i}}\). Thus, the distance from R_{i} to the singular set of \(E_{X, H_{X,i}}\) is at least r>0 for all i. This yields a contradiction. □
By Claim 6.26, as a subset of E_{i}, the rectangular branch R_{i} converges to the union of closed leaves {ℓ_{j}}_{j∈J} of \(H_{X, L_{X}}\). Thus the Hausdorff limit A of R_{i} in E_{X,H} must be a connected subset foliated by closed horizontal leaves. Therefore, A is either a flat cylinder or a single closed leaf.
First, suppose that the limit A is a flat cylinder. Then, the vertical edges of R_{i} are contained in the vertical edges of nonrectangular branches. The limit of the vertical edges of R_{i} are points on the different boundary components of A. Therefore, each boundary component of A must intersect a nonrectangular branch in its horizontal edge. Therefore, the cylinder is a branch of T_{X,H} by the construction of t_{X,H}.
If the limit A is a single leaf, similarly, one can show that the vertical oneskeleton of \(T_{X, H_{X}}\), since a loop can be regarded as a degeneration of a flat cylinder.
(2) First assume that the limit A is a flat cylinder. Since the \((r, \sqrt[4]{r})\)neighborhood of the singular set is disjoint from the interior of A, we can enlarge A to a maximal flat cylinder \(\hat{A}\) in E_{X,H} whose interior contains (the closure of) A. Then, each boundary component of \(\hat{A}\) contains at least one singular point. Since A is a cylindrical branch, each boundary component ℓ of A contains a horizontal edge of a nonrectangular branch P_{ℓ} of \(T_{X, H_{X}}\) which contains a singular point in the boundary of \(\hat{A}\).
Let P_{ℓ,1},…,P_{ℓ,n} be the nonrectangular branches of \(T_{X, H_{X}}\) whose boundary intersects ℓ. Recall that P_{ℓ,i} is the union of some branches of t_{X,H}. Although \(P_{\ell _{1}, i}\) itself may not be convex, a small neighborhood of the intersection P_{ℓ,i}∩ℓ in P_{ℓ,i} is convex. Let \(P_{i, 1}, \dots , P_{i, k_{i}}\) be all nonrectangular branches of T_{X,i}, such that their union \(P_{i, 1} \cup \cdots \cup P_{i, k_{i}}\) converges to the union of all nonrectangular branches of \(T_{X, H_{X}}\) which have horizontal edges contained in the boundary of A. Then, for sufficiently, large i, the vertical edges of \(R_{i,1}. \dots , R_{i, n_{i}}\) are contained in vertical edges of polygonal branches \(P_{i, 1}, \dots , P_{i, k_{i}}\). Then, by the convexity above, if the union of \(R_{i,1}, \dots , R_{i, n_{i}}\) intersects P_{i,j}, then its intersection is a monotone staircase curve. Therefore the union of \(R_{i,1}. \dots , R_{i, n_{i}}\) is a spiral cylinder. See Fig. 19. A similar argument holds in the case when the limit is a closed loop in a singular leaf. By Lemma 6.25 (1), (2), there is a constant c>0, such that, for H_{X}∈N, if a rectangular branch R of \(T_{X, H_{X}}\) has a horizontal edge of length more than c, then R is contained in a unique spiral cylinder, which may contain other rectangular branches. The diameter of such spiral cylinders is uniformly bounded from above by a constant depending only on X. Thus, replace all rectangular branches R of \(T_{X, H_{X}}\) with corresponding spiral cylinders, and we obtain a staircase train track \(\mathbf{T}_{X, H_{X}}\):
Corollary 6.27
There is c>0, such that, for all H=(H_{X},H_{Y})∈N, the diameters of the branches of the staircase train track \(\mathbf{T}_{X, H_{X}}\) are bounded by c.
6.8 Semidiffeomorphic bounded almost polygonal traintrack structures for Y
For ϵ>0, we have constructed, for all H=(H_{X},H_{Y}) in some compact neighborhood N of the diagonal Λ_{∞} in (PML×PML)∖Δ^{∗}, a staircase traintrack structure \(T_{X, H_{X}}\) of \(E_{X, H_{X}}\) and an ϵquasistaircase traintrack structure \(T_{Y, H_{Y}}\) of E_{Y,H}, such that \(T_{X, H_{X}}\) is semidiffeomorphic to \(T_{Y, H_{Y}}\). In §6.7, we modify \(T_{X, H_{X}}\) and obtain a uniformly bounded traintrack \(\mathbf{T}_{X, H_{X}}\) creating spiral cylinders. In this section, we accordingly modify \(T_{Y, H_{Y}}\) to a bounded ϵquasistaircase traintrack structure.
Lemma 6.28

(1)
For every spiral cylinder A of \(\mathbf{T}_{X, H_{X}}\), letting R_{X,1},R_{X,2},…,R_{X,n} be the rectangular branches of T_{X,H} whose union is A, there are corresponding branches R_{Y,1},R_{Y,2},…,R_{Y,n} of T_{Y,H}, such that

their union R_{Y,1}∪R_{Y,2}∪⋯∪R_{Y,n} is a spiral cylinder in \(E_{Y, H_{Y}}\), and

A is semidiffeomorphic to R_{Y,1}∪R_{Y,2}∪⋯∪R_{Y,n}.


(2)
Moreover, there is a constant c′>0, such that, if a rectangular branch of \(T_{Y, H_{Y}}\) has horizontal length more than c′, then it is contained in a spiral cylinder as above.
Proof
As (H_{X},H_{Y})∩Δ^{∗}=∅, the geodesic representative [V_{Y}]_{X} essentially intersects A. Thus, the realization W_{Y} has positive weights on R_{X,1},R_{X,2},…,R_{X,n}. Thus R_{X,j} corresponds to a rectangular branch R_{Y,j} of \(T_{Y, H_{Y}}\), and their union ∪_{j}R_{Y,j} is a spiral cylinder in \(T_{Y, H_{Y}}\) ((1)).
Let R_{X} and R_{Y} be corresponding rectangular branches of \(T_{X, H_{X}}\) and \(T_{Y, H_{Y}}\), respectively. As (H_{X},H_{Y}) varies only in a fixed compact neighborhood of the diagonal, the horizontal length of R_{X} is bilipschitz close to the length of the horizontal length of R_{Y} with a uniform bilipschitz constant for such all R_{X} and R_{Y}. Therefore, there is c′>0 such that if R_{Y} is more than c′, then the corresponding branch R_{X} has length more than the constant c (right before Corollary 6.27), then R_{X} is contained in a unique spiral cylinder (2). □
For every spiral cylinder A of \(\mathbf{T}_{X, H_{X}}\), by applying Lemma 6.28, we replace the branches R_{Y,1},R_{Y,2},…,R_{Y,n} of \(T_{Y, H_{Y}}\) with the spiral cylinder R_{X,1}∪R_{X,2}∪⋯∪R_{X,n} of \(T_{Y, H_{Y}}\). Then, we obtain an ϵquasistaircase traintrack decomposition \(\mathbf{T}_{Y, H_{Y}}\) without long rectangles:
Proposition 6.29
For every ϵ>0, there are c>0 and a neighborhood N of the diagonal in (PML×PML)∖Δ^{∗}, such that, for every H=(H_{X},H_{Y})∈N⊂PML×PML, there is an ϵquasistaircase traintrack decomposition \(\mathbf{T}_{Y, H_{Y}}\) of \(E_{Y, H_{Y}}\), such that

(1)
\(\mathbf{T}_{Y, H_{Y}}'\) is δHausdorff close to \(\mathbf{T}_{Y, H_{Y}}\) in \(E_{Y, H}^{1}\);

(2)
the diameters of \(\mathbf{T}_{Y, H_{Y}}\) and \(\mathbf{T}_{Y, H_{Y}}'\) are less than c;

(3)
\(\mathbf{T}_{X, H_{X}}\) is semidiffeomorphic with \(\mathbf{T}_{Y, H_{Y}}'\);

(4)
\(\mathbf{T}_{Y, H_{Y}}\) changes semicontinuously in (H_{X},H_{Y}) and the realization of [V_{Y}]_{X} on \(\mathbf{T}_{X, H_{X}}\).
Proof
Assertion (2) follows from Lemma 6.28 (2). Assertion (1) follows from Proposition 6.23. Assertion (3) follows from Proposition 6.23 and Lemma 6.28 (1). Assertion (4) holds, by Lemma 6.19, since \(T_{Y, H_{Y}}\) changes semicontinuously in (H_{X},H_{Y}) and the realization W_{Y} of [V_{Y}]_{X} on \(T_{X, H_{X}}\). □
7 Thurston laminations and vertical foliations
7.1 Model Euclidean polygons and projective circular polygons
A polygon with circular boundary is a projective structure on a polygon such that the development of each edge is contained in a round circle in \({\mathbb{C}{\mathrm{P}}}^{1}\). Let σ be an ideal hyperbolic ngon (n≥3). Let L be a measured lamination on σ except that each boundary geodesic of σ is a leaf of weight ∞. From a view point of the Thurston parameterization, it is natural to add such weightinfinity leaves. In fact, there is a unique \({\mathbb{C}{\mathrm{P}}}^{1}\)structure \(\mathcal{C}= \mathcal{C}(\sigma , L)\) on the complex plane \(\mathbb{C}\) whose Thurston’s parametrization is the pair (σ,L); see [GM21]. Let \(\mathcal{L}\) be the Thurston lamination on \(\mathcal{C}\). Denote, by \(\kappa \colon \mathcal{C}\to \sigma \), the collapsing map (§2.1.7).
For each boundary edge l of σ, pick a leaf ℓ of \(\mathcal{L}\) which is sent diffeomorphically onto l by κ. Then, those circular leaves bound a circular projective ngon \(\mathcal{P}\) in \(\mathcal{C}\), called an ideal projective polygon.
For each i=1,2,…,n, let v_{i} be an ideal vertex of σ, and let l_{i} and l_{i+1} be the edges of σ starting from v_{i}. Consider the geodesic g starting from v_{i} in the middle of l_{i} and l_{i+1}, so that the reflection about g exchanges l_{i} and l_{i+1}. Embed σ isometrically into a totally geodesic plane in \(\mathbb{H}^{3}\). Accordingly \(\mathcal{P}\) is embedding in \({\mathbb{C}{\mathrm{P}}}^{1}\) so that the restriction of κ to \(\mathcal{P}\) is the nearest point projection to σ in \(\mathbb{H}^{3}\).
Then, pick a round circle c_{i} on \({\mathbb{C}{\mathrm{P}}}^{1}\) such that the hyperbolic plane, Conv c_{i}, bounded by c_{i} is orthogonal to g, so that l_{i} and l_{i+1} are transversal to Conv c_{i}.
Let ℓ_{i} and ℓ_{i} be the edges of \(\mathcal{P}\) corresponding to l_{i} and l_{i+1}, respectively. If c_{i} is close to v_{i} enough, then there is a unique arc a_{i} in \(\mathcal{P}\) connecting ℓ_{i} to ℓ_{i+1} which is immersed into c_{i} by the developing map. Then, the region in \(\mathcal{P}\) bounded by a_{1}…a_{n} is called the truncated ideal projective polygon.
Definition 7.1
Let C be a \({\mathbb{C}{\mathrm{P}}}^{1}\)structure on S. Let E^{1} be the normalized flat surface of the Schwarzian parametrization of C. Let P be a staircase polygon in E^{1}. Then P is ϵclose to a truncated ideal projective polygon \(\mathcal{P}\), if \(\mathcal{P}\) isomorphically embeds onto a polygon in C which is ϵHausdorff close to P in the normalized Euclidean metric.
For X∈T⊔T^{∗}, recall that be the holonomy variety of the \({\mathbb{C}{\mathrm{P}}}^{1}\)structures on X. For , let C_{X,ρ} be the \({\mathbb{C}{\mathrm{P}}}^{1}\)structure on X with holonomy ρ, and let E_{X,ρ} be the flat surface given by the holomorphic quadratic differential of C_{X,ρ}. Similarly, for ϵ>0, let \(N_{\epsilon }^{1} Z_{X, \rho}\) be the ϵneighborhood of the singular set in the normalized flat surface \(E^{1}_{X, \rho}\). Let \(\mathcal{L}_{X, \rho}\) be the Thurston lamination of C_{X,ρ}.
Then, by combining what we have proved, we obtain the following.
Theorem 7.2
Let X∈T⊔T^{∗}. Then, for every ϵ>0, there is a bounded subset K=K(X,ϵ) of satisfying the following: Suppose that ρ is in , and that the flat surface E_{X,ρ} contains a staircase polygon P such that

∂P disjoint from \(N^{1}_{\epsilon }Z_{X, \rho}\) and

the diameter of P is less than \(\frac{1}{\epsilon }\).
Then

(1)
\(\mathcal{L}_{X, \rho}  P\) is (1+ϵ,ϵ)quasiisometric to V_{X,ρ}P up to an isotopy supported on \(N_{\epsilon }^{1} Z_{X, \rho} \cap P\), such that, in the normalized Euclidean metric \(E^{1}_{X, \rho}\),

(a)
on P, each leaf of V is ϵHausdorffclose to a leaf of \(\mathcal{L}\), and

(b)
the transversal measure of V is ϵclose to the transversal measure of \(\mathcal{L}\) for all transversal arcs whose lengths are less than one.

(a)

(2)
In the (unnormalized) Euclidean metric, P is ϵclose to a truncated circular polygon of the hyperbolic surface in the Thurston parameters.
Proof
The assertion (1a) follows from Lemma 4.4. The assertion (1b) is given by Proposition Theorem 4.3.
We shall prove (2). Set C_{X,ρ}=(τ,L)∈T×ML be the \({\mathbb{C}{\mathrm{P}}}^{1}\) structure on X with holonomy in Thurston coordinates, and let κ:C_{X,ρ}→τ be the collapsing map. Since sufficiently away from the zero, the developing map is wellapproximated by the exponential map (Lemma 3.2).
If K is sufficiently large, then for every vertical edge v of P, the restriction \(\operatorname{Ep}_{X, \rho}  v\) is a (1+ϵ)bilipschitz embedding on the Epstein surface. Therefore, by the closeness of \(\operatorname{Ep}_{X, \rho}\) and \(\hat{\beta}_{X, \rho}\), κ(v) is ϵclose to a geodesic segment s_{v} of length \(\sqrt{2} \operatorname{length}v\). By (1b), if K is large enough, L(s_{v})<ϵ.
Every horizontal edge h of P is very short on the Epstein surface (Lemma 3.1). As the developing map is approximated by the Exponential map and , it follows that, if κ(h) has length less than ϵ on τ. Therefore, the image of P on the hyperbolic surface is ϵclose to a truncated ideal polygon (see Fig. 20.)
7.2 Equivariant circle systems
For , we shall pick a system of a ρequivalent round circles on \({\mathbb{C}{\mathrm{P}}}^{1}\), which will be used to construct a circular traintrack structure of C_{X,ρ}. Let \(\tilde {\mathbf{T}}_{X, \rho}\) be the π_{1}(S)invariant traintrack structure on \(\tilde{E}_{X, \rho}\) obtained by lifting the traintrack structure T_{X,ρ} on E_{X,ρ}. Let \(\operatorname{Ep}^{\ast}_{X, \rho}\colon T \tilde{E}_{X, \rho} \to T \mathbb{H}^{3}\) be the differential of \(\operatorname{Ep}_{X, \rho}\colon \tilde{E}_{X, \rho} \to \mathbb{H}^{3}\).
Lemma 7.3
For every ϵ>0, there is a bounded subset K_{ϵ} of such that, if \(\rho \colon \pi _{1}(S) \to {\mathrm{PSL}}(2, \mathbb{C})\) belongs to , then, we can assign a round circle c_{h} to every minimal horizontal edge h of \(\tilde{\mathbf{T}}_{X, \rho}\) with the following properties:

(1)
The assignment h↦c_{h} is ρequivariant.

(2)
The hyperbolic plane bounded by c_{h} is ϵalmost orthogonal to the \(\operatorname{Ep}^{\ast}_{X, \rho}\)images of the vertical tangent vectors along h.

(3)
If h_{1}, h_{2} are horizontal edges of \(\tilde {\mathbf{T}}_{X, \rho}\) connected by a vertical edge v of length at least ϵ, then the round circles \(c_{h_{1}}\) and \(c_{h_{2}}\) are disjoint.

(4)
If h_{1}, h_{2}, h_{3} are “vertically consecutive” horizontal edges, such that

h_{1} and h_{2} are connected by a vertical edge v_{1} of E_{X,ρ}length at least ϵ;

h_{2} and h_{3} are connected by a vertical edge v_{3} of length at least ϵ;

h_{1} and h_{3} are on the different sides of h_{2}, i.e. the normal vectors of h_{2} in the direction of v_{1} and v_{3} are opposite,
then \(c_{h_{1}}\) and \(c_{h_{3}}\) are disjoint, and they bound a round cylinder whose interior contains \(c_{h_{2}}\).

