Constructing lattice surfaces with prescribed Veech groups: an algorithm

The Veech group of a translation surface is the group of Jacobians of orientation-preserving affine automorphisms of the surface. We present an algorithm which constructs all translation surfaces with a given lattice Veech group in any given stratum. In developing this algorithm, we give a new proof of a finiteness result of Smillie and Weiss, namely that there are only finitely many unit-area translation surfaces in any stratum with the same lattice Veech group. Our methods can be applied to obtain obstructions of lattices being realized as Veech groups in certain strata; in particular, we show that the square torus is the only translation surface in any minimal stratum whose Veech group is all of SL2Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{SL}_2\mathbb {Z}$$\end{document}.


Introduction
The study of translation surfaces may be approached from numerous different angles-complex analysis, differential geometry, algebraic topology and number theory, for instance, each offer unique insights into these objects. These different perspectives lead to a wealth of questions regarding translation surfaces and their dynamics. One such avenue of inquiry is the well-known action of SL 2 R on the stratum H 1 (d 1 , . . . , d κ ) of unit-area translation surfaces with singularities of orders d 1 , . . . , d κ . Stabilizers (i.e. Veech groups) and orbits of translation surfaces under this action have proven to be of great interest. Analogues of Ratner's theorems concerning unipotent flows on homogeneous spaces ( [Ra]) are proven in the celebrated work of Eskin, Mirzakhani and Mohammadi; in particular, SL 2 R-orbit closures are affine invariant submanifolds of H 1 (d 1 , . . . , d κ ) ( [EMM]). Despite having been well-studied since as early as the 1980's, some immediate questions regarding Veech groups have been non-trivial to address. General methods of computing Veech groups were unknown until recently ( [Bo], [BrJ], [Ed], [ESS], [Mu], [Ve3], and the special cases of [Fr] and [Sch]), and while some universal properties of Veech groups are known-for instance, they are necessarily discrete and non-cocompact ( [Ve])-a complete classification of which subgroups of SL 2 R are realized as Veech groups remains an open problem ( [HMSZ]).
The first non-arithmetic examples were discovered by Veech, who proved that the odd index Hecke triangle groups and an index two subgroup of the even index Hecke triangle groups are realized as Veech groups ( [Ve]). More general triangle groups are studied in [BM] and [Ho], where the authors show that there are Veech groups commensurable to all triangle groups ∆(m, n, ∞), 2 ≤ m < n < ∞. On the other hand, triangle groups whose orientation preserving subgroups are never contained in a Veech group are also given in [Ho]. Infinitely generated Veech groups are shown to exist in [HS] and [Mc], as are non-trivial Veech groups with no parabolic elements in [HL]. While this brief survey is far from an exhaustive overview of the literature, it does serve to highlight that the question of realizability of Veech groups has spanned decades and remains a large and interesting problem in the field.
Of particular interest within the space of all translation surfaces are so-called lattice surfaces, or Veech surfaces, which are those whose Veech groups are lattices (have finite covolume) in SL 2 R. These surfaces admit especially nice dynamics ( [Ve]), but the list of known families of lattice surfaces is relatively short ( [DPU]). Lattice surfaces are also interesting as they are precisely those surfaces whose SL 2 R-orbits are closed with respect to the analytic topology on the stratum ( [Ve2], [SmW]), and the projection of a closed orbit to the moduli space of Riemann surfaces is an algebraic curve-called a Teichmüller curve-isometrically immersed with respect to the Teichmüller metric (see, say, [Wr]).
We present an explicit algorithm which constructs all translation surfaces with a given lattice Veech group in any given stratum: Algorithm 1.1. Input: Non-negative integers d 1 ≤ · · · ≤ d κ whose sum is even and a finite set of generators of a lattice Γ ≤ SL 2 R.
An implicit algorithm is given in [SmW2] which enumerates all affine equivalence classes of lattice surfaces in terms of a parameter measuring the infimum of areas of triangles within the surface. Algorithm 1.1 similarly returns lattice surfaces, but it fundamentally differs both in its goal of finding surfaces with specific, prescribed Veech groups and in its method of doing so.
In developing Algorithm 1.1, we obtain a new proof of a finiteness result of Smillie and Weiss (Corollary 1.7, [SmW3]; see also [Mc2]): Theorem 1.2 (Smillie-Weiss). There are at most finitely many unit-area lattice surfaces with a given Veech group in any given stratum.
Our methods essentially reverse the algorithm for computing Veech groups presented in [Ed] and [ESS]. There the authors associate to each stratum a canonical (infinite area, disconnected in general) flat surface O, within which any closed, connected translation surface (X, ω) of the stratum may be naturally represented. Data regarding the translation surface and its saddle connections are recorded via pairs of points in O termed orientation-paired marked segments, and the authors show that (X, ω) may be recovered from a certain finite subset of these marked segments-the marked Voronoi staples ( §3.4 of [ESS]). Furthermore, a classification of the Veech group of (X, ω) in terms of its orientation-paired marked segments and affine automorphisms of O is given in Proposition 17 of [ESS].
Rather than beginning with a lattice surface (X, ω) and using its corresponding orientation-paired marked segments to compute elements of the Veech group Γ = SL(X, ω), Algorithm 1.1 begins with a lattice Γ ≤ SL 2 R, simulates normalized subsets of orientation-paired marked segments in O using Γ, and constructs candidate translation surfaces with Veech group Γ from finite unions of appropriately scaled versions of these 'simulations.' Theorem 1.2 follows from the facts that (i) a lattice Γ determines only finitely many such simulations (Lemma 4.8), (ii) the orientation-paired marked segments of any unit-area translation surface with Veech group Γ are realized as a finite union of scaled simulations (Theorem 4.9), and (iii) for any finite collection of simulations there are only finitely many scalars for which the union of scaled simulations could possibly coincide with the orientation-paired marked segments of a unit-area translation surface. For this latter point, we introduce the technical but important notions of permissible triples and permissible scalars of particular subsets of O, prove finiteness results regarding them (Proposition 3.5 and Lemma 3.7), and apply these results to orientation-paired marked segments and marked Voronoi staples (Proposition 4.11). Our novel proof differs from that of Smillie and Weiss, who use finiteness results of Markov partitions corresponding to hyperbolic affine automorphisms of surfaces ( [SmW3]).
This work should be viewed in light of the aforementioned open question regarding the realization of subgroups of SL 2 R as Veech groups: a full implementation 1 of Algorithm 1.1 could lead to the discovery of new lattice surfaces and experimental conjectures as to which groups are realized as Veech groups. Moreover, further work to determine a halting criterion for Algorithm 1.1 would give a general, explicit procedure to determine whether or not a given lattice is a Veech group in any particular stratum (see comments at the end of §5). At present, these ideas may be used in special cases to obtain obstructions for lattices being realized as Veech groups. In particular, we prove: Theorem 1.3. The square torus is the only translation surface in ∪ g>0 H(2g−2) with Veech group SL(X, ω) = SL 2 Z.
The paper is organized as follows. Section 2 gives a brief background on translation surfaces ( §2.1) and their Veech groups ( §2.2). Section 3 concerns the canonical surface O studied in [Ed] and [ESS]: definitions and immediate results regarding O are given in §3.1, permissible triples and scalars are introduced and studied in §3.2, and in §3.3 we recall definitions and results regarding marked segments and Voronoi staples of a translation surface. Section 4 further develops the machinery for Algorithm 1.1 and uses these ideas to give a new proof of Theorem 1.2: a characterization of lattice groups in terms of directions announced by linear transformations is given in §4.1; subsections 4.2 and 4.3 explore a group action on marked segments and how a lattice group may be used to simulate orbits of this action; subsection 4.4 relates marked Voronoi staples to permissible triples; and a new proof of Theorem 1.2 is given in §4.5. Algorithm 1.1 is presented in §5 and Theorem 1.3 is proven in §6.

Background
2.1. Translation surfaces. We begin with basic definitions and notation; see, say, surveys of [Wr] and [Zo] for more details.
2.1.1. Translation surfaces: various perspectives. A translation surface (X, ω) is a Riemann surface X together with a non-zero, holomorphic one-form ω. Equivalently, a translation surface is defined as a real surface X such that off of a finite set Σ, the surface X\Σ is equipped with a translation atlas, and the resulting flat structure extends to all of X to give conical singularities at points of Σ whose angles are positive integral multiples of 2π. Any translation surface may be polygonally represented as a collection of polygons in the plane together with an identification of edges in pairs, such that identified edges are parallel, equal length and of opposite orientation. A translation surface is closed if it is compact and has no boundary.
In these three perspectives-which we move fluidly between-the zeros of ω, elements of Σ of cone angles 2π(d + 1), d > 0, and, after identifying edges, the vertices of angles 2π(d + 1), d > 0, of a polygonally represented translation surface each correspond to the same set of non-removable singularities of the surface. Throughout, the singular set Σ denotes these singularities together with possibly finitely many removable singularities (or marked points), i.e. points of cone angle 2π. We refer to any element of Σ as a singularity (of cone angle 2π(d + 1), d ≥ 0), bearing in mind that it may in fact be removable. Points in the complement X\Σ are called regular.