Proof
Without loss of generality, we can assume that ϵ>0 is sufficiently small. With respect to the normalized Euclidean metric \(E_{X, \rho}^{1}\), the lengths of minimal horizontal edges of T_{X,ρ} are uniformly bounded from above by Corollary 6.27, and the distances of the horizontal edges from the singular set of \(E_{X, \rho}^{1}\) are uniformly bounded from below. Then, by Lemma 3.1, for every ϵ>0, if a bounded subset K_{ϵ} in χ is sufficiently large, then for every minimal horizontal edge h of T_{X,ρ}, the vertical tangent vectors along h on E_{X,ρ} of unit length map to ϵclose tangent vectors of \(\mathbb{H}^{3}\).
Therefore, if K_{ϵ} is large enough, for each minimal horizontal edge h of \(\tilde {\mathbf{T}}_{Y, \rho}\), we pick a round circle c_{h}, such that the assignment of c_{h} is holonomy equivariant and that the images of vertical tangent vectors along h are ϵ^{2}orthogonal to the hyperbolic plane bounded by c_{h}.
Then, if v is a vertical edge sharing an endpoint with h, then \(\operatorname{Ep}_{X, \rho} (v)\) is ϵ^{2}almost orthogonal to the hyperbolic plane bounded by c_{h}. For every sufficiently small ϵ>0, if K>0 is sufficiently large, then the geodesic segment of length, at least, ϵ connects the hyperbolic planes bounded by \(c_{h_{1}}\) and \(c_{h_{2}}\), and the geodesic segment is ϵ^{2}almost orthogonal to both hyperbolic planes. Therefore, if ϵ>0 is sufficiently small, then, by elementary hyperbolic geometry, the hyperbolic planes are disjoint, and (3) holds. By a similar argument, (4) also holds. □
The circle system in Lemma 7.3 is not unique, but unique up to an appropriate isotopy:
Proposition 7.4
For every ϵ_{1}>0, there is ϵ_{2}>0, such that, for every given two systems of round circles {c_{h}} and \(\{c_{h}'\}\) realizing Lemma 7.3 for ϵ_{2}>0, there is a oneparameter family of equivalent circles systems {c_{t,h}} (t∈[0,1]) realizing Lemma 7.3 for ϵ_{1}>0 which continuously connects {c_{h}} to \(\{c_{h}'\}\).
Proof
The proof is left for the reader. □
7.3 Pleated surfaces are close
The following gives a measuretheoretic notion of almost parallel measured laminations.
Definition 7.5
Quasiparallel
Let L_{1}, L_{2} be two measured geodesic laminations on a hyperbolic surface τ. Then, L_{1} and L_{2} are ϵquasi parallel, if a leaf ℓ_{1} of L_{1} and a leaf ℓ_{2} of L_{2} intersect at a point p and ∠_{p}(ℓ_{1},ℓ_{2})>ϵ, then letting s_{1} and s_{2} be the unit length segments in ℓ_{1} and ℓ_{2} centered at p,
Proposition 7.6
For every ϵ>0, if a bounded subset is sufficiently large, then L_{Y,ρ} is ϵquasiparallel to L_{X,ρ} on τ_{X,ρ} away from the nontransversal graph G_{Y}.
Proof
If K is sufficiently large, \(\angle _{E_{X, \rho}}(H_{X, \rho}, V_{Y, \rho}')\) is uniformly bounded from below by a positive number by Lemma 6.4. Then, the assertion follows from Lemma 3.1 and Theorem 7.2. □
In this section, we show that the pleated surfaces for C_{X,ρ} and C_{Y,ρ} are close away from the nontransversal graph. Recall that \(\hat{\beta}_{X, \rho}\colon \tilde{C}_{X, \rho} \to \mathbb{H}^{3}\) denotes the composition of the collapsing map and the bending map for C_{X,ρ}, and similarly \(\hat{\beta}_{Y, \rho}\colon \tilde{C}_{Y, \rho} \to \mathbb{H}^{3}\) denotes the composition of the collapsing map and the bending map for C_{Y,ρ}.
Theorem 7.7
Let X,Y∈T⊔T^{∗} with X≠Y. For every ϵ>0, there is a bounded subset K_{ϵ} in such that, for every , there are a homotopy equivalence map ϕ:E_{Y,ρ}→E_{Y,ρ} and a semidiffeomorphism \(\psi \colon \mathbf{T}_{X, \rho} \to \mathbf{T}_{Y, \rho}'\) given by Proposition 6.29 (3) satisfying the following:

(1)
\(d_{E^{1}_{Y, \rho}}(\phi (z), z) < \epsilon \);

(2)
the restriction of ϕ to \(E_{Y, \rho} \setminus N^{1}_{\epsilon }Z_{Y, \rho}\) can be transformed to the identity by a homotopy along vertical leaves of E_{X,ρ};

(3)
\(\hat{\beta}_{X, \rho}(z)\) is ϵclose to \(\hat{\beta}_{Y, \rho} \circ \tilde{\phi}\circ \tilde{\psi }(z)\) in \(\mathbb{H}^{3}\) for every point \(z \in \tilde{E}_{X, \rho}\) which are not in the interior of the nontransversal branches of T_{X,ρ}.
Using Lemma 3.1, one can prove the following.
Lemma 7.8
Let ϵ>0 and let X∈T⊔T^{∗}. Then, there is a compact subset K of such that, for every , if α is a monotone staircase closed curve in E_{X,ρ}, such that

the total vertical length of α is more than ϵ times the total horizontal length of α, and

α is disjoint from the ϵneighborhood of the singular set in the normalized metric \(E^{1}_{X, \rho}\),
then \(\operatorname{Ep}_{X, \rho} \tilde{\alpha}\) is a (1+ϵ,ϵ)quasigeodesic with respect to the vertical length.
Lemma 7.9
Let α_{X} be a staircase curve carried by t_{X,ρ} satisfying the conditions in Lemma 7.8. Then, there is a staircase geodesic closed curve α_{Y} carried by \(\mathbf{T}_{Y, \rho}'\) satisfying the conditions in Lemma 7.8, such that the image of α by the semidiffeomorphism \(\mathbf{T}_{X, \rho} \to \mathbf{T}_{Y, \rho}'\) is homotopic to α_{Y} in the traintrack \(\mathbf{T}_{Y, \rho}'\).
Proof
The proof is left for the reader. □
Let W_{Y} be a realization of [V_{Y}]_{X} on T_{X,ρ} (§6.3). Let x be a point of the intersection of the realization W_{Y} and a horizontal edge of h_{X} of T_{X,ρ}. Let y be a corresponding point of V_{Y,ρ} (on E_{Y,ρ}). Recall that r is the traintrack parameter, so that, in particular, horizontal edges are distance, at least, r away from the singular set in the normalized Euclidean metric. Let v_{x} be a vertical segment of length r/2 on \(E^{1}_{X, \rho}\) such that x is the middle point of v_{x}. Similarly, let v_{y} be the vertical segment of length r/2 on \(E^{1}_{Y, \rho}\) such that y is the middle point of v_{y}. We normalize the Epstein surfaces for C_{X,ρ} and C_{Y,ρ} so that they are ρequivariant for a fixed representation \(\rho \colon \pi _{1}(S) \to {\mathrm{PSL}}(2, \mathbb{C})\) (not a conjugacy class).
Proposition 7.10
Corresponding vertical edges are close in \(\mathbb{H}^{3}\)
For every ϵ>0, there is a compact subset K in , such that, for every , if v_{X} and v_{Y} are vertical segments of E_{X,ρ} and of E_{Y,ρ}, respectively, as above, then there is a (biinfinite) geodesic ℓ in \(\mathbb{H}^{3}\) satisfying the following:

\(\operatorname{Ep}_{X, \rho} v_{X}\) is ϵclose to a geodesic segment α_{X} of ℓ in C^{1}metric;

\(\operatorname{Ep}_{Y, \rho} v_{Y}\) is ϵclose to a geodesic segment α_{Y} of ℓ in C^{1}metric;

if p_{X} and p_{Y} are corresponding endpoints of α_{X} and α_{Y}, then the distance between \(\operatorname{Ep}_{X, \rho} p_{X}\) and \(\operatorname{Ep}_{Y, \rho} p_{Y}\) is at most ϵ times the diameters of E_{X,ρ} and E_{Y,ρ}.
Proof
Then, pick a L^{∞}geodesic staircase closed curves ℓ_{X,1}, ℓ_{X,2} on E_{X,ρ} containing v_{x} such that, for i=1,2, by taking appropriate lift \(\tilde{\ell}_{X, 1}\) and \(\tilde{\ell}_{X, 2}\) to \(\tilde{E}_{X, \rho}\),

(1)
ℓ_{X,i} is carried by T_{X,ρ};

(2)
\(\tilde{\ell}_{X, 1} \cap \tilde{\ell}_{X, 2}\) is a single staircase curve connecting singular points of \(\tilde{E}_{X, \rho}\), and the projection of \(\tilde{\ell}_{X, 1} \cap \tilde{\ell}_{X, 2}\) to E_{X,ρ} does not meet a branch of T_{X,ρ} more than twice;

(3)
if a branch B of \(\tilde{T}_{X, H_{X}}\) intersects both \(\tilde{\ell}_{X, 1}\) and \(\tilde{\ell}_{X, 2}\), then B intersects \(\tilde{\ell}_{X, 1} \cap \tilde{\ell}_{X, 2}\);

(4)
\(\tilde{\ell}_{X, 1}\) and \(\tilde{\ell}_{X, 2}\) intersect, in the normalized metric of \(\tilde{E}^{1}_{X, \rho}\), the ϵneighborhood of the singular set only in the near the endpoints of \(\tilde{\ell}_{X, 1} \cap \tilde{\ell}_{X, 2}\).
Then, there are homotopies of ℓ_{X,1}, ℓ_{X,2} to staircase verticallygeodesic closed curves \(\ell '_{X,1}\), \(\ell '_{X, 2}\) carried by T_{X,ρ}, such that the homotopies are supported on the 2ϵneighborhood of the singular set of \(E^{1}_{X, \rho}\) and that \(\ell '_{X,1}\), \(\ell '_{X, 2}\) are disjoint from the ϵneighborhood of the zero set. Then \(\operatorname{Ep}_{X, \rho} \tilde{\ell}_{X, 1}'\) and \(\operatorname{Ep}_{X, \rho} \tilde{\ell}_{X,2 }'\) are (1+ϵ,ϵ)quasigeodesics which are close only near the segment corresponding to \(\tilde{\ell}_{X, 1} \cap \tilde{\ell}_{X, 2}\).
Pick closed geodesic staircasecurves ℓ_{Y,1}, ℓ_{Y,2} on E_{Y,ρ}, such that

ℓ_{Y,i} contains v_{y};

the semidiffeomorphism T_{X,ρ}→T_{Y,ρ} takes \(\ell _{X, i}'\) to a curve homotopic to ℓ_{Y,i} on T_{Y,ρ};

ℓ_{Y,i} is carried by T_{Y,ρ};

ℓ_{Y,i} is disjoint from \(N_{\epsilon }^{1} Z_{X, \rho}\).
Let α be the geodesic such that a bounded neighborhood of α contains the quasigeodesic \(\operatorname{Ep}_{X, \rho} \tilde{\ell}_{X, i}'\). Let \(\tilde{\ell}_{Y, i}\) be a lift of ℓ_{Y,i} to \(\tilde{E}_{Y, \rho}\) corresponding to \(\tilde{\ell}_{X, i}'\) (connecting the same pair of points in the ideal boundary of \(\tilde{S}\)).
Lemma 7.11
For every ϵ>0, if a compact subset K of is sufficiently large and υ>0 is sufficiently small, then, for all , \(\operatorname{Ep}_{Y, \rho} \tilde{\ell}_{Y, i}\) is (1+ϵ,ϵ)quasiisometric with respect to the vertical length for both i=1,2.
Then \(\operatorname{Ep}_{X, \rho} \tilde{\ell}_{X, 1}' \cup \tilde{\ell}_{X, 2}'\) and \(\operatorname{Ep}_{Y, \rho} \tilde{\ell}_{Y, 1} \cup \tilde{\ell}_{Y, 2}\) are both ϵclose in the Hausdorff metric of \(\mathbb{H}^{3}\). Therefore, corresponding endpoints of \(\operatorname{Ep}_{Y, \rho} \tilde{\ell}_{X, 1}' \cap \tilde{\ell}_{X, 2}'\) and \(\operatorname{Ep}_{Y, \rho} \tilde{\ell}_{Y, 1}' \cap \tilde{\ell}_{Y, 2}'\) have distance, at most, ϵ times the diameters of E_{X,ρ} and E_{Y,ρ}. By (2), the length of \(\tilde{\ell}_{X, 1}' \cap \tilde{\ell}_{X, 2}'\) can not be too long relative to the diameter of E_{X,ρ}. Letting ℓ be the geodesic in \(\mathbb{H}^{3}\) fellowtraveling with \(\operatorname{Ep}_{X, \rho} \tilde{\ell}_{X, 1}'\) (or \(\operatorname{Ep}_{X, \rho} \tilde{\ell}_{X, 2}'\)), the vertical segment v_{x} and v_{y} have the desired property. □
Finally Theorem 7.7 follows from the next proposition.
Proposition 7.12
Suppose that a branch \(B_{Y}'\) of the train track \(\mathbf{T}_{Y, \rho}'\) corresponds transversally to a branch B_{X} of T_{X,ρ}.
Then, there is an ϵsmall isotopy of \(B_{Y}'\) in the normalized surface \(E_{Y, \rho}^{1}\) such that

in the complement of the \(\frac{r}{2}\)neighborhood of the zero set, every point of \(B_{Y}'\) moves along the vertical foliation V_{Y,ρ}, and

after the isotopy \(\hat{\beta}_{X, \rho}  B_{X} \) and \(\hat{\beta}_{Y, \rho} \vert B_{Y}'\) are ϵclose pointwise by a diffeomorphism \(\psi \colon B_{Y}' \to B_{X}\).
Proof
By Proposition 7.10, there is an ϵsmall isotopy of the boundary of \(B_{Y}'\) satisfying the conditions on the boundaries of the branches. Since the branches are transversal, by Theorem 4.3, if K is sufficiently large, then the restriction of L_{X,ρ} to B_{X} and L_{Y,ρ} on B_{Y} are ϵquasi parallel on the hyperbolic surface τ_{X,ρ} (Proposition 7.6). Therefore we can extend to the interior of the branch by taking an appropriate diffeomorphism \(\psi \colon B'_{Y} \to B_{X}\). □
8 Compatible circular traintracks
In §6, for every ρ in outside a large compact K, we constructed semidiffeomorphic traintrack structures T_{X,ρ} and \(\mathbf{T}_{Y, \rho}'\) of the flat surfaces E_{X,ρ} and E_{Y,ρ}, respectively. In this section, as E_{X,ρ} and E_{Y,ρ} are the flat structures on C_{X,ρ} and C_{Y,ρ}, using Theorem 7.7, we homotope T_{X,ρ} and \(\mathbf{T}_{Y, \rho}'\) to make them circular in a compatible manner.
8.1 Circular rectangles
A round cylinder is a cylinder on \({\mathbb{C}{\mathrm{P}}}^{1}\) bounded by two disjoint round circles. Given a round cylinder A, the boundary components of A bound unique (totally geodesic) hyperbolic planes in \(\mathbb{H}^{3}\), and there is a unique geodesic ℓ orthogonal to both hyperbolic planes. Moreover A is foliated by round circles which, in \(\mathbb{H}^{3}\), bound hyperbolic planes orthogonal to ℓ — we call this foliation the horizontal foliation. In addition, A is also foliated by circular arcs which are contained in round circles bounding hyperbolic planes, in \(\mathbb{H}^{3}\), containing ℓ — we call this foliation the vertical foliation. Clearly, the horizontal foliation is orthogonal to the vertical foliations of A.
Definition 8.1
Let \(\mathcal{R}\) be a \({\mathbb{C}{\mathrm{P}}}^{1}\)structure on a marked rectangle R, and let \(f\colon R \to {\mathbb{C}{\mathrm{P}}}^{1}\) be its developing map. Then \(\mathcal{R}\) is circular if there is a round cylinder A on \({\mathbb{C}{\mathrm{P}}}^{1}\) such that

the image of f is contained in A;

the horizontal edges of R are immersed into different boundary circles of A;

for each vertical edge v of R, its development f(v) is a simple arc on A transverse to the horizontal foliation.
Given a circular rectangle \(\mathcal{R}\), the support of \(\mathcal{R}\) consists of the round cylinder A and the simple arcs on A which are the developments of the vertical edges of \(\mathcal{R}\) in Definition 8.1. We denote the support by \(\operatorname{Supp}\mathcal{R}\). We can pullback the horizontal foliation on A to a foliation on \(\mathcal{R}\) by the developing map, and call it the horizontal foliation of \(\mathcal{R}\).
Given projective structures \(\mathcal{R}\) and \(\mathcal{Q}\) on a marked rectangle R, we say that \(\mathcal{P}\) and \(\mathcal{Q}\) are compatible if \(\operatorname{Supp}\mathcal{R}= \operatorname{Supp}\mathcal{Q}\). Let \(\mathcal{R}\) be a circular rectangle, such that the both vertical edges are supported on the same arc α on a circular cylinder. Then, we say that \(\mathcal{R}\) is semicompatible with α.
8.1.1 Grafting a circular rectangle
(See [Bab10].) Let \(\mathcal{R}\) be a circular \({\mathbb{C}{\mathrm{P}}}^{1}\)structure on a marked rectangle R. Let A be the round cylinder in \({\mathbb{C}{\mathrm{P}}}^{1}\) which supports \(\mathcal{R}\). Pick an arc α on \(\mathcal{R}\), such that α connects the horizontal edges and it is transversal to the horizontal foliation of \(\mathcal{R}\). Then α is embedded into A by \(\operatorname{dev}\mathcal{R}\) — we call such an arc α an admissible arc. By cutting and gluing A and \(\mathcal{R}\) along α in an alternating manner, we obtain a new circular \({\mathbb{C}{\mathrm{P}}}^{1}\)structure on R whose support still is \(\operatorname{Supp}\mathcal{R}\). This operation is the grafting of \(\mathcal{R}\) along α, and the resulting structure on R is denoted by \(\operatorname{Gr}_{\alpha }\mathcal{R}\).
One can easily show that \(\operatorname{Gr}_{\alpha }\mathcal{R}\) is independent of the choice of the admissible arc α, since an isotopy of α preserving its initial conditions does not change \(\operatorname{Gr}_{\alpha }\mathcal{R}\).
8.2 Circular staircase loops
Let C=(f,ρ) be a \({\mathbb{C}{\mathrm{P}}}^{1}\)structure on S. A topological staircase curve is a piecewise smooth curve, such that

its smooth segments are labeled by “horizontal” or “vertical” alternatively along the curve, and

at every singular point, the horizontal and vertical tangent directions are linearly independent in the tangent space.
Then, a topological staircase curve s on C is circular, if the following conditions are satisfied: Letting \(\tilde{s}\) be a lift of s to \(\tilde{S}\),

every horizontal segment h of \(\tilde{s}\) is immersed into a round circle in \({\mathbb{C}{\mathrm{P}}}^{1}\) by f, and

for every vertical segment v of \(\tilde{s}\), letting h_{1}, h_{2} be the horizontal edges starting from the endpoints of v,

the round circles c_{1}, c_{2} containing f(h_{1}) and f(h_{2}) are disjoint, and

fv is contained in the round cylinder bounded by c_{1}, c_{2} and, it is transverse to the horizontal foliation of the round cylinder.

8.3 Circular polygons
Let P be a marked polygon with even number of edges. Then, let e_{1},e_{2},…,e_{2n} denote its edges in the cyclic order so that the edges with odd indices are vertical edges and with even indices horizontal edges. Suppose that c_{2},c_{4}…c_{2n} are round circles in \({\mathbb{C}{\mathrm{P}}}^{1}\) such that, for every \(i \in \mathbb{Z}/ n \mathbb{Z}\),

c_{2i} and c_{2(i+1)} are disjoint, and

c_{2(i−1)} and c_{2(i+1)} are contained in the same component of \({\mathbb{C}{\mathrm{P}}}^{1} \setminus c_{2i}\).
Let \(\mathcal{A}_{i}\) denote the round cylinder bounded by c_{2i} and c_{2(i+1)}. A circular \({\mathbb{C}{\mathrm{P}}}^{1}\)structure \(\mathcal{P}\) on P is supported on \(\{c_{2i}\}_{i=1}^{n}\) if

e_{2i} is immersed into the round circles of c_{2i} by \(\operatorname{dev}\mathcal{P}\) for every i=1,…n, and

e_{2i+1} is immersed into \(\mathcal{A}_{i}\) and its image is transversal to the horizontal foliation of \(\mathcal{A}_{i}\) (Fig. 21) for every i=0,1,…,n−1.
Let \(\mathcal{P}\) be a circular \({\mathbb{C}{\mathrm{P}}}^{1}\)structure on a polygon P supported on a circle system \(\{ c_{2i}\}_{i}^{n}\). For ϵ>0, \(\mathcal{P}\) is ϵcircular, if

for every vertical edge v_{i} is ϵparallel to the vertical foliation \(\mathcal{V}\) of the support cylinder \(\mathcal{A}_{i}\), and

the total transversal measure of v given by the vertical foliation \(\mathcal{V}\) is less than ϵ.
(Here, by the “total” transversal measure, we mean that if v intersects a leaf of \(\mathcal{V}\) more than once, and the measure is counted with multiplicity.)
Let \(\mathcal{P}_{1}\), \(\mathcal{P}_{2}\) be circular \({\mathbb{C}{\mathrm{P}}}^{1}\)structures on a 2ngon P. Then \(\mathcal{P}_{1}\) and \(\mathcal{P}_{2}\) are compatible if, for each i=1,…,n, \(\operatorname{dev}\mathcal{P}_{1}\) and \(\operatorname{dev}\mathcal{P}_{2}\) take e_{2i} to the same round circle and the arcs f_{1}(v_{2i−1}) and f_{2}(v_{2i−1}) are the same.
Let A be a flat cylinder with geodesic boundary; then its universal cover \(\tilde{A}\) is an infinite Euclidean strip. A projective structure (f,ρ) on A is circular, if the developing map \(f\colon \tilde{A} \to {\mathbb{C}{\mathrm{P}}}^{1}\) is a covering map onto a round cylinder in \({\mathbb{C}{\mathrm{P}}}^{1}\).
Next, let A be a spiral cylinder. Then each boundary component b of A is a monotone staircase loop. Let \(\tilde{b}\) be the lift of b to the universal cover \(\tilde{A}\). Let \(\{e_{i} \}_{i \in \mathbb{Z}}\) be the segments of \(\tilde{b}\) linearly indexed so that e_{i} with an odd index is a vertical edge and with an even index is a horizontal edge; clearly \(\tilde{b} = \cup _{i \in \mathbb{Z}} e_{i}\). Then, a \({\mathbb{C}{\mathrm{P}}}^{1}\)structure (f,ρ) on A is circular, if, for each boundary staircase loop b of A and each \(i \in \mathbb{Z}\),

the horizontal edge e_{2i} is immersed into a round circle c_{i} on \({\mathbb{C}{\mathrm{P}}}^{1}\);

c_{i−1}, c_{i} and c_{i+1} are disjoint, and the round annulus bounded by c_{i−1} and c_{i+1} contains c_{i} in its interior;

f embeds v_{i} in the round cylinder \(\mathcal{A}_{i}\) bounded by c_{i} and c_{i+1}, and f(v_{i}) is transverse to the circular foliation of \(\mathcal{A}_{i}\).
Two circular \({\mathbb{C}{\mathrm{P}}}^{1}\)structures \(\mathcal{A}_{1} = (f_{1}, \rho _{1})\), \(\mathcal{A}_{2} = (f_{2}, \rho _{2})\) on a spiral cylinder A are compatible if

ρ_{1} is equal to ρ_{2} up to conjugation by an element of \({\mathrm{PSL}}(2, \mathbb{C})\) (thus we can assume ρ_{1}=ρ_{2});

for each boundary component h of \(\tilde{A}\), f_{1} and f_{2} take h to the same round circle;

for each vertical edge v of \(\tilde{A}\), f_{1}v=f_{2}v.
More generally, let \(\mathcal{F}= (f_{1}, \rho _{1})\) and \(\mathcal{F}' = (f_{2}, \rho _{2})\) be two circular \({\mathbb{C}{\mathrm{P}}}^{1}\)structures on staircase surfaces F and F′. First suppose that there is a diffeomorphism ϕ:F→F′, which takes the vertices of F bijectively to those of F′. Then \(\mathcal{F}\) is compatible with \(\mathcal{F}'\) if

ρ_{1} is conjugate to ρ_{2} (thus we can assume that ρ_{1}=ρ_{2});

for every vertex p_{1} of \(\mathcal{F}_{1}\), the development of p_{1} coincides with the development of ϕ(p_{2});

for every a horizontal edge h of \(\mathcal{F}_{1}\), letting h′ be its corresponding horizontal edge of \(\mathcal{F}'\), then the developments of h and h′ are contained in the same round circle;

for every vertical edge v of \(\mathcal{F}\), letting v′ be its corresponding edge v′ of \(\mathcal{F}'\), then the developments of v′ and v coincide.
Next, instead of a diffeomorphism, we suppose that there is a semidiffeomorphism ϕ:F→F′. Then \(\mathcal{F}\) is semicompatible with \(\mathcal{F}'\) if

ρ_{1} is conjugate to ρ_{2} (thus we can assume that ρ_{1}=ρ_{2});

for every vertex p_{1} of \(\mathcal{F}_{1}\), the development of p_{1} coincides with the development of ϕ(p_{2});

if a horizontal edge h of \(\mathcal{F}\) corresponds to a horizontal edge h′ of \(\mathcal{F}'\), then h and h′ are supported on the same round circle on \({\mathbb{C}{\mathrm{P}}}^{1}\);

for every vertical edge v of \(\mathcal{F}\), letting v′ be its corresponding vertical edge (segment) of \(\mathcal{F}'\), then the developments of v′ and v coincide.
8.4 Construction of circular train tracks \(\mathcal{T}_{Y,\rho}\)
In this section, if ρ is in minus a large compact subset, we construct a circular traintrack structure of C_{Y,ρ} related to the polygonal traintrack decomposition \(\mathbf{T}'_{Y, \rho}\).
Two traintrack structures T_{1}, T_{2} on a flat surface E is (p,q)quasiisometric for p>1 and q>0 if there is a continuous (p,q)quasiisometry ϕ:E→E homotopic to the identity such that ϕ(T_{1})=T_{2} and the restriction of ϕ to T_{1} is a homotopy equivalence between T_{1} and T_{2}.
Theorem 8.2
For every ϵ>0, there is a bounded subset K=K_{ϵ} in , such that, for every , there is an ϵcircular surface train track decomposition \(\mathcal{T}_{Y, \rho}\) of C_{Y,ρ} with the following properties:

(1)
\(\mathcal{T}_{Y, \rho}\) is diffeomorphic to \(\mathbf{T}_{Y, \rho}'\), and it is (1+ϵ,ϵ)quasiisometric to both T_{Y,ρ} and \(\mathbf{T}_{Y, \rho}'\) in the normalized metric \(E^{1}_{Y, \rho}\).

(2)
For every vertical edge v of \(\mathbf{T}_{Y, \rho}'\), its corresponding edge of \(\mathcal{T}_{Y, \rho}\) is contained in the leaf of the vertical foliation V_{Y,ρ}.

(3)
For a branch B_{X} of T_{X,ρ}, letting B_{Y} be its corresponding branch of \(\mathbf{T}_{Y, \rho}'\) and letting \(\mathcal{B}_{Y}\) be the branch of \(\mathcal{T}_{Y, \rho}\) corresponding to B_{Y}, the restriction of \(\hat{\beta}_{X, \rho}\) to \(\partial \tilde{B}_{X}\) is ϵclose to the restriction of \(\hat{\beta}_{Y, \rho}\) to \(\partial \tilde {\mathcal{B}}_{Y}\) pointwise; moreover, if B_{X} is a transversal branch, then \(\hat{\beta}_{X, \rho}  \tilde{B}_{X}\) is ϵclose to \(\hat{\beta}_{Y, \rho}  \tilde {\mathcal{B}}_{Y}\) pointwise.
We fix a metric on the unit tangent bundle of \(\mathbb{H}^{3}\) which is leftinvariant under \({\mathrm{PSL}}(2, \mathbb{C})\).
Proposition 8.3
For every ϵ>0, if a bounded subset K_{ϵ} of is sufficiently large, then, for every and every horizontal edge h of T_{X,ρ}, the \(\operatorname{Ep}^{\ast}_{X, \rho}\)images of the vertical unit tangent vectors of along h are ϵclose.
Proof
The assertion immediately follows from Theorem 3.9 (2). □
Recall that we have constructed a system of equivariant circles for horizontal edges of \(\tilde {\mathbf{T}}_{X, \rho}\) in Lemma 7.3. Let h=[u,w] denote the horizontal edge of T_{X,ρ} where u, w are the endpoints. We shall perturb the endpoints of each horizontal edge of T_{Y,ρ} so that the endpoints map to the corresponding round circle.
Proposition 8.4
For every ϵ>0, there are sufficiently small δ>0 and a (large) bounded subset K_{ϵ} of satisfying the following: For every , if c={c_{h}} is a circle system for horizontal edges h of T_{X,ρ} given by Lemma 7.3 for δ, then, for every horizontal edge h=[u,w] of \(\tilde {\mathbf{T}}_{Y, \rho}\), there are, with respect to the normalized metric \(E_{Y, \rho}^{1}\), ϵsmall perturbations u′ and w′ of u and w along V_{Y,ρ}, respectively, such that f_{Y,ρ}(u′) and f_{Y,ρ}(w′) are contained in the round circle c_{h}.
Proof
This follows from Theorem 7.7 and Lemma 7.3 (2). □
Proof of Theorem 8.2
By Proposition 8.4, for each horizontal edge h=[u,w] of \(\mathbf{T}_{Y, \rho}'\), there is an ϵhomotopy of h to the circular segment h′ the perturbations u′, w′ such that, letting \(\tilde{h}\) be a lift of h to \(\tilde{E}_{Y, \rho}\), the corresponding lift \(\tilde{h}'\) of h′ is immersed into the round circle \(c_{\tilde{h}}\). For each vertical edge v of \(\mathbf{T}_{Y, \rho}'\), at each endpoint of v, there is a horizontal edge of \(\mathbf{T}_{Y, \rho}'\) starting from the point; then the round circles corresponding to the horizontal edges bound a round cylinder.
Note that a vertex u of \(\mathbf{T}_{Y, \rho}'\) is often an endpoint of different horizontal edges h_{1} and h_{2}. Thus, if the perturbations \(u_{1}'\) and \(u_{2}'\) of u are different for h_{1} and h_{2}, then \(\mathcal{T}_{Y, \rho}\) has a new short vertical edge connecting \(u_{1}'\) and \(u_{2}'\), and \(\mathcal{T}_{Y, \rho}\) is nondiffeomorphic to \(\mathbf{T}_{Y, \rho}'\).
Recall that the δ/4neighborhood of the singular points of \(E_{Y, \rho}^{1}\) is disjoint from the oneskeleton of T_{Y,ρ} by Proposition 6.23. Thus, every vertical edge v of \(\mathbf{T}_{Y, \rho}'\) is ϵcircular with respect to the round cylinder by Corollary 3.7. Thus we have (2). Thus we obtained an ϵcircular traintrack decomposition \(\mathcal{T}_{Y, \rho}\) of E_{Y,ρ}.
As the applies homotopies are ϵsmall, \(\mathcal{T}_{Y, \rho}\) are ϵclose to \(\mathbf{T}_{Y, \rho}'\) (1). Thus we may, in addition, assume that \(\mathcal{T}_{Y, \rho}\) is ϵclose to \(\mathbf{T}_{Y, \rho}'\) by Proposition 6.23. Moreover, Theorem 7.7 give (3).
8.5 Construction of \(\mathcal{T}_{X, \rho}\)
Given a traintrack structure on a surface, the union of the edges of its branches is a locally finite graph embedded on the surface. An edge of a traintrack decomposition is an edge of the graph, which contains no vertex in its interior (whereas an edge interior of a branch may contain a vertex of the train track).
Definition 8.5
Let C, C′ be \({\mathbb{C}{\mathrm{P}}}^{1}\)structures on S with the same holonomy \(\rho \colon \pi _{1}(S) \to {\mathrm{PSL}}(2, \mathbb{C})\), so that \(\operatorname{dev}C\) and \(\operatorname{dev}C'\) are ρequivariant. A circular traintrack decomposition \(\mathcal{T}= \cup _{i} \mathcal{B}_{i}\) of C is semicompatible with a circular traintrack decomposition \(\mathcal{T}' = \cup \mathcal{B}_{j}'\) of C′ if there is a markingpreserving continuous map Θ:C→C′ such that, for each branch \(\mathcal{B}\) of \(\mathcal{T}\), Θ takes \(\mathcal{B}\) to a branch of \(\mathcal{B}'\) of \(\mathcal{T}'\), and that \(\mathcal{B}\) and \(\mathcal{B}'\) are compatible by Θ.
Theorem 8.6
For every ϵ>0, if a bounded subset K_{ϵ} in is sufficiently large, then, for every , there is an ϵcircular train track decomposition \(\mathcal{T}_{X, \rho}\) of C_{X,ρ}, such that