2.1.2. Metric, Lebesgue measure, saddle connections and holonomy vectors. A translation surface (X, ω) comes equipped with a metric defined as follows: given p, q ∈ X, the distance from p to q is given by where the infimum is over all piecewise-smooth curves γ ⊂ X from p to q. The infimum is realized by a straight line segment (with respect to the flat structure) on (X, ω) or by a union of such segments meeting at singularities. Lebesgue measure in the plane pulls back via coordinate maps of the translation atlas of (X, ω) to give a measure on X\Σ; we extend this measure to all of X by declaring Σ to be a null set. A separatrix is a straight line segment on (X, ω) emanating from a singularity and having no singularities in its interior. A saddle connection is a separatrix (of positive length) which also ends at a singularity. The holonomy vector of a saddle connection s is defined as hol(s) := s ω. The collection of all holonomy vectors of (X, ω) forms a subset of the plane with no limit points and whose directions are dense in S 1 (Proposition 3.1, [Vo]).
2.1.3. Voronoi decomposition. Every translation surface (X, ω) has a (unique) Voronoi decomposition subordinate to its singular set Σ, which is described as follows (see also [Bo], [MS]). For any x ∈ X, let d(x, Σ) denote the minimum of the set {d(x, σ)} σ∈Σ . To each σ ∈ Σ there is associated an open, connected 2-cell, C σ , comprised of the points x ∈ X for which d(x, Σ) is realized by a unique length-minimizing path ending at σ. The boundary of C σ is the union of 1-cells of points x for which d(x, Σ) is realized by precisely two length-minimizing paths, at least one of which ends at σ, and of 0-cells of points x for which d(x, Σ) is realized by three or more length-minimizing paths, at least one of which ends at σ.

Veech groups.
3 2.2.1. Affine diffeomorphisms and Veech groups. An affine diffeomorphism f : (X 1 , ω 1 ) → (X 2 , ω 2 ) between (not necessarily closed nor connected) translation surfaces is a diffeomorphism from X 1 to X 2 sending the singular set Σ 1 of X 1 into that of X 2 , and which-on the complement of Σ 1 -is locally an affine map of the plane of constant linear part 2 (i.e. a map of the form v → Av + b for some global A ∈ GL 2 R and local b ∈ R 2 ). An affine automorphism is an affine diffeomorphism from a translation surface (X, ω) to itself. The set of all affine automorphisms of (X, ω), denoted Aff(X, ω), forms a group under composition, as does the subset Aff + (X, ω) ⊂ Aff(X, ω) of orientation-preserving elements. The map der : Aff(X, ω) → GL 2 R sending an orientation-preserving affine automorphism to its linear part in local coordinates gives a group homomorphism. The image of Aff + (X, ω) under this map, denoted SL(X, ω), is called the Veech group of (X, ω), and its kernel is the group of translations denoted by Trans(X, ω). Veech shows in [Ve] that for closed, connected (X, ω), the Veech group SL(X, ω) is a discrete subgroup of SL 2 R-and thus its image in PSL 2 R is a Fuchsian group-and, moreover, SL(X, ω) is always non-cocompact.
2.2.2. Strata, a GL 2 R-action and lattices. We call (X 1 , ω 1 ) and (X 2 , ω 2 ) translation equivalent if there is an affine diffeomorphism between them with trivial linear part. For non-negative integers d 1 ≤ · · · ≤ d κ , let H(d 1 , . . . , d κ ) denote the stratum of all closed, connected translation surfaces-up to translation equivalence-with singularities of cone angles 2π(d 1 + 1) ≤ · · · ≤ 2π(d κ + 1) and no other singularities. (Recall that we allow for 'removable singularities,' so some d i may be zero, and with our definitions, SL(X, ω) depends on these marked points in Σ. In particular, Algorithm 1.1 applies not only to standard strata where each d i > 0, but it also allows for the construction of lattice surfaces (X, ω) with a specified number of marked points which remain invariant under Aff + (X, ω).) Abusing notation, we denote an element of a stratum by any of its translation-equivalent representatives. By the Riemann-Roch Theorem, every translation surface (X, ω) ∈ H(d 1 , . . . , d κ ) has the same genus g > 0, and the sum of the d i equals 2g − 2.
There is a natural action of GL 2 R on each stratum, given by post-composing the coordinate charts of (X, ω) ∈ H(d 1 , . . . , d κ ) with the usual action of a matrix in the plane via a linear transformation (one can verify that this action is well-defined with respect to translation-equivalence). Any translation surface may be normalized by the action of a diagonal matrix so as to have unit-area; we denote the collection of unitarea translation surfaces in H(d 1 , . . . , d κ ) by H 1 (d 1 , . . . , d κ ). There is a corresponding action of SL 2 R on H 1 (d 1 , . . . , d κ ), and the stabilizer of (X, ω) under this SL 2 R-action is isomorphic to the Veech group SL(X, ω) as defined in §2.2.1. We call (X, ω) a lattice surface if SL(X, ω) is a lattice group, i.e. SL(X, ω) has finite covolume in SL 2 R.
Remark. Recall from §2.2.1 that a Veech group is always non-cocompact, and thus a lattice Veech group is non-uniform (i.e. it is a non-cocompact discrete group of finite covolume). Other necessary properties of lattice Veech groups are known; for example, the trace field of SL(X, ω) is a totally real number field of degree at most g, where g is the genus of (X, ω) ( [GJ], [HL], [KS]). Throughout, when speaking of a lattice Γ ≤ SL 2 R the reader may wish to impose such known restrictions, though they are not strictly necessary for our statements. By a lattice we simply mean a discrete subgroup of SL 2 R of finite covolume.

A canonical surface for each stratum
3.1. Canonical surface. Here we recall the canonical (infinite area and disconnected in general) flat surface associated to each stratum H(d 1 , . . . , d κ ) studied in [Ed] and [ESS] and explore some immediate results. For each 1 ≤ i ≤ κ, let O i := (C, z di dz), and set Each component O i ⊂ O is an infinite translation surface with a sole singularity of cone angle 2π(d i + 1) at the origin, which we denote by 0 ∈ O i (the context will be clear as to which component an origin 0 belongs). Intuitively, we may think of O i as d i +1 copies of the plane, denoted c i 0 , . . . , c i di , each of which is slit along the non-negative real axis and glued so that the bottom edge, b i j , of the slit of c i j is identified with the top edge, 2 The translation structures guarantee that the linear part of a locally affine map is constant on components of (X 1 , ω 1 ); this latter statement in the definition is thus intended for disconnected translation surfaces.
, of the slit of c i j+1 for each j ∈ Z di+1 ; see Figure 1. After making these identifications, we (arbitrarily) consider t i j+1 ∼ b i j as part of c i j+1 and not of c i j . In particular, where γ is any piecewise-smooth curve from 0 to p (note that the integral is independent of path due to Cauchy's integral theorem), and extend to a map π : O → C by setting π| Oi := π i for each i ∈ {1, . . . , κ}. Observe that π i (·) is a degree d i + 1 branched covering with a single branch point 0 ∈ C, and the extended map π(·) is a degree κ i=1 (d i + 1) = 2g − 2 + κ branched covering with sole branch point 0 ∈ C. Throughout, we denote by z x and z y the real and imaginary parts, respectively of z ∈ C; hence π(p) = π(p) x + iπ(p) y . Note that on sufficiently small neighborhoods in O (and away from singularities), the map π(·) restricts to give a coordinate chart for the translation atlas associated to O. Moreover, each O i is naturally equipped with generalized polar coordinates: every point p ∈ c i j ⊂ O i is determined by a magnitude |p| := d(0, p) = |π(p)| and, for p = 0, a principal argument value arg(p) := arg(π(p)) + 2πj ∈ [2πj, 2π(j + 1)), where |z| and arg(z) ∈ [0, 2π) are the Euclidean norm and principal value of the argument function, respectively, of z ∈ C.
For each i ∈ {1, . . . , κ}, define an affine automorphism ρ i : O i → O i which acts as a counterclockwise rotation of angle 2π about the origin; note that ρ i belongs to Trans(O i ) since, in local coordinates given by π(·), ρ i has trivial linear part. We extend ρ i to O by acting as the identity on all other components and observe that ρ i ∈ Trans(O). For each integer d ≥ 0, let n(d) = #{1 ≤ i ≤ κ | d i = d}. Having fixed d, let i 1 , . . . , i n(d) be the distinct indices of the d 1 , . . . , d κ which equal d, and let S n(d) denote the symmetric group on the set {i 1 , . . . , i n(d) }. Extend S n(d) to act as the identity on all other indices, and for each α ∈ S n(d) , let f α : O → O permute the indices of O 1 , . . . , O κ by α, respecting polar coordinates. We again have that f α ∈ Trans(O) for each α ∈ S n(d) with n(d) ≥ 1.
where C di+1 is the cyclic group of order d i + 1.
Sketch. Lemma 2 of [ESS] shows that Trans(O) is generated by an action of d≥0, n(d)≥1 S n(d) together with an action of κ i=1 C di+1 on O, as naturally defined from the definitions of f α and ρ i above. That is, any element τ ∈ Trans(O) may be written as a finite composition of various f α 's and ρ i 's. We may rearrange the order of this composition (possibly altering the indices of some ρ i 's) to collect all f α 's with α belonging to the same S n(d) and all ρ i 's corresponding to the same O i . That τ may be uniquely written in such a way is verified by considering the indices of images of 2π-sectors c i j .
Lemma 3.2. Let f ∈ Aff(O) with der(f ) = A. Then π • f = A · π, where the notation on the right denotes the usual action of a matrix A on the plane as a linear transformation.