(1)
\(\mathcal{T}_{X, \rho}\) is semicompatible with \(\mathcal{T}_{Y, \rho}\), and

(2)
\(\mathcal{T}_{X,\rho}\) additively 2πHausdorffclose to T_{X,ρ} with respect to the (unnormalized) Euclidean metric E_{X,ρ}: More precisely, in the vertical direction, \(\mathcal{T}_{X, \rho}\) is ϵclose to T_{X,ρ}, and in the horizontal direction, 2πclose in the Euclidean metric of E_{X,ρ} for all .
Proof
First, we transform T_{X,ρ} by perturbing horizontal edges so that horizontal edges are circular. Recall that, the branches of \(\mathcal{T}_{Y, \rho}\) are circular with respect to a fixed system c of equivariant circles given by Lemma 7.3. Thus, the β_{X,ρ}images of vertical tangent vectors along h are ϵclose to a single vector orthogonal to the hyperbolic plane bounded by c_{h}. Therefore, similarly to Theorem 8.2, we can modify the traintrack structure T_{X,ρ} so that horizontal edges are circular and ϵHausdorff close to the original traintrack structure in the Euclidean metric of E_{X,ρ} (this process may create new short vertical edges). Thus we obtained an ϵcircular train track \(\mathbf{T}_{X, \rho}'\) whose horizontal edges map to their corresponding round circles of c.
Next, we make the vertical edges compatible with \(\mathcal{T}_{Y, \rho}\). Recall that \(\mathbf{T}_{X, H_{X}}\) has no rectangles with short vertical edges (Lemma 5.6). Therefore, we have the following.
Lemma 8.7
For every R>0, if the bounded subset K of is sufficiently large, then, for each vertical edge of \(\mathbf{T}_{X, \rho}'\), the horizontal distance to adjacent vertical edges is at least R.
Thus, by Lemma 8.7, there is enough room to move vertical edges, less than 2π, so that the traintrack is compatible with \(\mathcal{T}_{Y, \rho}\) along vertical edges as well.
Since T_{X,ρ} is semidiffeomorphic to T_{Y,ρ} (Proposition 6.29 (3)), \(\mathcal{T}_{X, \rho}\) is semicompatible with \(\mathcal{T}_{Y, \rho}\).
9 Grafting cocycles and intersection of holonomy varieties
In this section, given a pair of \({\mathbb{C}{\mathrm{P}}}^{1}\)structures on S with the same holonomy, we shall construct a \(\mathbb{Z}\)valued cocycle under the assumption that the holonomy is outside of an appropriately large compact subset of the character variety χ. Namely, we will construct a traintrack graph with compatible \(\mathbb{Z}\)valued weights on the branches and its immersion into S (see [PH92] for traintrack graphs). This embedding captures, in a way, the “difference” of the \({\mathbb{C}{\mathrm{P}}}^{1}\)structures sharing holonomy. If a smooth arc on S is transversal to the immersed traintrack graph, then the sum of the \(\mathbb{Z}\)weights at the transversal intersection points is an integer— this functional defined on transversal arcs is called a transversal cocycle. Note that this cocycle value does not change under the regular homotopy of the arc if it retains the transversality. In particular, given a simple closed curve on a surface, we first homotopy the loop so that it has a minimal geometric intersection with the immersed traintrack graph, and then consider its transversal cycle with the traintrack graph. In this manner, we obtain a functional on the set of homotopy classes on the simple closed curves, which we call a grafting cocycle.
Goldman showed that every \({\mathbb{C}{\mathrm{P}}}^{1}\)structure with Fuchsian holonomy \(\pi _{1}(S) \to {\mathrm{PSL}}_{2}\mathbb{C}\) is obtained by grafting the hyperbolic structure with the Fuchsian holonomy along a \(\mathbb{Z}\)weighted multiloop on S ([Gol87]). The grafting cocycles that we construct in this paper can be regarded as a generalization of such weighted multiloops.
9.1 Relative degree of rectangular \({\mathbb{C}{\mathrm{P}}}^{1}\)structures
Let a<b be real numbers. Let \(f, g\colon [a,b] \to \mathbb{S}^{1}\) be orientation preserving immersions or constant maps, such that f(a)=g(a) and f(b)=g(b).
DefinitionLemma 9.1
The integer ♯f^{−1}(x)−♯g^{−1}(x) is independent on \(x \in \mathbb{S}^{1} \setminus \{f(a), f(b)\}\), where ♯ denotes the cardinality. We call this integer the degree of f relative to g, or simply, the relative degree, and denote it by deg(f,g).
Clearly, it is not important that f and g are defined on the same interval as long as corresponding endpoints map to the same point on \(\mathbb{S}^{1}\). Moreover, the degree is additive in the following sense.
Lemma 9.2
Subdivision of relative degree
Suppose in addition that f(c)=g(c) for some c∈(a,b). Then
The proofs of the lemmas above are elementary. Let R, Q be circular projective structures on a marked rectangle, and suppose that R and Q are compatible: By their developing maps, corresponding horizontal edges of R and Q are immersed into the same round circle on \({\mathbb{C}{\mathrm{P}}}^{1}\), and the corresponding vertices map to the same point.
Then, the degree of R relative to Q is the degree of a horizontal edge of R relative to its corresponding horizontal edge of Q — we similarly denote the degree by \(\deg (R, Q) \in \mathbb{Z}\). Although R has two horizontal edges, this degree is welldefined:
Lemma 9.3
c.f. Lemma 6.2 in [Bab15]
The degree deg(R,Q) is independent of the choice of the horizontal edge.
Proof
Let A be the round cylinder on \({{\mathbb{C}{\mathrm{P}}}}^{1}\) supporting both R and Q. Then, the horizontal foliation \(\mathcal{F}_{A}\) of A by round circles c induces foliations \(\mathcal{F}_{R}\) and \(\mathcal{F}_{Q}\) on R and Q, respectively. Then, for each leaf c of \(\mathcal{F}_{A}\), the corresponding leaves ℓ_{R} and ℓ_{Q} of \(\mathcal{F}_{R}\) and \(\mathcal{F}_{Q}\), respectively, are immersed into c, and the endpoints of ℓ_{R} and ℓ_{Q} on the corresponding vertical edges of R and Q map to the same point on c. The degree of ℓ_{R} relative to ℓ_{Q} is an integer, and it changes continuously in the leaves c of \(\mathcal{F}_{A}\). Thus, the assertion follows immediately. □
Lemma 9.4
cf. Lemma 6.2 in [Bab15]
Let R, Q be \({\mathbb{C}{\mathrm{P}}}^{1}\)structures on a marked rectangle with \(\operatorname{Supp}R = \operatorname{Supp}Q\). Then

if deg(R,Q)>0, then R is obtained by grafting Q along an admissible arc deg(R,Q) times;

if deg(R,Q)<0, then Q is obtained by grafting R along an admissible arc −deg(R,Q) times;

if deg(R,Q)=0, then R is isomorphic to Q (as \({\mathbb{C}{\mathrm{P}}}^{1}\)structures).
By Lemma 9.4, the “difference” of \({\mathbb{C}{\mathrm{P}}}^{1}\)rectangles R and Q can be represented by an arc α with weight deg(R,Q) such that α sits on the base rectangle connecting the horizontal edges.
Lemma 9.5
Let R and Q be circular projective structures on a marked rectangle such that \(\operatorname{Supp}R = \operatorname{Supp}Q\). Let A be the round cylinder on \({\mathbb{C}{\mathrm{P}}}^{1}\) supporting R and Q. Suppose that there are admissible arcs α_{R} on R and α_{Q} on Q which develop onto the same arc on A (transversal to the horizontal foliation), so that the arcs decompose R and Q into two circular rectangles R_{1}, R_{2} and Q_{1}, Q_{2}, respectively and \(\operatorname{Supp}R_{1} = \operatorname{Supp}Q_{1}\) and \(\operatorname{Supp}R_{2} = \operatorname{Supp}Q_{2}\). Then
Proof
This follows from Lemma 9.2. □
9.2 Traintrack graphs for planar polygons
Let P be a L^{∞}convex staircase polygon in \(\mathbb{E}^{2}\), which contains no singular points. We can decompose P into finitely many rectangles P_{1},P_{2},…,P_{n} by cutting P along n−1 horizontal arcs each connecting a vertex and a point on a vertical edge. Let \(\mathcal{P}\), \(\mathcal{Q}\) be compatible circular projective structures on P such that the round circles supporting horizontal edges are all disjoint. The decomposition P into P_{1},P_{2},…P_{n} gives decompositions of \(\mathcal{P}\) into \(\mathcal{P}_{1}, \mathcal{P}_{2}, \dots , \mathcal{P}_{n}\) and \(\mathcal{Q}\) into \(\mathcal{Q}_{1}, \mathcal{Q}_{2}, \dots , \mathcal{Q}_{n}\) such that \(\operatorname{Supp}\mathcal{P}_{i} = \operatorname{Supp}\mathcal{Q}_{i}\) for i=1,2,…,n. As in §9.1, for each i, we obtain an arc α_{i} connecting horizontal edges of P_{i} with weight \(\deg (\mathcal{P}_{i}, \mathcal{Q}_{i})\). Then, by splitting and combining α_{1},α_{2},…,α_{n} appropriately, we obtain a \(\mathbb{Z}\)valued traintrack graph \(\Gamma (\mathcal{P}, \mathcal{Q})\) on P transversal to the decomposition (Fig. 22).
9.3 Traintrack graphs for cylinders
Let A_{X} be a cylindrical branch of T_{X,ρ}, and let A_{Y} be the corresponding cylindrical branch of T_{Y,ρ}.
Pick a monotone staircase curve α on A_{X}, such that

(1)
α connects different boundary components of A_{X}, and its endpoints are on horizontal edges (Fig. 23),

(2)
the restriction of [V_{Y,ρ}]_{X} to A_{X} has a leaf disjoint from A_{X}.
Then [V_{Y,ρ}]_{X} is essentially carried by A_{X}. Then, one can easily show that the choice of α is unique through an isotopy preserving the properties.
Lemma 9.6
Suppose there are two staircase curves α_{1}, α_{2} on A_{X} satisfying Conditions (1) and (2). Then, α_{1} and α_{2} are isotopic through staircase curves α_{t} satisfying Conditions (1) and (2).
In Proposition 6.7, pick a realization of [V_{Y}]_{X} on the decomposition (A_{X},α_{X}) by a homotopy of [V_{Y}]_{X} sweeping out triangles. This induces an ϵalmost staircase curve α_{Y}. Similarly to Lemma 7.3, pick a system of round circles c={c_{h}} corresponding to horizontal edges h of α_{X} so that the \(\operatorname{Ep}_{X, \rho}\)images of vertical tangent vectors along h are ϵclose to a single vector orthogonal to the hyperbolic plane bounded by c_{h}.
Then, as in §8.4 we can accordingly isotope the curve α_{Y} so that the horizontal edges are supported on their corresponding circle of c and vertical edges remain vertical— let \(\alpha _{Y}^{{\mathbf{c}}}\) denote the curve after this isotopy. Then \(\mathcal{A}_{Y} \setminus \alpha _{Y}^{{\mathbf{c}}}\) is a circular projective structure on a staircase polygon in \(\mathbb{E}^{2}\).
Then, (similarly to Theorem 8.6), we can isotope α_{X} to an ϵalmost circular staircase curve \(\alpha ^{{\mathbf{c}}}_{X}\) so that

α_{X} is 2πHausdorff close to \(\alpha _{X}^{{\mathbf{c}}}\);

the horizontal edge h of \(\alpha _{X}^{{\mathbf{c}}}\) is supported on c_{h};

\(\mathcal{A}_{X} \setminus \alpha _{X}^{\alpha}\) is an ϵalmost circular staircase polygon compatible with \(\mathcal{A}_{Y} \setminus \alpha _{Y}^{{\mathbf{c}}}\).
As in §9.2, \(\mathcal{A}_{X} \setminus \alpha _{X}^{{\mathbf{c}}}\) and \(\mathcal{A}_{Y} \setminus \alpha _{Y}^{{\mathbf{c}}}\) yield a \(\mathbb{Z}\)valued weighted train track \(\Gamma _{A \setminus \alpha _{X}}\) on the polygon A∖α_{X} such that \(\Gamma _{A_{X} \setminus \alpha _{X}}\) is transversal to the horizontal foliation. Up to a homotopy preserving endpoints on the horizontal edges, the endpoints of \(\Gamma _{A_{X} \setminus \alpha _{X}}\) match up along α_{X} as \(\mathbb{Z}\)weighted arcs. Thus, we obtain a weighted traintrack graph \(\Gamma _{A_{X}}\) on A_{X}.
Consider the subset of the boundary of the circular cylinder \(\mathcal{A}_{X}\) which is the union of the vertical boundary edges and the vertices of \(\mathcal{T}_{X, \rho}\) contained in \(\partial \mathcal{A}_{X}\). Let A be the homotopy class of arcs in \(\mathcal{A}_{X}\) connecting different points in this subset. Then \([\Gamma _{A_{X}}] \colon \mathsf{A}\to \mathbb{Z}\) be the map which takes an arc to its total signed intersection number with \(\Gamma _{A_{X}}\).
Then Lemma 9.6 gives a uniqueness of \([\Gamma _{A_{X}}]\):
Proposition 9.7
\([\Gamma _{A_{X}}] \colon \mathsf{A}\to \mathbb{Z}\) is independent on the choice of the staircase curve α and the realization of [V_{Y}]_{X} on \((\mathcal{A}_{X}, \alpha _{X})\).
9.4 Weighted train tracks and \({\mathbb{C}{\mathrm{P}}}^{1}\)structures with the same holonomy
In this section, we suppose that Riemann surfaces X, Y have the same orientation. Let C be the set of the homotopy classes of closed curves on S (which are not necessarily simple). Given a weighted traintrack graph immersed on S, it gives a cocycle taking γ∈C to its weighted intersection number with the graph.
Theorem 9.8
For all distinct X,Y∈T, there is a bounded subset K in , such that

(1)
for each , the semicompatible traintrack decompositions \(\mathcal{T}_{X, \rho}\) of C_{X,ρ} and \(\mathcal{T}_{Y, \rho}\) of C_{Y,ρ} in Theorem 8.6yield a \(\mathbb{Z}\)weighted train track graph Γ_{ρ} carried by \(\mathcal{T}_{X, \rho}\) (immersed in S);

(2)
the grafting cocycle \([\Gamma _{\rho}] \colon \mathsf{C}\to \mathbb{Z}\) is independent on the choices for the construction of \(\mathcal{T}_{X, \rho}\) and \(\mathcal{T}_{Y, \rho}\);

(3)
\([\Gamma _{\rho}] \colon \mathsf{C}\to \mathbb{Z}\) is continuous in .
Since [Γ_{ρ}] takes values in \(\mathbb{Z}\), the continuity immediately implies the following.
Corollary 9.9
For sufficiently large boundary subset K of , [Γ_{ρ}] is welldefined and constant on each connected component of .
We first construct a weighted traintrack in (1). Let h_{X,1}…h_{X,n} be the horizontal edges of branches of \(\mathcal{T}_{X, \rho}\).
Since \(\mathcal{T}_{X, \rho}\) and \(\mathcal{T}_{Y, \rho}\) are semicompatible (Proposition 8.6), for each i=1,2,…,n, letting h_{Y,i} be its corresponding edge of a branch of \(\mathcal{T}_{Y, \rho}\) or a vertex of \(\mathcal{T}_{Y, \rho}\). Then, h_{X,i} and h_{Y,i} develop into the same round circle on \({\mathbb{C}{\mathrm{P}}}^{1}\), and their corresponding endpoints map to the same point by the semicompatibility. Thus we have the degree of h_{X,i} relative to h_{Y,i} taking a value in \(\mathbb{Z}\) (Definition 9.1) for each horizontal edge:
We will construct a \(\mathbb{Z}\)weighted train track Γ_{ρ} carried by \(\mathcal{T}_{X, \rho}\) so that the intersection number with h_{X,i} is γ_{ρ}(h_{X,i}). The traintrack graph Γ_{ρ} will be constructed on each branch of \(\mathcal{T}_{X, \rho}\):

For each rectangular branch R of \(\mathcal{T}_{X, \rho}\), we will construct a \(\mathbb{Z}\)train track graph embedded in R (Proposition 9.10).

For each cylinder A of \(\mathcal{T}_{X, \rho}\), we have obtained a \(\mathbb{Z}\)weighted train track graph embedded in A (§9.3).