Proof. Since π(·) restricts on sufficiently small neighborhoods to give a (bijective) coordinate map for the translation atlas of O, we have-by definition of f -that for each v ∈ R 2 in the image of such a neighborhood, This composition agrees on the images of the intersections of such neighborhoods; in particular, b depends only on the component O i to which the neighborhood belongs. Choose some neighborhood as above which contains 0 ∈ O i on its boundary. The above composition (and each of the maps comprising it) extends continuously to the closure of its domain. Since f sends singularities to singularities, setting v = 0 gives b = 0. As O i was arbitrary, the result follows from the previous observations.
In Lemma 3 of [ESS], it is shown that der(Aff(O)) = GL 2 R. To each A ∈ GL + 2 R we associate a canonical f A ∈ Aff + (O) with der(f A ) = A as follows. Fix some g A ∈ Aff + (O) with der(g A ) = A, and choose τ ∈ Trans(O) so that the composition for each point p ∈ O i , the angle measured counterclockwise from p to f A (p) is non-negative and less than 2π. Existence of τ follows from Lemma 3.1. Notice that der(f A ) = A since der(·) is a group homomorphism, τ ∈ Trans(O) = ker(der) and der(g A ) = A. Furthermore, conditions (i) and (ii) guarantee that f A is unique, regardless of the initial choice of g A (and subsequent choice of τ ).
We also note that any g A ∈ Aff + (O) with der(g A ) = A ∈ GL + 2 R may be written uniquely as g A = τ ′ • f A for some τ ′ ∈ Trans(O) (namely τ ′ = τ −1 for τ as above), with f A the canonical affine automorphism associated to A.
While Trans(O) is generally non-abelian, we have the following commutativity result: Since der(τ • f A ) = der(f A • τ ) = A, both sides of the previous line are sent under π(·) to the same point z ∈ C (Lemma 3.2). Suppose p ∈ c i k and τ (O i ) = O j with τ (c i k ) = c j ℓ . Let r i k denote the set of points in O i with argument 2πk (i.e. r i k is the ray along the positive real axis in c i k ). Note that π(·) sends each of τ •f A (c i k ) and f A •τ (c i k ) bijectively onto C, and these sets are completely determined by the images τ •f A (r i k ) and f A • τ (r i k ) in O j : the former are the sets of points within angle 2π counterclockwise of the respective images of the ray r i k . Since there is only one point in each 2π-sector which maps to z ∈ C under π(·), it suffices to show that τ . By definition of f A and r i k , the image f A (r i k ) belongs to c i k , and thus τ • f A (r i k ) belongs to c j ℓ . But also τ (r i k ) belongs to c j ℓ by assumption, and again by definition of f A , the image f A • τ (r i k ) belongs to c j ℓ as well. Since the images τ • f A (r i k ) and f A • τ (r i k ) are rays emanating from 0 in c j ℓ and pointing in the same direction, we have τ • f A (r i k ) = f A • τ (r i k ) and the result follows.

Permissible triples.
Here we introduce terminology and present results regarding particular subsets of the canonical surface O = O(d 1 , . . . , d κ ). While the material of this subsection becomes technical, the underlying notions are rooted in elementary Euclidean geometry. As we shall see in §4.4 and §4.5, the results proven here will be crucial for our proof of Theorem 1.2. We begin with definitions and notation.
Note that H(p) is convex in the sense that the length-minimizing path between any two points in H(p) is also contained in H(p). Unless otherwise noted, a (closed or open) θ-sector of O i is a (closed or open) sector of infinite radius, centered at 0 ∈ O i , and of angle θ. For the following definition and subsequent discussion, see Figure 2.
Definition 3.3. Let p, q ∈ O i \{0} be two points with distinct arguments in the same open π-sector of O i . The triangle determined by p and q, denoted △(p, q), is the union of straight line segments from 0 ∈ O i to p, from p to q, and from q to 0. Let c(p, q) := ∂H(p) ∩ ∂H(q) denote the circumcenter determined by p and q, denote the circumcircle determined by p and q, and denote the ball determined by p and q.
Locally, and away from a non-removable singularity, the geometry on O i is Euclidean, so the circumcenter, circumcircle and ball of Definition 3.3 may be viewed as isometric copies of their Euclidean namesakes in the plane. To be more precise, Lemma 4 of [ESS] implies that for any two points p 1 , p 2 ∈ O i belonging to the same closed π-sector, the distance d(p 1 , p 2 ) between p 1 and p 2 in O i equals the distance between their respective images π(p 1 ) and π(p 2 ) in the plane. Hence on any such sector, π(·) is an isometry. Using Euclidean geometry, we find that π(c(p, q)) is the center of the Euclidean circle π(C(p, q)) of radius |c(p, q)|, and π(B(p, q)) is the open ball whose boundary is π(C(p, q)). Moreover, π(C(p, q)) is the circumcircle of the Euclidean triangle π(△(p, q)) with vertices 0, π(p), π(q), and 0, π(p), π(q) ∈ π(C(p, q)) implies 0, p, q ∈ C(p, q) = ∂B(p, q).
Remark. We use the same notation and terminology from Definition 3.3 for the analogous objects determined by two points in the Euclidean plane C.
and for any subset P ⊂ P(O) of oppositely projected pairs, define the forgotten version of P by Recall that π(·) is a degree 2g − 2 + κ branched covering with a single branch point 0 ∈ C. Thus for any p ∈ O with π(p) = 0, there are precisely 2g − 2 + κ points p − ∈ O in the fiber above −π(p). That is, The subsets P ⊂ P(O) of interest to us do not have this multiplicity of elements: Definition 3.5. Call a nonempty subset P ⊂ P(O) of oppositely projected pairs distinctive if for any p ∈ P F , or, equivalently, for any p ∈ P F , there is a unique point p − ∈ P F satisfying π(p − ) = −π(p). Now let (p, q) ∈ O 2 be an ordered pair of two regular points with different arguments which belong to the same open π-sector of the same component of O (see the left-hand side of Figure 3), and consider the difference π(q) − π(p) of their images under π(·) in C. As above, there are exactly 2g − 2 + κ points u ∈ O satisfying π(u) = π(q) − π(p). Each of the 2g − 2 + κ oppositely projected pairs {p, p − } ∈ P(O) naturally announces a unique such point u, namely the u lying nearest to p − . A distinctive set P determines a unique point p − and hence a unique point u. We fix special notation for this point u in the following: Remark. The reader should note that the point [p, q] crucially depends on the distinctive set P , as distinctiveness uniquely determines the point p − in Definition 3.6. However, as P shall be clear by context it is absent from the notation [p, q].
Notice from Definition 3.6 that [p, q] is defined if and only if [q, p] is defined. The following technical result is needed for Proposition 3.6 below and is due to the fact that for distinctive sets, the points p and q naturally determine three isometric triangles in O. See again Figure 3. Figure 4. A permissible triple (p, q, u) with permissible scalars (r, s, t). Points of rP F , sQ F and tU F are in cyan, yellow, and red, respectively. Condition (i) of Definition 3.7 is met, [p, q]] and [[p, q], p − ] are defined and equal q and q − , respectively.
Proof. Definition 3.6 gives that Since p and q belong to the same π-sector with arg(p) = arg(q), we have π Condition ( The following definition and subsequent results will prove to be essential in §4.4 and §4.5 below. Definition 3.7. Let P, Q and U be distinctive, limit-point free subsets of P(O) and ({p, [sq, rp] are defined and equal tu and tu − , respectively, and (ii) Figure 4.) For a collection P = {P i } i∈I of distinctive, limit-point free subsets of P(O), we let P(P) denote the set of all permissible triples arising from P; that is, One immediate, but important, observation regarding permissible triples and permissible scalars is the following: Proposition 3.5. Let (p, q, u) be a permissible triple with permissible scalars (r, s, t) ∈ R 3 + . The scalars s and t are uniquely determined by p, q, u ∈ O and r ∈ R + . In particular, the set of all permissible scalars for (p, q, u) is precisely the ray {(ar, as, at) | a ∈ R + }.
Proof. By Definitions 3.6 and 3.7, we have Rearranging, we find that s and t are solutions to The fact that rp and sq belong to the same open π-sector with different arguments implies that π(p) and π(q) are R-linearly independent. This, together with Equation (1), implies that also π(u) and π(q) are R-linearly independent. Hence the matrix on the left side of the previous equation is invertible, so s and t are uniquely determined. For the second statement, let P 1 , P 2 ⊂ P(O) be distinctive and a ∈ R + . From Definition 3.6, one finds is defined then [ap 1 , ap 2 ] is defined and equals a[p 1 , p 2 ]. It follows then from Definition 3.7 that if (r, s, t) are permissible scalars for (p, q, u), then so are (ar, as, at) for any a ∈ R + . On the other hand, if (r 1 , s 1 , t 1 ) and (r 2 , s 2 , t 2 ) are both permissible scalars for (p, q, u), then so are (1, s 1 /r 1 , t 1 /r 1 ) and (1, s 2 /r 2 , t 2 /r 2 ). The first statement of this proposition implies that (r 1 , s 1 , t 1 ) and (r 2 , s 2 , t 2 ) belong to a common ray.
Note that the order of entries of a permissible triple (p, q, u) is important in Definition 3.7; for instance, the first two entries p and q must necessarily belong to the same open π-sector of the same component of O for condition (i) to hold. Nevertheless, this ordering does admit some flexibility.
Proposition 3.6. With notation as in Definition 3.7, the following statements are equivalent: (a) (p, q, u) is a permissible triple with permissible scalars (r, s, t), Proof. We need only show (a) implies (b): the same argument will give (b) implies (c) and (c) implies (a). Let (p, q, u) be a permissible triple with permissible scalars (r, s, t). We claim that (q − , u − , p) is a permissible triple with permissible scalars (s, t, r).
One might suspect from Definition 3.7 that it is difficult for some (p, q, u) ∈ P F × Q F × U F to be a permissible triple. Indeed, this suspicion is confirmed by the following: be a finite collection of distinctive, limit-point free subsets of P(O). Then the set P(P) of all permissible triples arising from P is finite.
Proof. It suffices to show that for any distinctive, limit-point free P, Q, U ⊂ P(O), the set of permissible Since P is distinctive, these sets have the same cardinality and hence are both infinite. As the set of possible directions of points in each O i is compact and there are only finitely many components O i of O, we may also assume that each p k (resp. p − k ) belongs to some small sector of a fixed component of O and that lim k arg(π(p k )) = θ P (resp. lim k arg(π(p − k )) = θ − P ) exists. Rotating each of P, Q and U by −θ P , assume for simplicity that θ P = 0 (and hence θ − P = π). Rescaling each (r k , s k , t k ) as necessary (see Proposition 3.5), we further assume that |r k p k | = 1 for all k. Under these assumptions, we find that {r k π(p k )} k∈N ⊂ S 1 with r k π(p k ) → 1 ∈ C. Also note that since P is limit-point free, |p k | → ∞ and hence r k = 1/|p k | → 0. Figure 5. Illustration of the proof of Lemma 3.7. The vertical dashed line represents x = 1/2, and the shaded region is the subset S k of the ball B k = B(r k π(p k ), s k π(q k )).
Condition (ii) of Definition 3.7 (together with the fact that π(·) is an isometry on π-sectors) guarantees that in the plane, we have for each k ∈ N both For each k, let c k := c(r k π(p k ), s k π(q k )) and B k := B(r k π(p k ), s k π(q k )) be the circumcenter and ball determined by r k π(p k ) and s k π(q k ). The circumcenter c k is the intersection of the perpendicular bisectors of the straight line segments from the origin to r k π(p k ) and from the origin to s k π(q k ). Since arg(π(p k )) → 0, for large k the former perpendicular bisector does not intersect the negative real axis, so for all large k we have arg(c k ) ∈ [0, π) ∪ (π, 2π). Passing to a subsequence, assume without loss of generality that arg(c k ) ∈ [0, π) for all k.
We consider two cases (see Figure 5): (i) Suppose there is a subsequence for which π(p k ) y ≥ 0 for all k. Since arg(π(p k )) → 0, we may pass to a subsequence to assume arg(π(p k+1 )) ≤ arg(π(p k )) for all k. For large enough k, the x-coordinate of r k π(p k ) ∈ S 1 is greater than 1/2. Since both 0 and r k π(p k ) belong to C k := ∂B k , we find that the vertical line x = 1/2 intersects C k at two distinct points; let b + k denote the point of intersection on the upper-semicircle of C k . Note that b y := inf k {(b + k ) y } > 0; otherwise arg(c k ) ∈ (π, 2π) contrary to our assumption. Set b := (1/2, b y ) and note that for large enough k, we have arg(π(p k )) < arg(b). Let S k = {p ∈ C | |p| < 1/2, arg(p) ∈ (arg(π(p k )), arg(b))}.
Note that S k is contained in the triangle with vertices 0, r k π(p k ) and b + k , which is contained in B k ; hence S k ⊂ B k for each k. Choose some N ∈ N large enough that arg(π(p N )) ∈ [0, arg(b)). For each k > N , we have arg(π(p k )) ≤ arg(π(p N )) by assumption, so arg(π(p N )) ∈ [arg(π(p k )), arg(b)). Taking k large enough, we also have |r k π(p N )| = |π(p N )|/|π(p k )| < 1/2. Hence r k π(p N ) ∈ S k ⊂ B k = B(r k π(p k ), s k π(q k )), contradicting Equation 2. (ii) Suppose there is no subsequence for which π(p k ) y ≥ 0 for all k. Then there is some subsequence for which π(p k ) y < 0-and hence −π(p k ) y > 0-for all k. Notice that c(t k π(u k ), −r k π(p k )) = c k − r k π(p k ), so the argument of this circumcenter belongs to (0, π). An analogous proof (reflecting about the x-axis) to that of case (i) implies that for some fixed N and large enough k, −r k π(p N ) ∈ B(t k π(u k ), −r k π(p k )), which contradicts Equation 3.
We conclude that the set of permissible triples must be finite.
3.3. Marked segments and Voronoi staples. In this subsection we fix a stratum H(d 1 , . . . , d κ ), its corresponding canonical surface O = O(d 1 , . . . , d κ ) and a translation surface (X, ω) ∈ H(d 1 , . . . , d κ ). Label the singularities of (X, ω) as σ 1 , . . . , σ κ so that σ i has cone angle 2π(d i + 1). In a sufficiently small neighborhood of σ i define generalized polar coordinates in a fashion analogous to that on O i (see §3.1 and the proof of Lemma 5 of [ESS]). Let s be a separatrix emanating from σ i of length |s| > 0 and corresponding angle θ ∈ [0, 2π(d i + 1)), and denote byŝ the point of O i with polar coordinates (|s|, θ). If |s| = 0, setŝ := 0 ∈ O i . If s is in fact a saddle connection, we callŝ the marked segment determined by s; note that in this case, hol(s) = π(ŝ) ∈ C. Denote by s ′ the identical, but oppositely-oriented saddle connection to s. Let M(X, ω) be the set of all pairs {s, s ′ } of oppositely-oriented saddle connections of (X, ω) and  Proof. Suppose {ŝ,ŝ ′ } ∈ M(X, ω). Since the saddle connections s and s ′ are identical but oppositelyoriented, it is clear that π(ŝ ′ ) = −π(ŝ), so {ŝ,ŝ ′ } ∈ P(O) and M(X, ω) ⊂ P(O). The image of ( M(X, ω)) F under π(·) equals the image of M F (X, ω) under hol(·). Since the set of holonomy vectors has no limit points ( §2.1.2) and π(·) is a homeomorphism on sufficiently small neighborhoods of regular points, we have that ( M(X, ω)) F has no limit points and hence M(X, ω) is limit-point free.
Now supposeŝ andt are marked segments with identical arguments in the same component O i ⊂ O. Then the underlying saddle connections s and t both emanate in the same direction from σ i ∈ Σ. If their lengths differ, say |ŝ| < |t|, then |s| < |t|. This implies that t has a singularity in its interior, which is impossible. If |ŝ| = |t|, then in fact s = t, and so {ŝ,ŝ ′ } = {t,t ′ }. It follows that M(X, ω) is distinctive.
Let M F (X, ω) := ( M(X, ω)) F denote the set of all marked segments determined by saddle connections on (X, ω). The star domain for σ i ∈ Σ is star i (X, ω) := {ŝ | s is a separatrix on (X, ω) emanating from σ i } ⊂ O i .
Note that star i (X, ω) consists of the union of closed rays emanating from 0 ∈ O i which stop only when meeting a marked segment (and thus almost every such ray is infinite). The star domain for (X, ω) is Define a map η : star(X, ω) → (X, ω), where for each point p ∈ star i (X, ω), if s is the separatrix for which p =ŝ, then η(p) is the endpoint of s on (X, ω). In other words, if p ∈ star(X, ω) has polar coordinates (|p|, θ), then η(p) is the endpoint of the separatrix of length |p| emanating from σ i ∈ Σ with angle θ.
For each x in the Voronoi 2-cell C i := C σi , let s x be the unique length-minimizing separatrix from σ i to x. Note that η is injective-and thus invertible-on the set if η(ŝ x ) = η(ŝ y ) for x, y ∈ C i , then x = y and s x = s y by the aforementioned uniqueness of these separatrices. Henceŝ x =ŝ y . Let ι i : C i → {ŝ x | x ∈ C i } denote the corresponding inverse, namely x →ŝ x for each x ∈ C i , and define ι : κ i=1 C i → O by setting ι| Ci = ι i for each i; see Figure 6. The translation surface (X, ω) is isometric to the quotient space of κ i=1 C i under the equivalence relation defined by identifying shared edges of Voronoi 2-cells. Proposition 7 of [ESS] shows that in a similar fashion, (X, ω) may be recovered from the closure of the image under ι of its Voronoi 2-cells by identifying appropriate edges of the various ι(C i ). We provide a brief overview of the method by which these edge identifications Figure 6. Left: Voronoi decomposition of a translation surface subordinate to two removable singularities, σ 1 (white) and σ 2 (black). The 2-cell C 1 := C σ1 (resp. C 2 := C σ2 ) is the open region in white (resp. gray). Middle: The image ι(C 1 ) in O 1 , along with three marked segments. Right: The image ι(C 2 ) in O 2 , along with three marked segments.
are made. Recall that the half-space H(p) determined by p ∈ O i is convex, and thus so is any intersection of such half-spaces (see Definition 3.2). When S = M F (X, ω) is the set of all marked segments of (X, ω), we use the suppressed notation Ω i := Ω i ( M F (X, ω)) and E i := E i ( M F (X, ω)). Call distinct points p, q ∈ S adjacent within S if p, q = 0 belong to the same component O i , and either of the two open sectors centered at 0 between p and q contains no points of S.