For each transversal branch, we will construct a \(\mathbb{Z}\)weighted train track graph embedded in the branch (Proposition 9.12).

For each nontransversal branch of \(\mathcal{T}_{X, \rho}\), we will construct a \(\mathbb{Z}\)weighted train track immersed in the branch (Lemma 9.15).
9.4.1 Train tracks for rectangular branches
Given an ordered pair of compatible \({\mathbb{C}{\mathrm{P}}}^{1}\)structures on a rectangle, Lemma 9.4 gives a \(\mathbb{Z}\)weighted arc connecting the horizontal edges of the rectangle. Since the horizontal edges of a rectangular branch of \(\mathcal{T}_{X, \rho}\) may contain a vertex, we transform the weighted arc to a weighted traintrack graph so that it matches with γ_{ρ}.
Proposition 9.10
For every ϵ>0, there is a bounded subset K of , such that, for every , for each rectangular branch \(\mathcal{R}_{X}\) of \(\mathcal{T}_{X, \rho}\), there is a \(\mathbb{Z}\)weighted train track graph \(\Gamma _{\rho , \mathcal{R}_{X}}\) embedded in \(\mathcal{R}_{X}\) satisfying the following:

\(\Gamma _{\mathcal{R}_{X}}\) induces γ_{ρ};

\(\Gamma _{\mathcal{R}_{X}}\) is transversal to the horizontal foliation on \({\mathcal{R}_{X}}\);

each horizontal edge of \(\mathcal{T}_{X, \rho}\) in \(\partial \mathcal{R}_{X}\) contains, at most, one endpoint of \(\Gamma _{\rho , \mathcal{R}_{X}}\);

As cocycles, Γ_{R} is (1+ϵ,ϵ)quasiisometric to (V_{X,ρ}R_{X})−(V_{Y,ρ}R_{Y}), where R_{X} is a branch of T_{X,ρ} corresponding to \(\mathcal{R}_{X}\) and R_{Y} is the branch of \(\mathbf{T}_{Y, \rho}'\) corresponding to R_{X}.
Proof
Since \(\mathcal{R}_{X}\) and \(\mathcal{R}_{Y}\) share their support, let \(n = \deg (\mathcal{R}_{X}, \mathcal{R}_{Y})\) as seem in Lemma 9.4. Let h_{X} and h_{Y} be corresponding horizontal edges of \(\mathcal{R}_{X}\) and \(\mathcal{R}_{Y}\). Then h_{X}=h_{X,1}∪⋯∪h_{X,m} be the decomposition of h_{X} into horizontal edges of \(\mathcal{T}_{X}\), and let h_{Y}=h_{Y,1}∪⋯∪h_{Y,m} be the corresponding decomposition into horizontal edges and vertices of \(\mathcal{T}_{X}\) compatible with the semidiffeomorphism \(\mathcal{T}_{X, \rho} \to \mathcal{T}_{Y, \rho}\). Let \(n_{i} \in \mathbb{Z}\) be deg(h_{X,i},h_{Y,i}). Then, by Lemma 9.2, n=n_{1}+⋯+n_{m}. Then it is easy to construct a desired \(\mathbb{Z}\)weighted train track realizing such decomposition for both pairs of corresponding horizontal edges (see Fig. 24).
The last assertion follows from Lemma 4.1 and Theorem 7.7. □
9.4.2 Train tracks for cylinders
For each cylindrical branch \(\mathcal{A}_{X}\) of \(\mathcal{T}_{X, \rho}\), in §9.3, we have constructed a traintrack graph \(\Gamma _{\rho , \mathcal{A}_{X}}\) on \(\mathcal{A}_{X}\), representing the difference between \(\mathcal{A}_{X}\) and its corresponding cylindrical branch \(\mathcal{A}_{Y}\) of \(\mathcal{T}_{Y, \rho}\).
Proposition 9.11
For every ϵ>0, if a bounded subset K of is sufficiently large, then, for each cylindrical branch \(\mathcal{A}_{X}\) of \(\mathcal{T}_{X, \rho}\), the induced cocycle \([\Gamma _{\mathcal{A}_{X}}]\colon \mathsf{A}\to \mathbb{Z}\) times 2π is (1+ϵ,ϵ)quasiisometric to V_{Y,ρ}A_{X}−V_{X,ρ}A_{Y}, where A_{X} and A_{Y} are the corresponding cylindrical branches of T_{X,ρ} and \(\mathbf{T}_{Y, \rho}'\).
Proof
Recall that \(\Gamma _{\mathcal{A}_{X}}\) is obtained from \(\mathbb{Z}\)weighted traintrack graphs on the rectangles. There is a uniform upper bound, which depends only on S, for the number of the rectangles used to define \([\Gamma _{\mathcal{A}_{X}}]\), since the decomposition was along horizontal arcs starting from singular points. Then, on each rectangle, the weighted graph is (1+ϵ,ϵ)quasiisometric to the difference of V_{X,ρ} and V_{Y,ρ} by Proposition 9.10. Thus \([\Gamma _{\mathcal{A}_{X}}]\) is also (1+ϵ,ϵ)quasiisometric to the difference of V_{X,ρ} and V_{Y,ρ} if K is sufficiently large. □
9.4.3 Traintrack graphs for transversal polygonal branches
Recall that all transversal branches are polygonal or cylindrical (Lemma 6.18), i.e. their Euler characteristics are nonnegative.
Proposition 9.12
For every ϵ>0, there is a bounded subset K in such that, for each transversal polygonal branch \(\mathcal{P}_{X}\) of \(\mathcal{T}_{X, \rho}\), there is a \(\mathbb{Z}\)weighted train track \(\Gamma _{\rho , \mathcal{P}_{X}}\) embedded in \(\mathcal{P}_{X}\), letting P_{X} and P_{Y} be the branches of T_{X,ρ} and T_{Y,ρ}, respectively, corresponding to \(\mathcal{P}_{X}\), respectively, such that

(1)
each horizontal edge h of \(\mathcal{P}_{X}\) contains, at most, one endpoint of Γ_{ρ,P};

(2)
\([\Gamma _{\mathcal{P}_{X}}]\) agrees with γ_{ρ} on the horizontal edges of \(\mathcal{T}_{X, \rho}\) contained in \(\partial \mathcal{P}_{X}\);

(3)
Γ_{P} is transversal to the horizontal foliation H_{X,ρ} on P_{X};

(4)
2π[Γ_{P}] is (1+ϵ,ϵ)quasiisometric to (V_{X,ρ}P_{X}−V_{Y,ρ}P_{Y}).
For every ϵ>0, if K is sufficiently large, then, by Theorem 7.2, let \(\hat{\mathcal{Q}}_{X}\) be an ideal circular polygon whose truncation \({\mathcal{Q}}_{X}\) is ϵclose to \(\mathcal{P}_{X}\) in C_{X,ρ}. Similarly, \(\hat{\mathcal{Q}}_{Y}\) be an ideal circular polygon whose truncation \(\mathcal{Q}_{Y}\) is ϵclose to \(\mathcal{P}_{Y}\). Since \(\operatorname{Supp}\mathcal{P}_{X} = \operatorname{Supp}\mathcal{P}_{Y}\) as circular polygons, we may in addition assume that \(\operatorname{Supp}\mathcal{Q}_{X} = \operatorname{Supp}\mathcal{Q}_{Y}\) as truncated idea polygons.
Let \(\overline{\mathcal{Q}}_{X}\) be the canonical polynomial \({\mathbb{C}{\mathrm{P}}}^{1}\)structure on \(\mathbb{C}\) which contains \(\hat{\mathcal{Q}}_{X}\) (§7.1). Let \(\hat{\mathcal{L}}_{X}\) be the restriction of the Thurston lamination of \(\overline {\mathcal{Q}}_{X}\) to \(\hat{\mathcal{Q}}_{X}\). Similarly, let \(\overline{\mathcal{Q}}_{Y}\) be the canonical polynomial \({\mathbb{C}{\mathrm{P}}}^{1}\)structure on \(\mathbb{C}\) which contains \(\mathcal{Q}_{Y}\). Let \(\hat{\mathcal{L}}_{Y}\) be the restriction of the Thurston lamination of \(\overline {\mathcal{Q}}_{Y}\) to \(\hat{\mathcal{Q}}_{Y}\). As \(\operatorname{Supp}\mathcal{Q}_{X} = \operatorname{Supp}\mathcal{Q}_{Y}\), thus \(\overline{\mathcal{Q}}_{X}\) and \(\overline{\mathcal{Q}}_{Y}\) share their ideal vertices. Then \(\mathcal{Q}_{X}\) and \(\mathcal{Q}_{Y}\) are ϵclose to \(\mathcal{P}_{X}\) and \(\mathcal{P}_{Y}\), respectively. Thus, since γ_{ρ} takes values in \(\mathbb{Z}\), \(\hat{\mathcal{L}}_{X}  \hat{\mathcal{L}}_{Y}\) satisfies (2).
Theorem 7.2 (1) implies that \(\hat{\mathcal{L}}_{X}\) is (1+ϵ,ϵ)quasiisometric to V_{X}P_{X} and \(\hat{\mathcal{L}}_{Y}\) is (1+ϵ,ϵ)quasiisometric to V_{Y}P_{Y}, and therefore \(\hat{\mathcal{L}}_{X}  \hat{\mathcal{L}}_{Y}\) satisfies (4).
(1) is easy to be realized by homotopy combining the edges of the traintrack graph ending on the same horizontal edge. We show that there is a \(\mathbb{Z}\)weighted train track Γ which is ϵclose to \(\hat{\mathcal{L}}_{X}  \hat{\mathcal{L}}_{Y}\), satisfying (3).
Let \(\Gamma _{X}^{P}\) be a weighted train track graph on P which represents \(\hat{\mathcal{L}}_{X}\) a Let \(\Gamma _{Y}^{P}\) be the weighted traintrack graph which represents \(\hat{\mathcal{L}}_{Y}\). By Theorem 7.7, the pleated surface of \(\mathcal{P}_{X}\) is ϵclose to the pleated surface of \(\mathcal{P}_{Y}\), because of the quasiparallelism in Proposition 7.6. Let \(\check{\Gamma }_{X}^{P}\) be the subgraph of \(\Gamma _{X}^{P}\) obtained by eliminating the edges of weights less than a sufficiently small ϵ. Similarly, let \(\check{\Gamma }_{Y}^{P}\) be the subgraph of \(\Gamma _{Y}^{P}\) obtained by eliminating the edges of weight less than ϵ.
Then, there is a minimal traintrack graph Γ^{P} containing both \(\check{\Gamma }_{Y}^{P}\) and \(\check{\Gamma }_{X}^{P}\) and satisfying (1). Since the pleated surfaces are sufficiently close, by approximating the weights of \(\Gamma _{X}^{P}  \Gamma _{Y}^{P}\) by integers, we obtain a desired \(\mathbb{Z}\)weighted traintrack graph supported on Γ^{P}.
Since \(\mathcal{P}_{X}\) is a transversal branch, \(\Gamma _{X}^{P}\) and \(\Gamma _{Y}^{P}\) are both transversal to H_{X,ρ}P, thus Γ^{P} is transversal to H_{X,ρ}P (3).
9.4.4 Traintrack graphs for nontransversal branches
Let P_{X} and P_{Y} be the corresponding branches of T_{X,ρ} and T_{Y,ρ}, respectively, which are nontransversal. Let \(\mathcal{P}_{X}\) and \(\mathcal{P}_{Y}\) be the branches of \(\mathcal{T}_{X, \rho}\) and \(\mathcal{T}_{Y, \rho}\) corresponding to P_{X} and P_{Y}, respectively. Then, by Theorem 8.2, \(\hat{\beta}_{X, \rho}  \partial \mathcal{P}_{X}\) is ϵclose to \(\hat{\beta}_{Y, \rho}  \partial \mathcal{P}_{Y}\) in the C^{0}metric and C^{1}close along the vertical edges. Let σ_{X} be a pleated surface with crownshaped boundary whose truncation approximates \(\hat{\beta}_{X}  \mathcal{P}_{X}\), and let σ_{Y} be the pleated surfaced boundary with crownshaped boundary whose truncation approximates \(\hat{\beta}_{Y}  \mathcal{P}_{Y}\), such that σ_{X} and σ_{Y} share their boundary (in \(\mathbb{H}^{3}\)). In particular, the base hyperbolic surfaces for σ_{X} and σ_{Y} are diffeomorphic preserving markings and spikes.
Let ν_{X} and ν_{Y} be the bending measured laminations for σ_{X} and σ_{Y}, respectively; then ν_{X} and ν_{Y} contain only finitely many leaves whose connect ideal points.
In Thurston coordinates, the developing map and the pleated surface of a \({\mathbb{C}{\mathrm{P}}}^{1}\)structure are related by the nearest point projection to the supporting planes of the pleated surface ([KP94, Bab20]). For small υ>0, let F_{X}, F_{Y} be the surfaces in \(\mathbb{H}^{3}\) which are at distance υ from σ_{X} and σ_{Y} in the direction of the nearest point projections of P_{X} and P_{Y} ([EM87, Chapter II.2]).
Thus, if necessary, refining ν_{X} to ν_{Y} to ideal triangulations of the hyperbolic surfaces appropriately, pick an (irreducible) sequence of flips w_{i} which connects ν_{X} to ν_{Y}. Clearly the sequence w_{i} corresponds to a sequence of triangulations.
Lemma 9.13
If a bounded subset is sufficiently large, for every and all nontransversal branches P_{X} and P_{Y}, there is a uniform upper bound on the length of the flip sequence which depends only on X,Y∈T, or appropriate refinements into triangulations.
Proof
This follows from the length bound in Proposition 6.2. □
A triangulation in the sequence given by w_{i} is realizable, if there is an equivariant pleated surface homotopic to σ_{X} (and σ_{Y}) relative to the boundary such that the pleating locus agrees with the triangulation. In general, a triangulation in the sequence is not realizable when the endpoints of edges develop to the same point on \({\mathbb{C}{\mathrm{P}}}^{1}\). However a generic perturbation makes the triangulation realizable:
Lemma 9.14
For almost every perturbation of the holonomy of P_{X} and holonomy equivariant perturbation of the (ideal) vertices of \(\tilde{\sigma }_{X}\) (and \(\tilde{\sigma }_{Y}\)) in \({\mathbb{C}{\mathrm{P}}}^{1}\), all triangulations in the flip sequence w_{i} are realizable. Moreover, the set of realizable perturbations is connected.
Proof
If σ_{X} is an ideal polygon, the holonomy is trivial. Then, since there are only finitely many vertices and \({\mathbb{C}{\mathrm{P}}}^{1}\) has real dimension two, almost every perturbation is realizable.
If σ_{X} is not a polygon, an edge of a triangulation forms a loop if the endpoints are at the same spike of τ_{X}. For each loop ℓ of τ_{X}, the condition that the holonomy of ℓ is the identity is a complex codimension, at least, one in the character variety (and also in the representation variety). Since the flip sequence is finite, for almost all perturbations of the holonomy, if an edge of a triangulation in the sequence forms a loop, then its holonomy is nontrivial. Clearly, such a perturbation is connected. Then, for every such perturbation of the holonomy, it is easy to see that, for almost all equivariant perturbations of the ideal points, the triangulations in the sequence are realizable. □
For every perturbation of the holonomy and the ideal vertices given by Lemma 9.14, the flip sequence w_{i} gives the sequence of pleated surfaces \(\sigma _{X} = \sigma _{1} \xrightarrow{w_{1}} \sigma _{2} \xrightarrow{w_{2}} \dots \xrightarrow{w_{n1}} \sigma _{n} = \sigma _{Y}\) in \(\mathbb{H}^{3}\) connecting σ_{Y} to σ_{X}, such that σ_{i}’s share their boundary geodesics and ideal vertices.
For each flip w_{i}, the pairs of triangles of the adjacent pleated surfaces σ_{i} and σ_{i+1} bound a tetrahedron in \(\mathbb{H}^{3}\). To be precise, if the four vertices are contained in a plane, the tetrahedron is collapsed into a quadrangle, but it does not affect the following argument. The edges exchanged by the flip correspond to the opposite edges of the tetrahedron. Then pick a geodesic segment connecting those opposite edges. Then there is a path σ_{t} (i≤t≤i+1) of pleated surfaces with a single cone point of angle more than 2π such that