Proposition 14 of [ESS] shows that the convex body Ω i is precisely the set ι(C i ). Furthermore, Proposition 14 and Definition 15 of [ESS] show that there is a subset S(X, ω) of M(X, ω) for which the union of essential points E := ⊔ κ i=1 E i equals S F (X, ω) := ( S(X, ω)) F , i.e. the essential points come equipped with a natural pairing. Elements {ŝ,ŝ ′ } of S(X, ω) are called marked Voronoi staples and their underlying pairs of saddle connections {s, s ′ } in M(X, ω) are called Voronoi staples. It follows from the definition of E i that each edge on the boundary of Ω i = ι(C i ) belongs to the boundary of a half-space H(ŝ) for someŝ ∈ E i ⊂ S F (X, ω); conversely, the boundary of the half-space determined by eachŝ ∈ S F (X, ω) contains an edge on the boundary of some Ω i = ι(C i ). Propositions 7 and 14 of [ESS] show that the edges of the convex bodies of ⊔ κ i=1 Ω i corresponding toŝ and its orientation-pairedŝ ′ are equal length, and that (X, ω) is isometric to the quotient space of ⊔ κ i=1 Ω i under the equivalence relation given by identifying these edges via translation. Of import for our purposes is the observation that a translation surface is uniquely determined by its marked Voronoi staples S(X, ω), and that S F (X, ω) is precisely the union of essential points E = ⊔ κ i=1 E i . Remark. Under this identification of edges, the vertices of the various Ω i are regarded as regular points on the resulting translation surface (they correspond to the Voronoi 0-cells of (X, ω)). The origins 0 ∈ Ω i ⊂ O i become singularities of the resulting translation surface; we require this also if O i = (C, dz), in which case the singularity is a marked point.
Example 3.9. In Figure 6, the marked Voronoi staples are {â,â ′ }, {b,b ′ } and {ĉ,ĉ ′ }. The hexagonal translation surface on the left is recovered by identifying the edges of the convex bodies corresponding to these respective pairs. Figure 7. The set of eighth roots of unity contains Θ Γ with Γ = S, T 2 and S and T defined as in Example 4.1.
We conclude this subsection a brief observation: Proposition 3.10. Any two adjacent elementsŝ andt of S F (X, ω) belong to the same open π-sector.
Proof. If not, then ∂H(ŝ) ∩ ∂H(t) = ∅, and the corresponding convex body has infinite area. This would imply that the closed translation surface (X, ω) has infinite area, which is a contradiction.

Fanning groups, simulations and finiteness of lattices in strata
4.1. Fanning groups and lattices. In this subsection we consider a set of directions Θ Γ ⊂ S 1 announced by a discrete subset Γ ⊂ SL 2 R and give an equivalent definition of lattice groups in terms of this set of directions (Lemma 4.3). Abusing notation, we write Γ for both a discrete subset of SL 2 R whose elements act as linear transformations on R 2 and for its image as a discrete subset of PSL 2 R whose elements act as Möbius transformations on the closure of the upper-half plane H. Similarly, we denote elements of Γ in both of these settings by the same notation A ∈ Γ. When there is risk of confusion, we shall explicitly mention the setting in which to consider Γ.
Definition 4.1. Let Γ be a discrete subset of SL 2 R and ∆ the open unit disk. Set where A · ∆ denotes the usual action of a matrix on R 2 as a linear transformation. A group Γ for which Θ Γ is finite is called a fanning group. We call (X, ω) a fanning surface if its Veech group SL(X, ω) is fanning.
As we shall see in Corollary 4.6 below, Θ Γ contains the directions of certain short marked segments of any translation surface with Veech group Γ. These directions will be instrumental in our construction of such surfaces.  Figure 7 shows that Θ Γ is contained in the set of eighth roots of unity and thus Γ a fanning group. One verifies that in fact Θ Γ equals this set.
We introduce notation to help us relate lattices to fanning groups. For A ∈ PSL 2 R, set where d H denotes the hyperbolic metric on H, and for a discrete subset Γ ⊂ PSL 2 R, define D(Γ) := A∈Γ H(A) (compare with Definitions 3.2 and 3.8). Recall that if Γ is a Fuchsian group which trivially stabilizes i ∈ Hor, equivalently, Γ ∩ PSO 2 R = {Id}-then D(Γ) is the Dirichlet region centered at i for Γ. In this case D(Γ) is a convex fundamental polygon for Γ; see, say, [Be] and [Ka]. For any subset S ⊂ H, let S and ∂S denote the closure and boundary, respectively, of S inĈ = C ∪ {∞}. Define Proposition 4.2. For any discrete Γ ⊂ SL 2 R,

Proof. We first show that for any
If A ∈ SO 2 R, then both sets are all of ∂H, so suppose A / ∈ SO 2 R. Assume that A satisfies Ai = i + 2t 0 for some t 0 ∈ R + . Then H(A) consists of all points τ ∈ H with Re(τ ) ≤ t 0 , and and, moreover, 1 = det(A) = ad − bc = c 2 + d 2 . Notice that (x, y) T ∈ S 1 \A · ∆ if and only if (x, y) T ∈ S 1 and A −1 (x, y) T / ∈ ∆. The latter requirement may be rewritten as |A −1 (x, y) T | ≥ |(x, y) T |. We claim that (1, 0) T and 1 √ t 2 0 +1 (t 0 , 1) T satisfy this with equality.
We compute so the claim holds. Note that A · ∆ is the open region whose boundary is the ellipse A · S 1 centered at the origin, so S 1 ∩ A · ∆ consists of all points in S 1 with slope strictly between 0 and 1/t 0 (see, for instance, Figure 7, setting A = T 2 and t 0 = 1). It follows that Now consider general A ∈ Γ\SO 2 R, and fix B ∈ SO 2 R such that BAi = i + 2t 0 for some t 0 ∈ R + . From the argument above, we have S(BA) = H(BA) ∩ ∂H.
We claim that S(BA) = BS(A) and H(BA) = BH(A). By definition, t ∈ S(BA) if and only if there is some (x, y) T ∈ S 1 \BA · ∆ with x/y = t. The latter inclusion is equivalent to B −1 (x, y) T ∈ B −1 (S 1 \BA · ∆) = S 1 \A · ∆ since B −1 ∈ SO 2 R. Writing B −1 = ( e f g h ), we find B −1 (x, y) T = (ex + f y, gx + hy) T . But B −1 t = B −1 (x/y) = (ex + f y)/(gx + hy) is the reciprocal of the slope of B −1 (x, y) T ∈ S 1 \A · ∆, so t ∈ S(BA) if and only if B −1 t ∈ S(A), as desired. For the second claim, note that τ ∈ H(BA) if and only if d H (i, τ ) ≤ d H (BAi, τ ). Since B −1 is an isometry fixing i, this inequality is equivalent to Next we show that The forward inclusion holds as the closure of an intersection of sets is always contained in the intersection of the closures of the sets. For the reverse inclusion, suppose τ ∈ A∈Γ H(A)\{i}, and let γ be a geodesic segment in H without endpoints, for which γ has endpoints i and τ . Since τ ∈ H(A) for each A ∈ Γ and each H(A) is convex with i in its interior, we have γ ⊂ A∈Γ H(A) = D(Γ). Hence τ ∈ γ ⊂ D(Γ), so the claim holds. It follows that With the aid of Proposition 4.2, lattices may be characterized as the finitely generated discrete groups Γ for which the set of directions Θ Γ is finite. Furthermore, for a lattice Γ, the set of directions Θ Γ may be computed in finite time: Lemma 4.3. A discrete subgroup Γ ⊂ SL 2 R is a lattice if and only if it is a finitely generated fanning group. In this case there is some finite subset Γ n ⊂ Γ for which Θ Γ = Θ Γn .
Proof. Let Λ denote the image in PSL 2 R of Γ ∩ SO 2 R. As Λ contains only elliptic elements, it is a finite cyclic group of order, say, n (Corollary 2.4.2 of [Ka]). Let A 0 ∈ Λ be a generator of this group. Recall that if n = 1, then D(Γ) is the Dirichlet region centered at i for Γ and is thus a convex fundamental polygon. If n > 1, then D(Γ) is no longer a fundamental domain: one finds that A 0 D(Γ) = D(Γ), and the interior of D(Γ) contains n points from each orbit Γτ, τ ∈ H. Let γ 0 denote a geodesic segment beginning at i and ending on ∂D(Γ). For each 0 ≤ j < n, the geodesic segment γ j := A j 0 γ 0 also begins and ends at i and ∂D(Γ). Let F 0 ⊂ D(Γ) denote the convex polygon bounded by γ 0 , γ 1 and ∂D(Γ). A slight generalization of the proof of Theorem 3.2.2 of [Ka] shows that F 0 is a convex fundamental polygon for Γ. In particular, where µ H denotes hyperbolic area. Hence Γ is a lattice if and only if µ H (D(Γ)) is finite.
Suppose Γ is a lattice. Then Γ is geometrically finite and hence finitely generated. If Γ is not a fanning group, then Θ Γ -and, consequently, S Γ -is infinite. By Proposition 4.2, D(Γ) ∩ ∂H is infinite, and by Gauss-Bonnet, D(Γ) has infinite hyperbolic area. This is a contradiction, so Γ is a finitely generated fanning group. Now assume Γ is a finitely generated fanning group. Again by Proposition 4.2, this implies that D(Γ)∩∂H is finite, so F 0 ⊂ D(Γ) has no free sides and at most finitely many cusps. Since Γ is finitely generated, it is geometrically finite and thus every convex fundamental polygon for Γ has finitely many sides (Theorem 10.1.2 of [Be]). By Gauss-Bonnet, Γ is a lattice, and the first statement is proven.
For the second statement, suppose Γ is a lattice. From the previous paragraph, we know that F 0 has finitely many sides, and thus the same is true of D(Γ). But then there is some finite subset Γ n ⊂ Γ for which D(Γ n ) = D(Γ), and by Proposition 4.2, S Γ = S Γn , which implies Θ Γ = Θ Γn .

The group Aff +
O (X, ω) and its action on marked segments. Let (X, ω) ∈ H(d 1 , . . . , d κ ) and O = O(d 1 , . . . , d κ ). A central result of [Ed] (Theorem 18) and [ESS] (Proposition 17) is that a matrix A ∈ SL 2 R belongs to SL(X, ω) if and only if there is some f ∈ Aff + (O) with der(f ) = A satisfying f ( S(X, ω)) ⊂ M(X, ω). Theorem 17 of [Ed] shows that the same statement holds with the latter subset inclusion replaced by the equality f ( M(X, ω)) = M(X, ω). Note, in particular, that these statements require the affine automorphism f to respect the orientation-pairing of marked segments-it is not enough that the set of marked segments M F (X, ω) is invariant under f to conclude that der(f ) ∈ SL(X, ω) (see Example 21 of [ESS]). We introduce the following notation for the collection of such affine automorphisms of O: . From the comments above we see that the image of Aff + O (X, ω) under der(·) is precisely the Veech group Γ = SL(X, ω) (though der(·) restricted to Aff + O (X, ω) need not be injective, so Aff + O (X, ω) and SL(X, ω) are in general non-isomorphic). The group Aff + O (X, ω) acts on M(X, ω) via Recall from §3.3 that (X, ω) may be reconstructed from its marked Voronoi staples S(X, ω) ⊂ M(X, ω). The goal of Algorithm 1.1 is to construct-using only Γ = SL(X, ω) and d 1 ≤ · · · ≤ d κ -increasing subsets of Aff + O (X, ω)-orbits [{ŝ,ŝ ′ }] whose union eventually contains S(X, ω). To this end, the two main obstacles we face are (i) determining Aff + O (X, ω), and (ii) determining a representative {ŝ,ŝ ′ } of each of the Aff + O (X, ω)-orbits [{ŝ,ŝ ′ }] intersecting S(X, ω) using only the data Γ and d 1 ≤ · · · ≤ d κ . The first challenge is addressed by the following: Lemma 4.4. Let G be a set of generators of Γ ≤ SL 2 R. Then for any (X, ω) ∈ H(d 1 , . . . , d κ ) with SL(X, ω) = Γ, the group Aff + O (X, ω) is generated by a subset of In particular, if G is finite then Aff + O (X, ω) is generated by a subset of the finite set G O . Proof. For each A ∈ G, fix g A ∈ Aff + O (X, ω) with der(g A ) = A. Recall that g A may be written uniquely as , so the set of such g A is contained in G O . Let Λ be the subgroup of Aff + O (X, ω) generated by these g A . Since Trans(O) is closed under composition, it suffices to show that Aff + O (X, ω) = ∪ τ Λ, where the union is over some collection of τ ∈ Trans(O). Now let f ∈ Aff + O (X, ω) be arbitrary and A := der(f ) ∈ SL(X, ω). The matrix A may be written as a product A = A δ1 1 · · · A δm m , δ j ∈ {±1}, of elements A j in the generating set G and their inverses. Set g := g δ1 A1 . . . g δm Am ∈ Λ, with each g Aj a chosen generator of Λ as above. Then der(f • g −1 ) = Id, so τ := f • g −1 ∈ Trans(O) and f ∈ τ Λ as desired.
The final statement of the Lemma follows immediately from finiteness of both Trans(O) (Lemma 3.1) and G.
For challenge (ii) mentioned above, we wish to determine the generalized polar coordinates of representatives of Aff + O (X, ω)-orbits. In the proof of the following Lemma-which is also crucial for Algorithm 1.1-it is shown that the directions of the shortest representatives {ŝ,ŝ ′ } of each orbit [{ŝ,ŝ ′ }] are announced by the set Θ SL(X,ω) . The more delicate procedure of determining the lengths of such representatives is addressed in §4.5.
For each v ∈ S 1 , let r v denote the infinite open ray emanating from 0 ∈ C in the direction of v. Note that the preimage π −1 (r v ) ⊂ O consists of κ i=1 (d i + 1) = 2g − 2 + κ infinite open rays emanating from the origins of the various components of O, and each of these rays contains at most one marked segment of (X, ω) (Proposition 3.8). From the claim above, the minimal-length representatives of each [{ŝ,ŝ ′ }] ∈ M(X, ω)/Aff + O (X, ω) belong to one of the 2g − 2 + κ open rays of π −1 (r v ) for some v ∈ Θ SL(X,ω) . Hence where | · | denotes cardinality. Since (X, ω) is a fanning surface, the right-hand side is finite.

Simulating normalized Aff +
O (X, ω)-orbits. The set of orientation-paired marked segments M(X, ω) (and the set of Aff + O (X, ω)-orbits into which it partitions) depends intrinsically on (X, ω) and its geometry. The results of the previous subsection suggest that in the case of a fanning surface, much of this information is encoded in the Veech group SL(X, ω). This subsection further explores these ideas by introducing simulations of Aff + O (X, ω)-orbits which are constructed via generators of a fanning group. We again fix a stratum H(d 1 , . . . , d κ ).
Definition 4.3. Let Γ ≤ SL 2 R be a fanning group generated by G ⊂ Γ and set 1 ) and arguments arg(p 0 ) = 0 and arg(p − 0 ) = 5π. The map ρ 2 1 • f S acts as a counterclockwise rotation of O by an angle of π/2 + 4π = 9π/2, and f T 2 is a bijective horizontal shear on each c 1 j . A subset of the simulation sim(s 0 ) is shown in cyan in Figure 8. In red and yellow are subsets of sim(s 1 ) and sim(s 2 ), respectively, for s 1 := (H, {p 1 , p − 1 }) ∈ S G and s 2 := (H, {p 2 , p − 2 }) ∈ S G , where arg(p 1 ) = 4π, arg(p − 1 ) = π, arg(p 2 ) = π/4 and arg(p − 2 ) = 13π/4. Observe that Definition 4.3 and Lemmas 4.3 and 4.4 immediately imply: Lemma 4.8. If Γ is a lattice, then there exists a set of generators G of Γ for which the set of simulations Sims G is finite.
Remark. Recall from Lemma 4.3 that for a lattice Γ, the set of directions Θ Γ may be computed in finite time. Indeed, letting Γ n be the subset of Γ consisting of all words of length no greater than n in the generators G and their inverses, there is some n ∈ N for which Θ Γn = Θ Γ . In general, Θ Γn ⊃ Θ Γ , and it may be difficult to determine when equality holds. Thus in practice, the set Θ Γ in the definition of S G (Definition 4.3) is replaced by some finite Θ Γn containing Θ Γ . Such a replacement does not affect the statements of the results in the remainder of the paper, though it could increase the size of the-still finite-set of simulations Sims G . Theorem 4.9. If (X, ω) is a fanning surface, then M(X, ω) is a finite union of scaled simulations. In particular, if Γ ≤ SL 2 R is a fanning group generated by G ⊂ Γ, then Proof. We will prove the second statement; the first follows from this together with Lemma 4.5. Let (X, ω) be a translation surface with SL(X, ω) = Γ, and let G be a (finite or infinite) set of generators for Γ. By Lemma 4.4, there is some subset Set p := n(ŝ) and p − := n(ŝ) ′ . By Corollary 4.6, π(p) = π(ŝ)/|π(ŝ)| and π(p − ) = π(ŝ ′ )/|π(ŝ ′ )| belong to the finite set Θ Γ , so s : Theorem 4.9 implies that for any fanning group Γ, the orientation-paired marked segments M(X, ω) of any (X, ω) with Veech group Γ are given by a finite collection of simulations, each of which is appropriately scaled. Recall that the marked Voronoi staples S(X, ω)-from which (X, ω) may be reconstructed-are determined by M(X, ω) ( §3.3). In §4.5 , we show that for any finite collection of simulations, there are at most finitely many scalars for which the union of scaled simulations agrees with the marked segments of a unit-area translation surface. Together with Lemma 4.8, this will prove Theorem 1.2.

4.4.