σ_{t} connects σ_{i} to σ_{i+1};

the pleated surfaces σ share their quadrangular boundary, which corresponds to the ideal quadrangle supporting the flip w_{i};

by the homotopy, σ_{t} sweeps out the tetrahedron;

the cone point on the geodesic segment (see Fig. 25).
In this manner, this sequence of pleated surfaces σ_{i} continuously extends to a homotopy of the pleated surfaces with, at most, one singular point of cone angle greater than 2π. This interpolation also connects a bending cocycle on σ_{i} to a bending cocycle on σ_{i+1} continuous, although the induced cocycle on σ_{i+1} may correspond to a measured lamination only immersed on the surface, since the edges of the triangulations transversally intersect. Thus, ν_{X} induces a sequence of the bending (immersed) measured laminations ν_{i} of σ_{i} supported a union of the pleating loci of σ_{1},…,σ_{i}.
For each i, the difference ν_{i+1}−ν_{i} of the transverse cocycles is supported on the geodesics corresponding to the edges of the tetrahedron, so that, on the surface, the edges form an ideal rectangle with both diagonals. Let μ_{i} be the difference cocycle ν_{i+1}−ν_{i}. From each vertex of the ideal rectangle, there are three leaves of ν_{i+1}−ν_{i} starting, and the sum of their weights is zero by Euclidean geometry (Fig. 26). Note that P_{X} can be identified with σ_{X} by collapsing each horizontal edge of P_{X} to a point. Hence, for every i, if α is a closed curve on P_{X} or an arc connecting vertical edges of P_{X}, then μ_{i}(α)=0. By regarding ν_{j} is a geodesic lamination on σ_{X}, their union \(\cup _{j = 1}^{i} \nu _{j}\) is a graph on σ_{X} whose vertices are the transversal intersection points of the triangulations. A small regular neighborhood N of \(\cup _{j = 1}^{i} \nu _{j}\) is decomposed into a small regular neighborhood N_{0} of the vertices and a small regular neighborhood of the edges minus N_{0} in N∖N_{0}.
Since, after the Whitehead moves, the pleated surface σ_{X} is transformed into a pleated surface σ_{Y}. Thus ν_{n}−ν_{Y} gives a \(\mathbb{Z}\)valued transversal cocycle.
By the construction of the regular homotopy, we have the following.
Proposition 9.15
Train tracks for nontransversal branches
For every nontransversally compatible branches \(\mathcal{P}_{X}\) of \(\mathcal{T}_{X, \rho}\) and \(\mathcal{P}_{Y}\) of \(\mathcal{T}_{Y, \rho}\), there is a \(\mathbb{Z}\)weighted immersed traintrack graph \(\Gamma _{P_{X}}\) representing the transversal cocycle supported on \(\cup _{j = 1}^{i} \nu _{j}\). Moreover, the traintrack cocycle is independent of the choice of the flip sequence w_{i}.
Proof
Given two flip sequences (w_{i}), \((w_{j}')\) connecting the triangulations of σ_{X} to σ_{Y}, there are connected by a sequence of sequences \((v_{i}^{k})\) of triangulations connecting σ_{X} to σ_{Y}, such that \((v_{i}^{k})\) and \((v_{i}^{k+1})\) differ by either an involutivity, a commutativity or a pentagon relation ([Pen12, Chap. 5, Corollary 1.2]). Clearly, the difference by an involutivity and a commutativity do not affect the resulting cocycle. Also by the pentagon relation, the pleated surface does not change including the bending measure since each flip preserves the total bending along the vertices. Therefore \((v_{i}^{k})\) and \((v_{i}^{k+1})\) give the same traintrack cocycle. □
Therefore we obtain \(\mathsf{A}\to \mathbb{Z}\). By continuity and the connectedness in Lemma 9.14. we have the following.
Corollary 9.16
[Γ_{P}] is independent on the choice of the perturbation in Lemma 9.14.
There are only finitely many combinatorial types of the traintracks T_{X,ρ}. We say that a branch B_{X} of T_{X,ρ} and a branch \(B_{X}'\) of T_{X,ρ′} are isotopic if they are diffeomorphic and isotopic on S. Then there are only finitely many isotopy classes of branches of T_{X,ρ} for all . Let α be an arc α on a branch B_{X}, such that each endpoint of α is at either on a vertical edge or a vertex of T_{X,ρ}.
Proposition 9.17
Let X,Y∈T. For every ϵ>0, there a compact subset K in with the following property: For every pair (B_{X},α) of an isotopy class of a branch B_{X} and an arc α as above, there is a constant k_{α}>0 such that, if B_{X} is a (nontransversal) branch of T_{X,ρ} for some , then, \(2\pi [\Gamma _{P_{X}}] (\alpha )\) is (1+ϵ,k_{α})quasiisometric to \(\sqrt{2}(V_{X}  P_{X}  V_{Y}  P_{Y}) (\alpha )\).
Proof
Since the length of the flip sequence is bounded from above, the difference between ν_{X} and ν_{n} is uniformly bounded in the space of transverse cocycles on S. Then the assertion follows. □
9.4.5 Independency of the transverse cocycle
From the traintrack decompositions \(\mathcal{T}_{X, \rho}\) and \(\mathcal{T}_{Y, \rho}\), we have constructed a weighted traintrack graph Γ_{ρ} (Theorem 9.8 (1)). Next we show its cocycle is independent on the traintrack decompositions \(\mathcal{T}_{X, \rho}\) and \(\mathcal{T}_{Y, \rho}\) (Theorem 9.8(2)).
Recall that the traintrack decompositions \(\mathcal{T}_{X, \rho}\) and \(\mathcal{T}_{Y, \rho}\) are determined by

(1)
the holonomy equivariant circle system c={c_{h}} indexed by horizontal edges h of \(\tilde {\mathbf{T}}_{X, \rho}\) (given by Lemma 7.3),

(2)
the realization W_{Y} of [V_{Y,ρ}]_{X,ρ} on T_{X,ρ} (Lemma 6.7), and

(3)
the choice of vertical edges of \(\mathcal{T}_{X, \rho}\) (Theorem 8.6 (2)).
Proposition 9.18
The cocycle \([\Gamma _{\rho}]\colon \mathsf{C}\to \mathbb{Z}\) constructed above is independent of the construction for \(\mathcal{T}_{X, \rho}\) and \(\mathcal{T}_{Y, \rho}\), i.e. (1), (2), (3).
Proof
(1) By Proposition 7.4, given two appropriate circle systems {c_{h}} and \(\{c'_{h}\}\), there is an equivariant isotopy of circles systems {c_{t,h}} connecting {c_{h}} to \(\{c'_{h}\}\). Then accordingly, we obtain a continuous family of cocycles \([\Gamma _{t, \rho}]\colon \mathsf{C}\to \mathbb{Z}\). As it takes discrete values, [Γ_{t,ρ}] must remain the same.
By the different choices for (2) and (3), the \(\mathbb{Z}\)weights on Γ shift across bigon regions corresponding vertical edges of T_{Y,ρ} by integer values. These weight shifts clearly preserve the cocycle \([\Gamma _{\rho}]\colon \mathsf{C}\to \mathbb{Z}\). □
9.4.6 Continuity of the transverse cocycle
Next we prove the continuity of [Γ_{ρ}] in ρ claimed in (3).
Definition 9.19
Convergence and semiconvergence of train tracks
Suppose that C_{i}∈P converges to C∈P. In addition, suppose, for each i, there are a traintrack structure \(\mathcal{T}_{i}\) of C_{i} and a traintrack structure \(\mathcal{T}\) for C. Then,

\(\mathcal{T}_{i}\) converges to \(\mathcal{T}\) if

for every branch \(\mathcal{P}\) of \(\mathcal{T}\), there is a sequence of branches \(\mathcal{P}_{i}\) of \(\mathcal{T}_{i}\) converging to \(\mathcal{P}\), and

for every sequence of branches \(\mathcal{P}_{i}\) of \(\mathcal{T}_{i}\), up to a subsequence, converges to either a branch of \(\mathcal{T}\) or an edge of a branch of \(\mathcal{T}\).


\(\mathcal{T}_{i}\) semiconverges to \(\mathcal{T}\) if there is a subdivision of \(\mathcal{T}\) into another circular traintrack structure \(\mathcal{T}'\) so that \(\mathcal{T}_{i}\) converges to \(\mathcal{T}'\).
Lemma 9.20
Let ρ_{i} be a sequence in converging to , where K is a sufficiently large compact (as in Theorem 8.6). Pick an equivariant circle system c_{i} for \(\mathbf{T}_{X, \rho _{i}}\) by Lemma 7.3 which converges to a circle system c for T_{X,ρ}. Then, up to a subsequence,

the circular train track \(\mathcal{T}_{X, \rho _{i}}\) of \(C_{X, \rho _{i}}\) semiconverges to a circular train track \(\mathcal{T}_{X, \rho}\) of C_{X,ρ};

the circular train track \(\mathcal{T}_{Y, \rho _{i}}\) of \(C_{Y, \rho _{i}}\) semiconverges to a circular train track \(\mathcal{T}_{Y, \rho}\) of C_{Y,ρ};

\(\mathcal{T}_{X, \rho}\) is semicompatible with \(\mathcal{T}_{Y, \rho}\).
Proof
By Lemma 6.19, \(\mathbf{T}_{X, \rho _{i}}\) semiconverges to T_{X,ρ}. Therefore \(\mathbf{T}_{X, \rho _{i}}\) converges to a subdivision \(\mathbf{T}_{X, \rho}'\) of T_{X,ρ} as i→∞. Then, if \(\mathbf{T}_{X, \rho} \neq \mathbf{T}_{X, \rho}'\), then T_{X,ρ} is obtained from \(\mathbf{T}_{X, \rho}'\) by gluing nonrectangular branches with rectangular branches of small width or replacing long rectangles into spiral cylinders (as in §5.3 and §6.7).
Recall that 7 the realization of \([V_{Y, \rho _{i}}]_{X,\rho _{i}}\) in the train track \(\mathbf{T}_{X, \rho _{i}}\) is unique up to shifting across vertical edges of nonrectangular branches (Proposition 6.12 (2)). Therefore, up to a subsequence, the realization of \([V_{Y, \rho _{i}}]_{X,\rho _{i}}\) on \(\mathbf{T}_{X, \rho _{i}}\) converges to a realization of [V_{Y,ρ}]_{X} on \(\mathbf{T}_{X, \rho}'\), Since \(\mathbf{T}_{X, \rho}'\) is a subdivision of T_{X,ρ}, the limit can be regarded as also a realization on T_{X,ρ}. Since the realization determines the traintrack structure of E_{Y,ρ}, up to a subsequence, \(\mathbf{T}_{Y, \rho _{i}}\) converges to a bounded traintrack \(\mathbf{T}_{Y, \rho}'\). Then T_{Y,ρ} is transformed to \(\mathbf{T}_{Y, \rho}'\) by possibly sliding vertical edges and subdividing spiral cylinders to wide rectangles. Moreover, by Theorem 8.2 (1), \(\mathcal{T}_{Y, \rho}\) is (1+ϵ,ϵ)quasiisometric to T_{Y,ρ}. Therefore, up to a subsequence, \(\mathcal{T}_{Y, \rho _{i}}\) converges to a circular traintrack structure \(\mathcal{T}_{Y}'\) of C_{Y,ρ}. If \(\mathcal{T}_{Y, \rho}\) is different from \(\mathcal{T}_{Y, \rho}'\), then \(\mathcal{T}_{Y, \rho}\) can be transformed to \(\mathcal{T}_{Y, \rho}'\) by sliding vertical edges and subdividing spiral cylinders into rectangles.
By Theorem 8.6, \(\mathcal{T}_{X, \rho _{i}}\) is additively 2πclose to \(\mathbf{T}_{X, \rho _{i}}\) in the Hausdorff metric of \(E^{1}_{X, \rho}\). Therefore, up to a subsequence \(\mathcal{T}_{X, \rho _{i}}\) converges to a circular train track decomposition \(\mathcal{T}_{X, \rho}'\) semidiffeomorphic to \(\mathcal{T}_{Y, \rho}\). Moreover \(\mathcal{T}_{X, \rho}\) can be transformed to \(\mathcal{T}_{X, \rho}'\) possibly by subdividing and sliding by 2π or 4π.
We have already shown that \(\mathcal{T}_{X, \rho}\) is semidiffeomorphic to \(\mathcal{T}_{Y, \rho}\) (Theorem 8.6).
Finally we have the continuity (3).
Corollary 9.21
\([\Gamma _{\rho _{i}}]\colon \mathsf{C}\to \mathbb{Z}\) converges to \([\Gamma _{\rho}]\colon \mathsf{C}\to \mathbb{Z}\) as i→∞.
Proof
Since \(\mathcal{T}_{X, \rho _{i}}\) semiconverges to \(\mathcal{T}_{X, \rho}\), up to taking a subsequence, \(\mathcal{T}_{X,i}\) converges to a subdivision \(\mathcal{T}'_{X, \rho}\) of \(\mathcal{T}_{X, \rho}\). Accordingly, there is a subdivision \(\mathcal{T}'_{Y, \rho}\) of \(\mathcal{T}_{Y, \rho}\), such that, to up a subsequence, \(\mathcal{T}_{Y,i}\) converges to \(\mathcal{T}'_{Y, \rho}\) and that \(\mathcal{T}'_{X, \rho}\) is semidiffeomorphic to \(\mathcal{T}'_{Y, \rho}\).
Let \(\Gamma _{\rho _{i}}\) be the \(\mathbb{Z}\)weighted traintrack given by \(\mathcal{T}_{X, \rho _{i}}\) and \(\mathcal{T}_{Y, \rho _{i}}\). Let \(\Gamma _{\rho}'\) be the \(\mathbb{Z}\)weighted train track given \(\mathcal{T}'_{X, \rho}\) and \(\mathcal{T}'_{Y, \rho}\). Then, by the convergence of the train tracks, \(\Gamma _{\rho _{i}}\) converges to \(\Gamma _{\rho}'\) as i→∞. Since \(\mathcal{T}'_{X, \rho}\) and \(\mathcal{T}'_{Y, \rho}\) are obtained by sliding and subdividing \(\mathcal{T}_{X, \rho}\) and \(\mathcal{T}_{Y, \rho}\) respectively, thus \(\Gamma _{\rho}'\) and Γ_{ρ} yield the same cocycle \(\mathsf{C}\to \mathbb{Z}\). □
9.5 Approximation of the grafting cocycle [Γ_{ρ}] by vertical foliations
Suppose that X, Y be distinct marked Riemann surfaces homeomorphic to S such that X and Y have the same orientation. For a branch B_{X} of T_{X,ρ}, let A(B_{X}) be the homotopy class of arcs α on \(\mathcal{R}_{X}\) such that every endpoint of α is either on a vertical edge or a vertex of T_{X,ρ}.
Theorem 9.22
Let c_{1},…,c_{n} be essential closed curves on S. Then, for every ϵ>0, there is a bounded subset K_{ϵ} of such that, for every , the grafting cocycle [Γ_{ρ}] times 2π is (1+ϵ,q)quasiisometric to \(\sqrt{2} (V_{X, \rho}  V_{Y, \rho})\) along c_{1},…,c_{n}. That is,
for all i=1,2,…,n.
Proof
Let H∈PML. Recall that \(E^{1}_{X, H}\) is the flat surface conformal to X, such that the horizontal foliation is H and AreaE^{1}(X,H)=1. Let T_{X,H} be the bounded traintrack structure of E_{X,H}.
Then, every closed curve c can be isotoped to a closed curve c′ so that, for each branch of B of T_{X,ρ}, c′B is an arc connecting different vertices. Let \(c'_{1}, \dots , c'_{m}\) be the decomposition into subarcs. By the finiteness of possible traintracks, the number m of the subarcs is bounded from above for all ρ. Then \(2\pi \Gamma _{\rho } c'_{j} \) is, if B is a transversal branch, (1+ϵ,ϵ)quasiisometric to \(\sqrt{2}(V_{X, \rho}  B  V_{Y, \rho}  B) c_{k}'\) by Proposition 9.12(4), Proposition 9.11, Proposition 9.10, and, if nontransversal, (1+ϵ,q)quasiisometric by Proposition 9.17. As the number of subarcs is bounded, the assertion follows. □
10 The discreteness
10.1 The discreteness of the intersection of holonomy varieties
Theorem 10.1
Suppose that X, Y are marked Riemann surface structures on S with the same orientation. Then every (connected) component of is bounded.
Proof
Let K be a component of . Suppose, to the contrary, that V is unbounded in . Then, there is a path ρ_{t} in which leaves every compact subset. Then, by Corollary 4.6, by taking a diverging sequence t_{1}<t_{2}<⋯, there are \(k_{i}, k_{i}' \in \mathbb{R}_{> 0}\) such that \(\frac{k_{i}}{k_{i}'} \to 1\) as i→∞ and
By taking a subsequence, we may, in addition, assume that their vertical foliations \([V_{X, \rho _{t_{i}}}]\) and \([V_{Y, \rho _{t_{i}}}]\) converge in PML. Thus let [V_{X,∞}] and [V_{Y,∞}] be their respective limits in PML. Since X≠Y, V_{X,∞} and V_{Y,∞} can not be asymptotically the same, in comparison to their horizontal foliations. Then \(V_{X, \rho _{t_{i}}}  V_{Y, \rho _{t_{i}}}\) “diverges to ∞”. That is, there is a closed curve α on S, such that
as i→∞.
Let \([\Gamma _{\rho _{t}}]\colon \mathsf{C}\to \mathbb{Z}\) be the function given by Theorem 9.8. As \([\Gamma _{\rho _{t}}]\) is continuous (Theorem 9.8 (3)), \([\Gamma _{\rho _{t}}]\colon \mathsf{C}\to \mathbb{Z}\) is a constant function (for t≫0). On the other hand, by Theorem 9.22, there is q>0 such that \(\sqrt{2} (V_{X, \rho _{t_{i}}}  V_{Y, \rho _{t_{i}}})(\alpha )\) is (1+ϵ_{i},q)quasiisometrically close to \(2\pi [\Gamma _{\rho _{t_{i}}}] (\alpha )\), and ϵ_{i}→0 as i→∞. We thus obtain a contradiction.
Since and are complex analytic, thus their intersection is also a complex analytic set (Theorem 5.4 in [FG02]). Therefore, since every bounded connected analytic set is a singleton (Proposition 2.7), Theorem 10.1 implies the following.
Corollary 10.2
is a discrete set.
We will, moreover, show that this intersection is nonempty in §12.
10.2 A weak simultaneous uniformization theorem
In this section, using Corollary 10.2, we prove a weak version of a simultaneous uniformization theorem for general representations. Let \(\rho \colon \pi _{1}(S) \to {\mathrm{PSL}}(2, \mathbb{C})\) be any nonelementary representation which lifts to \({\mathrm{SL}}(2, \mathbb{C})\). Let C, D be \({\mathbb{C}{\mathrm{P}}}^{1}\)structures on S^{+} with the holonomy ρ. Then, if a neighborhood U_{ρ} of ρ in is sufficiently small, then there are (unique) neighborhoods V_{C} and W_{D} of C and D in P, respectively, which are biholomorphic to U_{ρ} by . Then, for every η∈U_{ρ}, there are unique \({\mathbb{C}{\mathrm{P}}}^{1}\)structures C_{η} in V_{C} and D_{η} in W_{D} with holonomy η. Let Φ_{ρ,U}=Φ:U_{ρ}→T×T be the map which takes η∈U_{ρ} to the pair of the marked Riemann surface structures of C_{η} and D_{η}.
Theorem 10.3
Φ_{ρ,U} is a finitetoone open mapping.
Proof
By Corollary 10.2, the fiber of Φ is discrete. In addition, Φ is holomorphic and dimU_{ρ}=2dimT. Therefore, by Theorem 2.8, Φ is an open map. □
11 Opposite orientations
In this section, when the orientations of the Riemann surfaces are opposite, we show the discreteness of analogous to Theorem 11.1 and the local uniformization theorem analogous to Theorem 10.3.
Theorem 11.1
Fix X∈T and Y∈T^{∗}. Then, is a nonempty discrete set.
Since the proof is similar to the case when the orientations coincide, we simply outline the proof, yet explain how some parts are modified. We leave the details to the reader.
Recall that we have constructed compatible train track decomposition regardless of the orientation of XY (§8.4, §8.5). In summary, we have the following (in the case of opposite orientataions):
Proposition 11.2
Fix X∈T and Y∈T^{∗}. For every ϵ>0, there is a bounded subset K_{ϵ} in , such that, if , then there are circular polygonal traintrack decompositions \(\mathcal{T}_{X, \rho}\) of C_{X,ρ} and \(\mathcal{T}_{Y, \rho}\) of C_{Y,ρ}, such that