Marked Voronoi staples determine permissible triples. Let (X, ω) ∈ H(d 1 , . . . , d κ ) and O = O(d 1 , . . . , d κ ). In this subsection we apply the results of §3.2 to orientation-paired marked segments and their Aff + O (X, ω)-orbits. In particular, we shall see that adjacent elements of S F (X, ω) naturally determine permissible triples (Proposition 4.11). To this end, we begin with the following: Lemma 4.10. For any two adjacentŝ,t ∈ S F (X, ω), we have Proof. Let i be the index for whichŝ,t ∈ O i , and recall from Definition 3.3 that the circumcenter c := c(ŝ,t) determined byŝ,t ∈ O i is the intersection of ∂H(ŝ) and ∂H(t). Sinceŝ,t ∈ S F (X, ω) are essential points of Ω i , there are two (adjacent) edges of Ω i contained in ∂H(ŝ) and ∂H(t), respectively. The intersection of these edges is precisely the circumcenter c, so c ∈ Ω i . Suppose for the sake of contradiction that there is some marked segmentû ∈ B(ŝ,t). Then d(û, c) < |c| = d(0, c), and hence c / ∈ H(û). This contradicts the fact that c ∈ Ω i , so we have This result-together with the convexity of B(ŝ,t) and the fact that 0 ∈ ∂B(ŝ,t)-gives that B(ŝ,t) ⊂ star i (X, ω), so η is defined on all of B(ŝ,t). Let γ be the straight line segment in O i fromŝ tot, and let σ j , σ k ∈ Σ be the singularities at which the oriented saddle connections s and t end, respectively. Then, recycling notation, u := η(γ) is a saddle connection from σ j to σ k , andû ∈ M F (X, ω). We claim that u = [ŝ,t].
Recall that permissible triples are defined in terms of three distinctive, limit-point free subsets of P(O) (Definition 3.7). We introduce notation for a set of-a priori non-permissible-triples determined by adjacent s,t ∈ S F (X, ω) and subsequently prove that these are in fact permissible triples belonging to the set P(Sim G ), where SL(X, ω) = G .  Proposition 4.11. For any fanning group Γ ≤ SL 2 R generated by G ⊂ Γ and any (X, ω) ∈ H(d 1 , . . . , d κ ) with Veech group SL(X, ω) = Γ, we have P(X, ω) ⊂ P(Sims G ). Moreover, if G is finite (i.e. Γ is a lattice), then there exists some N ∈ N such that for any (X, ω) ∈ H(d 1 , . . . , d κ ) with Veech group SL(X, ω) = Γ, we have P(X, ω) ⊂ P(Sims G ) ⊂ P(Sims N G ). Proof. Let (n(ŝ), n(t), n([ŝ,t])) ∈ P(X, ω) for some (X, ω) with Veech group Γ. By Theorem 4.9, there are sŝ, st, s [ŝ,t] and we must show that (n(ŝ), n(t), n([ŝ,t])) is a permissible triple. The corresponding permissible scalars we shall use are the minimal lengths (rŝ, rt, r [ŝ,t]  . We now verify the two criteria of Definition 3.7 on the triple (n(ŝ), n(t), n([ŝ,t])) and the scalars (rŝ, rt, r [ŝ,t] ). (ii) We must show that  Now suppose for the sake of contradiction that and let j be the index for whichŝ ′ ∈ O j . Since M F (X, ω) has no limit points, there are at most finitely many marked segments belonging to this intersection, and by Proposition 3.8, none of these marked segments have the same argument asŝ ′ . Choose someû in the intersection with argument nearest that ofŝ ′ . Then the straight-line segment fromŝ ′ toû belongs to star j (X, ω), and a similar argument to that in the proof of Lemma 4.10 gives that [ŝ ′ ,û] ∈ M(X, ω). By definition of [ŝ ′ ,û], this marked segment belongs to the same open π-sector asŝ and hence to B(ŝ,t), contradicting Lemma 4.10 (see Figure 10). A similar argument gives Thus (n(ŝ), n(t), n([ŝ,t])) is a permissible triple, and P(X, ω) ⊂ P(Sims G ). Now suppose that G is finite. Then the set Sims G of all simulations is finite, and by Lemma 3.7, the set P(Sims G ) is finite. Hence for each (p, q, u) ∈ P(Sims G ) there is some n for which (p, q, u) ∈ P(Sims n G ). Taking N to be the maximum of these, we have P(X, ω) ⊂ P(Sims G ) ⊂ P(Sims N G ).

4.5.
Finiteness of lattice surfaces with given Veech groups. Throughout this subsection, let G be a set of generators of a fanning group Γ. As noted at the end of §4.3, to prove Theorem 1.2 it suffices to show that for any finite collection of simulations determined by G, there are at most finitely many scalars for which the union of scaled simulations coincides with the marked segments of a unit-area translation surface. Propositions 3.5, 3.6 and 4.11 suggest that for adjacent elementsŝ,t ∈ S F (X, ω) of a translation surface (X, ω) with Veech group Γ, the desired scalars for the simulations corresponding toŝ,t and [ŝ,t] are uniquely determined by one another. Hence if the elements of S F (X, ω) are all-loosely speaking-'transitively adjacent' to one another (or their orientation-paired inverses), then an arbitrary choice of one scalar uniquely determines all other scalars. We make these vague notions precise by introducing a permissible triples graph corresponding to a set P of permissible triples whose vertices are the distinctive, limit-point free sets corresponding to the triples and whose edges are defined between the vertices corresponding to entries of the same permissible triple in P . The 'transitive adjacency' alluded to above refers to the connectedness of such a graph.
Definition 4.6. Let Q = {Q i } i∈I be a collection of distinctive, limit-point free subsets of P(O). For a subset P ⊂ P(Q) of permissible triples, define the permissible triples graph determined by P to be the graph G P = (V P , E P ) with vertex set consisting of the sets in Q corresponding to the permissible triples in P , and edge set where an edge connects vertices Q ℓ and Q m if there is some permissible triple in P to which both Q ℓ and Q m correspond. We call (r i ) i∈I ∈ R |I| + consistent scalars for the graph G P if (r i , r j , r k ) are permissible scalars  Figure 11. Subsets of the scaled simulations sim(s 0 ), sim(s 1 ) and √ 2sim(s 2 ), with s 0 , s 1 s 2 ∈ S G as defined in Example 4.7, and balls showing that (p, q, u) ∈ P(Sims G ) is a permissible triple with permissible scalars (1, 1, √ 2).
We call the graph G P consistent if it admits consistent scalars.
Proposition 4.13. Let (X, ω) be a fanning surface with Veech group SL(X, ω) = Γ, and let P := P(X, ω) ⊂ P(Sims G ). Then the permissible triples graph G P is connected and consistent, and if (r s ) sim(s)∈VP are consistent scalars for G P , then any other consistent scalars are of the form (ar s ) sim(s)∈VP with a ∈ R + .
Proof. Theorem 4.9 implies that the vertex set V P is finite, say V P = {sim(s i )} n i=1 . The same theorem also gives that n i=1 r i sim(s i ) ⊂ M(X, ω) for some (r i ) n i=1 ∈ R n + . In fact, each r i is the minimal length of a pair in the Aff + O (X, ω)-orbit for which r i sim(s i ) = [{ŝ,ŝ ′ }]. The proof of Proposition 4.11 shows that the scalars (r i ) n i=1 ∈ R n + are consistent, so the graph G P is consistent. Now suppose that (r ′ i ) n i=1 ∈ R n + are also consistent scalars for G P , and recall the definition of the edge set E P . Induction with Propositions 3.5 and 3.6 implies that any of the scalars r ′ i corresponding to a simulation of a connected component of G P uniquely determines the remaining scalars r ′ j of the simulations of the component. In particular, if r ′ i = ar i for some a ∈ R + , then r ′ j = ar j for each of these scalars. Hence it suffices to show that G P is connected.
Consider the subgraph (V, E) of (V P , E P ) with vertex set V consisting of all simulations in V P which contain a pair of normalized marked Voronoi staples and edge set E consisting of edges {sim(s i ), sim(s j )}, where sim(s i ) and sim(s j ) contain {n(ŝ), n(ŝ) ′ } and {n(t), n(t) ′ }, respectively, for adjacentŝ,t ∈ S F (X, ω). Recall that (X, ω) is isometric to the quotient space of κ i=1 Ω i under the equivalence relation given by identifying the two edges of convex bodies which correspond to a pair of marked Voronoi staples ( §3.3 above, and also §3 of [ESS]). If (V, E) were disconnected, then after this identification of edges of convex bodies the resulting surface-and hence (X, ω)-would be disconnected. This is a contradiction, so the subgraph (V, E) is connected. By definition of P = P(X, ω) and V P , any vertex sim(s i ) ∈ V P \V must contain {n([ŝ,t]), n([ŝ,t]) ′ } for some adjacentŝ,t ∈ S F (X, ω). By definition of E P , there are edges of G P from sim(s i ) to the simulations containing {n(ŝ), n(ŝ) ′ } and {n(t), n(t) ′ }, both of which belong to V ⊂ V P . Thus G P is connected, and the result follows.