\(\mathcal{T}_{X, \rho}\) and \(\mathcal{T}_{Y, \rho}\) are semidiffeomorphic, and

\(\mathcal{T}_{X, \rho}\) and \(\mathcal{T}_{Y, \rho}\) are (1+ϵ,ϵ)quasiisometric to the train track decompositions T_{X,ρ} of the flat surface \(E_{X, \rho}^{1}\) and T_{Y,ρ} of the flat surface \(E_{Y, \rho}^{1}\), respectively, with respect to the normalized metrics.
In the case when the orientation of X and Y are the same, in Theorem 9.8, we constructed a \(\mathbb{Z}\)weighted traintrack graph representing the “difference” of projective structures on X and Y with the same holonomy. As the orientations of X and Y are different, we shall construct a \(\mathbb{Z}\)weighted train track graph representing, in this case, the “sum” of the \({\mathbb{C}{\mathrm{P}}}^{1}\)structures on X and Y with the same holonomy.
Let \(\mathcal{T}_{X, \rho}\) and \(\mathcal{T}_{Y, \rho}\) be circular train track decompositions of C_{X,ρ} and C_{Y,ρ} given by Proposition. 11.2 Let \(\{h_{X, 1}, h_{X_{2}}, \dots , h_{X_{n}}\}\) be the horizontal edges of \(\mathcal{T}_{X, \rho}\). Similarly to §9.4, we first define the \(\mathbb{Z}\)valued function on the set of horizontal edges. For each i=1,…,n, let c_{i} be the round circle on \({\mathbb{C}{\mathrm{P}}}^{1}\) supporting the development of h_{X,i}. First, suppose that h_{X,i} corresponds to an edge h_{Y,i} of \(\mathcal{T}_{Y, \rho}\) by the collapsing map \(\mathcal{T}_{X, \rho} \to \mathcal{T}_{Y, \rho}\). Since \(\mathcal{T}_{X, \rho}\) and \(\mathcal{T}_{Y, \rho}\) are compatible, the corresponding endpoints of h_{X,i} and h_{Y,i} map to the same point on c_{i} by their developing maps. Thus, by identifying the endpoints, we obtain a covering map from a circle h_{X,i}∪h_{Y,i} onto c_{i}. Then, define \(\gamma _{X, i}(h_{X, i}) \in \mathbb{Z}_{> 0}\) to be the covering degree.
Next, suppose that h_{X,i} corresponds to a vertex of \(\mathcal{T}_{Y, i}\). Then the endpoints of h_{X,i} develop to the same point on c_{i}. The circle obtained by identifying endpoints of h_{X,i} covers c_{i}. Thus, let \(\gamma _{X, i}(h_{X, i}) \in \mathbb{Z}_{> 0}\) be the covering degree.
Similarly to §9.4, we shall construct a \(\mathbb{Z}\)weighted traintrack graph Γ_{ρ} immersed in \(\mathcal{T}_{X, \rho}\). On each branch \(\mathcal{P}_{X}\) of \(\mathcal{T}_{X, \rho}\), we construct a \(\mathbb{Z}\)weighted traintrack graph \(\Gamma _{\mathcal{P}_{X}}\) on \(\mathcal{P}_{X}\) such that, for every ϵ>0, there is a compact subset K of satisfying the following:

The endpoints of \(\Gamma _{\mathcal{P}_{X}}\) are on horizontal edges of \(\mathcal{P}_{X}\).

They agree with γ_{X,ρ} along the horizontal edges.

If \(\mathcal{P}_{X}\) is a transversal branch, then, for , then \(2\pi \Gamma _{\mathcal{P}_{X}} (\alpha )\) is (1+ϵ,ϵ)quasiisometric to \(\sqrt{2}(V_{X, \rho}  P_{X}) (\alpha ) +\sqrt{2} (V_{Y, \rho}  P_{Y})( \alpha )\) for all \(\alpha \in \mathcal{A}(\mathcal{P}_{X})\).

For every smooth isotopy class of a staircase surface B on a flat surface homeomorphic to S and every arc α on B connecting points on horizontal edges or vertices, there is a positive constant q(B,α) such that, if \(\mathcal{T}_{X, \rho}\) contains a nontransversal branch \(\mathcal{B}_{X}\) smoothly isotopic to B on S, then \(2\pi \Gamma _{\mathcal{B}_{X}} (\alpha )\) is (1+ϵ,q(B,α))quasiisometric to \(\sqrt{2}\, V_{X, \rho}  B_{X} (\alpha )+ \sqrt{2}\, V_{Y, \rho}  B_{Y} (\alpha )\).
Theorem 11.3
Let X∈T and Y∈T^{∗}. For every ϵ>0, there is a bounded subset such that, for every , there is a \(\mathbb{Z}\)weighted graph Γ_{ρ} carried in \(\mathcal{T}_{X, \rho}\) such that

(1)
the induced cocycle \([\Gamma _{\rho}]\colon \mathsf{C}\to \mathbb{Z}\) changes continuously in ,

(2)
for every loop α on S, there is q_{α}>0, such that 2πΓ_{ρ}(α) is (1+ϵ,q_{α})quasiisometric to \(\sqrt{2}( V_{X, \rho} (\alpha )+ V_{Y, \rho} (\alpha )) \) for all .
Proof
The proof is similar to Theorem 9.8 (3) and Theorem 9.22. □
Then, Theorem 11.3 implies, similarly to Theorem 10.1, the following:
Theorem 11.4
Each connected component of is bounded.
12 The completeness
In this section, we prove the completeness in Theorem A. Let Q be a connected component of the Bers’ space B; then Q is a complex submanifold of (P⊔P^{∗})×(P⊔P^{∗}), and \(\dim _{\mathbb{C}}Q = 6g6\). We call that ψ:P⊔P^{∗}→T⊔T^{∗} is the uniformization map and Ψ:Q→(T⊔T^{∗})^{2} is defined by Ψ(C,D)=(ψ(C),ψ(D)). Then, by Theorem 10.3, Ψ is an open holomorphic map. In this section, we prove the completeness of Ψ.
Lemma 12.1
The open map Ψ:Q→(T⊔T^{∗})^{2} has a local path lifting property. That is, for every z∈Q, there is a neighborhood W of Ψ(z) such that if path α_{t}, 0≤t≤1 in W satisfies ζ(z)=α_{0}, then there is a lift \(\tilde{\alpha}_{t}\) of α_{t} to Q with \(\tilde{\alpha}_{0} = z\).
Proof
Since Ψ is an open map and dimQ=dim(T⊔T^{∗})^{2}, Ψ is a locally branched covering map. Then, for every z∈Q, there is an open neighborhood V of z in Q and a finite group G_{z} biholomorphically acting V, such that Ψ is G_{z}invariant, and Ψ induces the biholomorphic map V/G_{z}→Ψ(V).
For g∈G_{z}∖{id}, let F_{g}⊂W be the (pointwise) fixed point set of g. Clearly F_{g} is a proper analytic subset, and thus \(F := \cup _{g \in G_{z} \setminus \{id\}} F_{g}\) is an analytic subset strictly contained in W.
For every path α:[0,1]→V with α(0)=Ψ(z), we can take a oneparameter family of paths α_{t} (t∈[0,1]) in W with α_{t}(0)=Ψ(z) such that α_{1}=α and, for t<1, α_{t} is disjoint from Ψ(F) (since Ψ(F) has complex codimension, at least, one.)
Then, for t<1, α_{t} continuously lifts a path \(\tilde{\alpha}_{t}\colon [0,1] \to Q\), and α_{t} converges to a desired lift of α_{1} as t→1. □
Now we are ready to prove the completeness.
Theorem 12.2
Ψ:Q→(T⊔T^{∗})^{2}∖Δ is a complete map, where Δ is the diagonal set.
The completeness of Theorem 12.2 immediately implies the following:
Corollary 12.3
Ψ:Q→(T⊔T^{∗})^{2}∖Δ is surjective onto a connected component of (T⊔T^{∗})^{2}∖Δ.
Proof of Theorem 12.2
By Lemma 12.1, Ψ has a local path lifting property. Thus, suppose that (X_{t},Y_{t}):[0,1]→(T⊔T^{∗})^{2}∖Δ (0≤t≤1) be a path and there is a (partial) lift (C_{t},D_{t}):[0,1)→Q of the path (X_{t},Y_{t}). For each t∈[0,1], let denote the common holonomy of C_{t} and D_{t}. By the continuity, the orientations of Riemann surfaces in the pairs obviously remain the same along ρ_{t} for all t>0.
First we, in addition, suppose that there is an increasing sequence 0≤t_{1}<t_{2}<⋯ converging to 1, such that \(\rho _{t_{i}}\) converges to a representation in . By Corollary 10.2 or, in the case of opposite orientations, Theorem 11.1, is a discrete subset of . Then, since (X_{t},Y_{t}) converges to a point (X_{1},Y_{1}) in T×T, and and change continuously in t∈[0,1], every neighborhood contains \(\rho _{t_{i}}\) for all sufficiently large i. Thus the sequence \(\rho _{t_{i}}\) converges to a point in . Since is a discrete set in which continuously changes in t∈[0,1], we indeed have a genuine convergence.
Lemma 12.4
ρ_{t} converges to ρ_{1} as t→1.
By this lemma, (C_{t},D_{t}) converges in (P⊔P^{∗})×(P⊔P^{∗}) as t→1, so that the partial lift (C_{t},D_{t}) extends to t=1.
Thus it is suffices to show the addition assumption always holds:
Proposition 12.5
There is a compact subset K in , such that, for every t>0, there is t′>0 such that ρ_{t′}∈K.
Proof
The proof is essentially the same as the proof of Theorem 10.1 or Theorem 11.1, which states that each component of is a bounded subset of .
For 0≤t<1, let \(V_{C_{t}}\) and \(V_{D_{t}} \) be the vertical measured foliations of C_{t} and D_{t}, respectively. Suppose, to the contrary, that ρ_{t} leaves every compact subset of . As (X_{t},Y_{t}) converges to (X,Y)∈(T⊔T^{∗})^{2}∖Δ and \(\operatorname{Hol}(C_{t}) = \operatorname{Hol}(D_{t})\), similarly to Theorem 9.8 or Theorem 11.3, for t close to 1, we can construct a \(\mathbb{Z}\)weighed train track Γ_{t} on S, such that

the intersection function \([\Gamma _{t}]\colon \mathsf{C}\to \mathbb{Z}\) is continuous in t, and

for every closed curve α on S, there is a constant q_{α}>0 and a function ϵ_{t}>0 converging to 0, such that, for all sufficiently large t>0, [Γ_{t}](α) is (1+ϵ_{t},q_{α})quasiisometric to \(V_{C_{t}} (\alpha )  V_{D_{t}} (\alpha )\) if the orientation of X and Y are the same and to \(V_{C_{t}} (\alpha ) + V_{D_{t}} (\alpha )\) if the orientation of X and Y are different.
The first condition implies that the intersection number is constant in t, whereas the second condition implies that the intersection number with some closed curve α diverges to infinity as t→1. This is a contradiction. □
Last we remark the behavior of \(\operatorname{Hol}\) near the diagonal Δ.
Proposition 12.6
Let (C_{t},D_{t}), 0≤t<1 be a path in B, such that Ψ(C_{t},D_{t}) converges to a diagonal point (X,X) of (T⊔T^{∗})^{2}. Then \(\operatorname{Hol}C_{t} = \operatorname{Hol}D_{t}\) leaves every compact set in as t→1.
Proof
Fix arbitrary X∈T⊔T^{∗} and a bounded open subset U in χ. Recall that is a locally biholomorphic map, and that, for all Y∈T⊔T^{∗}, the set P_{Y} of \({\mathbb{C}{\mathrm{P}}}^{1}\)structures on Y is properly embedded in χ by \(\operatorname{Hol}\). Therefore, if an open neighborhood V of X in T⊔T^{∗} is sufficiently small, then, letting P_{V} denotes all \({\mathbb{C}{\mathrm{P}}}^{1}\)structure on Riemann surfaces in V, \(\mathsf{P}_{V} \cap \operatorname{Hol}^{1} (U)\) embeds into U by the holonomy map \(\operatorname{Hol}\). In particular, for every Y,Z∈V, is disjoint from U. Then the assertion is immediate. □
12.1 Cardinalities of the intersections
By the surjectivity of Corollary 12.3 and the existence of nonquasiFuchsian components of B in P⊔P^{∗} in Lemma 2.6, we immediately have the following:
Corollary 12.7
Let X, Y be distinct marked Riemann surface structures on S with any orientations. If the orientations of X and Y are opposite, the intersection contains, at least, two points, if the orientations of X and Y are the same, contains, at least, one point.
13 A proof of the simultaneous uniformization theorem
In this section, using Theorem A, we give a new proof of the simultaneous uniformization theorem without using the measurable Riemann mapping theorem. Recall QF is the quasiFuchsian space, and it is embedded in \(\mathsf{B}/\mathbb{Z}_{2}\).
Given (C,D) in QF, the universal covers \(\tilde{C}\), \(\tilde{D}\) are the connected components of \({\mathbb{C}{\mathrm{P}}}^{1}\) minus its equivariant Jordan curve equivariant via \(\operatorname{Hol}C = \operatorname{Hol}D\).
Lemma 13.1
QF is a union of connected components of \(\mathsf{B}/\mathbb{Z}_{2}\).
Proof
As being a quasiisometric embedding is an open condition, QF is an open subset of B. Thus, it suffices to show that QF is closed.
Let (C_{i},D_{i})∈P×P^{∗} be a sequence in QF which converges to (C,D)∈P×P^{∗}. Let \(\rho _{i}\colon \pi _{1}(S) \to {\mathrm{PSL}}(2, \mathbb{C})\) be the quasiFuchsian representation of C_{i} and D_{i}. We show that the holonomy \(\rho \colon \pi _{1}(S) \to {\mathrm{PSL}}(2, \mathbb{C})\) of the limits C and D is also quasiFuchsian. Let \(\tilde{C}_{i}\) and \(\tilde{D}_{i}\) be the universal covers of C_{i} and D_{i}, respectively. Then \(\tilde{C}_{i}\) and \(\tilde{D}_{i}\) are the components of \({\mathbb{C}{\mathrm{P}}}^{1}\) minus the ρ_{i}equivariant Jordan curve. Let \(f_{i} \colon \tilde{C_{i}} \cup \mathbb{S}^{1} \to {\mathbb{C}{\mathrm{P}}}^{1}\) and \(g_{i} \colon \tilde{D_{i}} \cup \mathbb{S}^{1} \to {\mathbb{C}{\mathrm{P}}}^{1}\) be the extensions of the embeddings to their boundary circles by a theorem of Carathéodory. Let h_{i} be the homeomorphism \(\tilde{C}_{i} \cup \mathbb{S}^{1} \cup \tilde{D}_{i} \to { \mathbb{C}{\mathrm{P}}}^{1}\).
Since embeddings \(\operatorname{dev}C_{i}\) converge to \(\operatorname{dev}C\) uniformly on compact as i→∞, the limit \(\operatorname{dev}C\) is also an embedding. (Suppose, to the contrary, that \(\operatorname{dev}C\colon \tilde{C} \to {\mathbb{C}{\mathrm{P}}}^{1}\) is not embedding. Then there are distinct open subsets in \(\tilde{C}\) which homeomorphically map onto the same open subset in \({\mathbb{C}{\mathrm{P}}}^{1}\). Then \(\operatorname{dev}C_{i}\) is also not embedding for all sufficiently large i against the assumption.) Thus, by the convergence of corresponding convex pleated surfaces in \(\mathbb{H}^{3}\), the equivariant property implies that \(\operatorname{dev}C\) extends to the boundary circle continuously and equivariantly. Similarly, since the embedding \(\operatorname{dev}D_{i}\) converges to \(\operatorname{dev}D\), then \(\operatorname{dev}D\) is also an embedding, and \(\operatorname{dev}D\) extends to the boundary circle continuously and equivariantly. Therefore h_{i} converges to a continuous map
such that the restriction of h to \(\tilde{C} \sqcup \tilde{D}\) is an embedding.
The domain and the target of h are both homeomorphic to \(\mathbb{S}^{2}\). Therefore, if \(h  \mathbb{S}^{1}\) is not a Jordan curve on \({\mathbb{C}{\mathrm{P}}}^{1}\), then there is a point \(z \in {\mathbb{C}{\mathrm{P}}}^{1}\), such that h^{−1}(z) is a single segment of \(\mathbb{S}^{1}\). By the equivariant property, \(h^{1} (z) = \mathbb{S}^{1}\), and Im h is a wedge of two copies of \(\mathbb{S}^{2}\), which is a contradiction. □
The following asserts that the diagonal of T×T^{∗} corresponds to the Fuchsian representations.
Lemma 13.2
Let X∈T. Let \(\eta \colon \pi _{1}(S) \to {\mathrm{PSL}}(2, \mathbb{C})\) be a quasiFuchsian representation, such that the ideal boundary of \(\mathbb{H}^{3} / {\mathrm{Im}}\rho \) realizes the marked Riemann surface X and its complex conjugate X^{∗}∈T^{∗}. Then η is the Fuchsian representation \(\pi _{1}(S) \to {\mathrm{PSL}}(2, \mathbb{R})\) such that \(X = \mathbb{H}^{2}/ {\mathrm{Im}}\eta \).
Proof
By the Riemann uniformization theorem, the universal covers \(\tilde{X}\) and \(\tilde{X}^{\ast}\) are the upper and the lower half planes. Then, by identifying their ideal boundaries equivariantly, we obtain \({\mathbb{C}{\mathrm{P}}}^{1}\) so that the universal covers \(\tilde{X}\) and \(\tilde{X}^{\ast}\) are round open disks. Let (C,D)∈P×P^{∗} be the pair corresponding to η, such that the complex structure of C is X and the complex structure of D is on X^{∗}.
On the other hand, the universal covers \(\tilde{C}\) and \(\tilde{D}\) are connected components of \({\mathbb{C}{\mathrm{P}}}^{1} \setminus \Lambda (\eta )\), where Λ(η) is the ηequivariant Jordan curve in \({\mathbb{C}{\mathrm{P}}}^{1}\). Thus, there is a ηequivariant homeomorphism \(\phi \colon {\mathbb{C}{\mathrm{P}}}^{1} \to {\mathbb{C}{\mathrm{P}}}^{1}\), such that ϕ restricts to a biholomorphism from \({\mathbb{C}{\mathrm{P}}}^{1} \setminus {\mathbb{R}\cup \{\infty \}} \to { \mathbb{C}{\mathrm{P}}}^{1} \setminus \Lambda (\eta )\).
Then, by Morera’s theorem for the line integral along triangles (see [SS03] for example), ϕ is a genuine biholomorphic map \({\mathbb{C}{\mathrm{P}}}^{1} \to {\mathbb{C}{\mathrm{P}}}^{1}\). Therefore ϕ is a Möbius transformation, and therefore η is conformally conjugate to the Fuchsian representation uniformizing X. □
Proposition 13.3
QF is a single connected component of \(\mathsf{B}/\mathbb{Z}_{2}\).
Proof
By Lemma 13.1, QF is the union of some connected components of B. By Theorem A, for every component Q of QF, the image Ψ(Q) contains the diagonal {(X,X^{∗})} of T×T^{∗}. Then, by Lemma 13.2, every diagonal pair (X,X^{∗})∈T×T^{∗} corresponds to a unique point in QF. Therefore QF is connected. □
Last we reprove the simultaneous uniformization theorem.
Theorem 13.4
QF is biholomorphic to T×T^{∗} by Ψ.
Proof
By Theorem A, Ψ is a complete local branched covering map. Since Ψ is surjective, by Lemma 13.2, Ψ is a degreeone over the diagonal \(\{(X, \overline{X} ) \mid X \in \mathsf{T}\}\), and the diagonal corresponds to the Fuchsian space.
The set of ramification points of Ψ is an analytic set. The Fuchsian space is a totally real subspace of dimension 6g−6. Therefore, if the ramification locus contains the Fuchsian space, then the locus must have the complex dimension 6g−6, the full dimension. This is a contradiction as Ψ is a locally branched covering map. Therefore, QF→T×T^{∗} has degree one, and thus it is biholomorphic. □
References
Ahlfors, L.V., Sario, L.: Riemann Surfaces. Princeton Mathematical Series, vol. 26. Princeton University Press, Princeton (1960)
Baba, S.: Neckpinching of \(\mathbb{C}{\mathrm{P}}^{1}\)structures in the \({\mathrm{PSL}}(2,\mathbb{C})\)character variety. arXiv:1907.00092. Preprint
Baba, S.: A Schottky decomposition theorem for complex projective structures. Geom. Topol. 14(1), 117–151 (2010)
Baba, S.: 2πGrafting and complex projective structures, I. Geom. Topol. 19(6), 3233–3287 (2015)
Baba, S.: 2πGrafting and complex projective structures with generic holonomy. Geom. Funct. Anal. 27(5), 1017–1069 (2017)
Baba, S.: On Thurston’s parametrization of \(\mathbb{C}{\mathrm{P}}^{1}\)structures. In: The Tradition of Thurston, pp. 241–254. Springer, Berlin (2020)
Bers, L.: Simultaneous uniformization. Bull. Am. Math. Soc. 66, 94–97 (1960)
Bromberg, K.: Hyperbolic conemanifolds, short geodesics, and Schwarzian derivatives. J. Am. Math. Soc. 17(4), 783–826 (2004a)
Bromberg, K.: Rigidity of geometrically finite hyperbolic conemanifolds. Geom. Dedic. 105, 143–170 (2004b)
Bromberg, K.: Projective structures with degenerate holonomy and the Bers density conjecture. Ann. Math. (2) 166(1), 77–93 (2007)
Canary, R.D., Epstein, D.B.A., Green, P.: Notes on notes of Thurston. In: Analytical and Geometric Aspects of Hyperbolic Space, Coventry/Durham, 1984. London Math. Soc. Lecture Note Ser., vol. 111, pp. 3–92. Cambridge University Press, Cambridge (1987)
Culler, M., Shalen, P.B.: Varieties of group representations and splittings of 3manifolds. Ann. Math. (2) 117(1), 109–146 (1983)
Dumas, D.: Grafting, pruning, and the antipodal map on measured laminations. J. Differ. Geom. 74(1), 93–118 (2006)
Dumas, D.: The Schwarzian derivative and measured laminations on Riemann surfaces. Duke Math. J. 140(2), 203–243 (2007)
Dumas, D.: Complex projective structures. In: Handbook of Teichmüller Theory. Vol. II. IRMA Lect. Math. Theor. Phys., vol. 13, pp. 455–508. Eur. Math. Soc., Zürich (2009)
Dumas, D.: Skinning maps are finitetoone. Acta Math. 215(1), 55–126 (2015)
Dumas, D.: Holonomy limits of complex projective structures. Adv. Math. 315, 427–473 (2017)
Dumas, D., Wolf, M.: Projective structures, grafting and measured laminations. Geom. Topol. 12(1), 351–386 (2008)
Earle, C.J.: On variation of projective structures. In: Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference, State Univ. New York, Stony Brook, N.Y., 1978. Ann. of Math. Stud., vol. 97, pp. 87–99. Princeton Univ. Press, Princeton (1981)
Earle, C.J., Kra, I.: A supplement to Ahlfors’s lecture. In: Lectures on Quasiconformal Mappings, 2nd edn. University Lecture Series, vol. 38, pp. viii+162. Am. Math. Soc., Providence (2006). With supplemental chapters by C. J. Earle, I. Kra, M. Shishikura and J. H. Hubbard
Epstein, D.B.A., Marden, A.: Convex hulls in hyperbolic space, a theorem of Sullivan, and measured pleated surfaces. In: Analytical and Geometric Aspects of Hyperbolic Space, Coventry/Durham, 1984. London Math. Soc. Lecture Note Ser., vol. 111, pp. 113–253. Cambridge University Press, Cambridge (1987)
Epstein, C.: Envelopes of horospheres and weingarten surfaces in hyperbolic 3space. Preprint
Faltings, G.: Real projective structures on Riemann surfaces. Compos. Math. 48(2), 223–269 (1983)
Fritzsche, K., Grauert, H.: From Holomorphic Functions to Complex Manifolds. Graduate Texts in Mathematics, vol. 213. Springer, New York (2002)
Farb, B., Dan, M.: A Primer on Mapping Class Groups. Princeton Mathematical Series, vol. 49. Princeton University Press, Princeton (2012)
Gallo, D., Kapovich, M., Marden, A.: The monodromy groups of Schwarzian equations on closed Riemann surfaces. Ann. Math. (2) 151(2), 625–704 (2000)
Gupta, S., Mj, M.: Meromorphic projective structures, grafting and the monodromy map. Adv. Math. 383, 107673 (2021)
Goldman, W.M.: Projective structures with Fuchsian holonomy. J. Differ. Geom. 25(3), 297–326 (1987)
Goldman, W.M.: Topological components of spaces of representations. Invent. Math. 93(3), 557–607 (1988)
Grauert, H., Remmert, R.: Coherent Analytic Sheaves. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 265. Springer, Berlin (1984)
Hejhal, D.A.: Monodromy groups and linearly polymorphic functions. Acta Math. 135(1), 1–55 (1975)
Hodgson, C.D., Kerckhoff, S.P.: Rigidity of hyperbolic conemanifolds and hyperbolic Dehn surgery. J. Differ. Geom. 48(1), 1–59 (1998)
Hodgson, C.D., Kerckhoff, S.P.: Universal bounds for hyperbolic Dehn surgery. Ann. Math. (2) 162(1), 367–421 (2005)
Hodgson, C.D., Kerckhoff, S.P.: The shape of hyperbolic Dehn surgery space. Geom. Topol. 12(2), 1033–1090 (2008)
Hubbard, J., Masur, H.: Quadratic differentials and foliations. Acta Math. 142(3–4), 221–274 (1979)
Hubbard, J.H.: The monodromy of projective structures. In: Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference, State Univ. New York, Stony Brook, N.Y., 1978. Ann. of Math. Stud., vol. 97, pp. 257–275. Princeton Univ. Press, Princeton (1981)
Hubbard, J.H.: Teichmüller Theory and Applications to Geometry, Topology. Matrix Editions, Ithaca (2006)
Hu, P.C., Yang, C.C.: Differentiable and Complex Dynamics of Several Variables. Mathematics and Its Applications., vol. 483. Kluwer Academic, Dordrecht (1999)
Kapovich, M.: On monodromy of complex projective structures. Invent. Math. 119(2), 243–265 (1995)
Kapovich, M.: Hyperbolic Manifolds and Discrete Groups. Progress in Mathematics, vol. 183. Birkhäuser, Boston (2001)
Kulkarni, R.S., Pinkall, U.: A canonical metric for Möbius structures and its applications. Math. Z. 216(1), 89–129 (1994)
Kamishima, Y., Tan, S.P.: Deformation spaces on geometric structures. In: Aspects of LowDimensional Manifolds. Adv. Stud. Pure Math., vol. 20, pp. 263–299. Kinokuniya, Tokyo (1992)
Lehto, O.: Univalent Functions and Teichmüller Spaces. Graduate Texts in Mathematics, vol. 109. Springer, New York (1987)
McMullen, C.T.: Complex earthquakes and Teichmüller theory. J. Am. Math. Soc. 11(2), 283–320 (1998)
Morgan, J.W., Shalen, P.B.: Valuations, trees, and degenerations of hyperbolic structures. I. Ann. Math. (2) 120(3), 401–476 (1984)
Newstead, P.: Geometric Invariant Theory 3space. CIMAT Lectures (2006)
Ott, A., Swoboda, J., Wentworth, R., Wolf, M.: Higgs bundles, harmonic maps, and pleated surfaces. Geom. Topol. To appear
Penner, R.C.: Decorated Teichmüller Theory. QGM Master Class Series. European Mathematical Society (EMS), Zürich (2012). With a foreword by Yuri I. Manin
Penner, R.C., Harer, J.L.: Combinatorics of Train Tracks. Annals of Mathematics Studies., vol. 125. Princeton University Press, Princeton (1992)
Poincaré, H.: Sur les groupes des équations linéaires. Acta Math. 4(1), 201–312 (1884)
Stein, E.M., Shakarchi, R.: Complex Analysis. Princeton Lectures in Analysis, vol. 2. Princeton University Press, Princeton (2003)
Strebel, K.: Quadratic Differentials. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 5. Springer, Berlin (1984).
Tanigawa, H.: Divergence of projective structures and lengths of measured laminations. Duke Math. J. 98(2), 209–215 (1999)
Thurston, W.P.: The Geometry and Topology of ThreeManifolds. Princeton University Lecture Notes (1978–1981)
Thurston, W.P.: ThreeDimensional Geometry and Topology. Princeton Mathematical Series, vol. 1. Princeton University Press, Princeton (1997). Edited by Silvio Levy
Wolf, M.: High energy degeneration of harmonic maps between surfaces and rays in Teichmüller space. Topology 30(4), 517–540 (1991)
Wolpert, S.A.: Families of Riemann Surfaces and WeilPetersson Geometry. CBMS Regional Conference Series in Mathematics, vol. 113. Am. Math. Soc., Providence (2010). Published for the Conference Board of the Mathematical Sciences, Washington, DC
Acknowledgements
I thank Ken Bromberg, David Dumas, Misha Kapovich, and Shinnosuke Okawa for their helpful conversations. I appreciate the anonymous referee for many valuable comments and for pointing out some important inaccuracies in the original version.
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Open access funding provided by Osaka University. The author was partially supported by the GrantinAid for Scientific Research (20K03610).
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Dedicated to Misha Kapovich on the occasion of his 60th birthday
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Baba, S. Bers’ simultaneous uniformization and the intersection of Poincaré holonomy varieties. Geom. Funct. Anal. 33, 1379–1453 (2023). https://doi.org/10.1007/s00039023006538
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DOI: https://doi.org/10.1007/s00039023006538