We are now ready to prove Theorem 1.2: Proof of Theorem 1.2. Let Γ ≤ SL 2 R be a lattice. By Lemma 4.3, Γ is a finitely generated fanning group. Let G be a finite set of generators of Γ and fix H 1 (d 1 , . . . , d κ ). We must show that the set of all (X, ω) ∈ H 1 (d 1 , . . . , d κ ) for which SL(X, ω) = Γ is finite. By Proposition 4.11, each of these translation surfaces belongs to the set {(X, ω) ∈ H 1 (d 1 , . . . , d κ ) | P(X, ω) ⊂ P(Sims G )}, so it suffices to show that this latter set is finite. By Lemmas 3.7 and 4.8, the set P(Sims G ) of all permissible triples is finite, so it suffices to show for any P ⊂ P(Sims G ) that the set be the vertex set of the permissible triples graph G P . As in the proof of Proposition 4.13, for each (X, ω) with P(X, ω) = P , there are consistent scalars ( By definition of G P and its vertex set V P , we have that the set of marked Voronoi staples, S(X, ω)-from which (X, ω) may be recovered-is contained in this union of scaled simulations. Again from Proposition 4.13, if (Y, η) ∈ H 1 (d 1 , . . . , d κ ) also satisfies P(Y, η) = P , then its corresponding consistent scalars are of the form (ar i ) n i=1 for some a ∈ R n + . From §3.3, we have Since (X, ω) and (Y, η) are isometric to the quotient spaces of these respective sets under an equivalence relation identifying edges, and since both translation surfaces have area one, we conclude that a = 1 and thus (X, ω) = (Y, η).

The algorithm
The constructive results of the previous sections suggest an algorithm (Algorithm 1.1) which returns all translation surfaces with a given lattice Veech group Γ in any given stratum H 1 (d 1 , . . . , d κ ). We highlight the main steps of this algorithm here, providing necessary references and further commentary in italics. Recall that G is a given finite set of generators of the lattice Γ ≤ SL 2 R.
(1) Compute a finite set Θ Γm ⊂ S 1 containing Θ Γ , and let (See Definition 4.3 and the remark following Lemma 4.8.) (2) For each n ∈ N: (a) Compute the set Sims n G = {sim n (s) | s ∈ S G } of all stage n distinctive simulations determined by G. (See Defintion 4.3.) (b) Compute the set P(Sims n G ) of all permissible triples arising from Sims n G along with their respective rays of permissible scalars. (See Definition 3.7 and Proposition 3.5. Recall from Definition 4.5 and Proposition 4.11 that there is some N ∈ N so that for any (Y, η) ∈ H 1 (d 1 , . . . , d κ ) with SL(Y, η) = Γ, the set P(Y, η) determined by the marked Voronoi staples S(Y, η) is contained in the finite set of permissible triples P(Sims N G ).) (c) For each P ⊂ P(Sims n G )\{∅}: (For n = N and any (Y, η) as above, this loop will set P = P(Y, η) for some P .)  Figure 12. The convex body and resulting translation surface (X, ω) (with opposite sides identified) using the scaled simulations sim(s 0 ), sim(s 1 ) and √ 2sim(s 2 ) from Examples 4.7 and 4.12. The arcs about the singularity of O and of (X, ω) indicate arguments of generalized polar coordinates.
(i) Construct the permissible triples graph G P = (V P , E P ). If G P is connected and consistent with consistent scalars (r s ) sim n (s)∈V n P , then set S := sim n (s)∈VP r s sim n (s), construct the convex bodies Ω i (S F ) and determine the corresponding essential points E i (S F ) for each i ∈ {1, . . . , κ}.
(See Definition 4.6 and §3.3.) (ii) If each Ω i (S F ) is compact, the essential points ⊔ κ i=1 E i (S F ) come in pairs {p, p − } determined by the simulations sim n (s) ∈ V P , and the edges of the convex bodies determined by p and p − are equal length, then construct the translation surface (X, ω) by identifying these edges via translation. (See §3.3. When P = P(Y, η), the proof of Proposition 4.13 shows that for some a ∈ R + , the set aS contains the set of marked Voronoi staples S(Y, η), and thus the union of scaled essential points ⊔ κ i=1 aE i (S F ) is precisely S F (Y, η). Hence (X, ω) is a scaled version of (Y, η).) (iii) Rescale (X, ω) if necessary so that it has area one, and verify that SL(X, ω) = Γ.
(Computing SL(X, ω) is done using one of the algorithms mentioned in §1.) Example 5.1. Examples 4.7 and 4.12 illustrate (parts of ) Steps 1, 2.a, 2.b and 2.c.i of the algorithm. Figure  12 illustrates the construction of convex bodies and corresponding essential points in Step 2.c.i as well as the resulting translation surface (X, ω) constructed in Step 2.c.ii. Notice that the essential points come only from sim(s 0 ) and sim(s 1 ); however, sim(s 2 ) was necessary to determine the permissible scalars (1, 1, √ 2) (Example 4.12). Rescaling to unit-area, one verifies that indeed SL(X, ω) = S, T 2 .
A number of improvements to Algorithm 1.1 can be made for computational efficiency. We mention three in particular: • It is optimal to choose a minimal set of generators G for Γ, as the size of S G -and hence of Sims Gdepends on |G|. • Recall from Proposition 3.8 that ifŝ,t ∈ M F (X, ω) are two marked segments with identical arguments in the same component O i ⊂ O, then in fact {ŝ,ŝ ′ } = {t,t ′ }. This requirement restricts the permissible triples graphs that one needs to consider in step 2.c. • In Step 2.b, one need only consider permissible triples (p, q, u) ∈ sim n F (r) × sim n F (s) × sim n F (t) where r, s and t are defined using the same subset H ⊂ G O . A similar restriction applies to the subsets P ⊂ P(Sims n G ) considered in Step 2.c. While Algorithm 1.1 does produce the finite set of lattice surfaces asserted by Theorem 1.2 in finite time, it does not give a halting criterion to determine when the set of returned translation surfaces is exhaustive. An explicit test to determine the N ∈ N for which P(Sims G ) ⊂ P(Sims N G ) would give such a halting criterion (see Proposition 4.11). In particular, Algorithm 1.1 together with this halting criterion would give a general procedure to determine whether or not any given lattice group is realized as a Veech group in any given stratum. However, even without this criterion, the ideas of this paper may still be used in special cases to show that certain lattice groups are not realized in certain strata; see §6.
As the set of all strata is countable, Algorithm 1.1-if allowed to run indefinitely-is easily adapted to produce all unit-area translation surfaces with a given lattice Veech group in any stratum. A full implementation could also provide experimental evidence for lattices which are never realized as Veech groups, though-even with the aforementioned stratum-wise halting criterion-a criterion to definitively assert that a lattice is not a Veech group in any stratum seems to the author to be elusive.
We also mention that as the set of generators for any fanning group is at most countable, Algorithm 1.1 may be adapted to construct surfaces (X, ω) with given fanning Veech groups in given strata. However, for countably infinite generating sets G, the set S G is uncountable, so a construction of such surfaces may not be exhaustive.

The modular group in minimal strata
Here we give an example showing how the ideas of the previous sections may be used-in certain casesto give obstructions for lattices being realized as Veech groups in strata. 3 In particular, we show that the square torus is the only translation surface with Veech group SL 2 Z in the collection of all minimal strata H(2g − 2), g > 0. Throughout this section, fix g > 0 and (X, ω) ∈ H(2g − 2). Let σ := σ 1 be the sole singularity of (X, ω), Ω := Ω 1 the convex body subordinate to M F (X, ω) and O = O(2g − 2) the canonical surface associated to H(2g − 2). We begin with two lemmas.
Lemma 6.1. Let S(X, ω) = {{ŝ i ,ŝ ′ i }} n i=1 be the marked Voronoi staples of (X, ω). If the counterclockwise angle measured fromŝ i toŝ ′ i equals that measured fromŝ ′ i toŝ i for each i, then n = 2g for n even and n = 2g + 1 for n odd.
Proof. Since S F (X, ω) is the set of all essential points of Ω, the convex body Ω has 2n edges. Decompose Ω into 2n closed triangular regions with disjoint interiors, where each triangle has one vertex at the origin 0 ∈ O, and remaining two vertices at the endpoints of an edge of Ω. The sum of the interior angles of all triangles in this decomposition is 2nπ. The origin 0 ∈ O is a singularity of cone angle 2(2g − 1)π, so the sum of the interior angles of Ω is 2nπ − 2(2g − 1)π = 2(n − 2g + 1)π.
Recall that (X, ω) is isometric to the quotient space of Ω under the equivalence relation given by identifying edges determined by the marked Voronoi staples S(X, ω), and the vertices of Ω under this identification correspond to Voronoi 0-cells of (X, ω) ( §3.3). In particular, the sum of interior angles of Ω which are identified as a single point must equal 2π. Our assumptions on the counterclockwise angles betweenŝ i andŝ ′ i and betweenŝ ′ i andŝ i imply that for n even, the vertices of Ω become a single point under the identification, while for n odd, the vertices of Ω become two distinct points under the identification (the argument here is analogous to that which shows the translation surface constructed by identifying opposite edges of a regular 2m-gon has one singularity when m is even and two singularities when m is odd). In particular, for n even, 2(n − 2g + 1)π = 2π implies n = 2g, while for n odd, 2(n − 2g + 1)π = 4π implies n = 2g + 1.
Lemma 6.2. Suppose that the minimal length of a saddle connection on (X, ω) is one. If the set P := {p ∈ O | |p| = 1, arg(π(p)) ∈ {0, π/2, π, 3π/2}} belongs to the set of marked segments M F (X, ω), then in fact P = S F (X, ω). Proof. It suffices to show that P is the set of essential points of the convex body Ω, i.e. that (i) Ω = ∩ p∈P H(p) and (ii) for any q ∈ P , Ω ∩ p∈P \{q} H(p). Condition (ii) is immediate from the fact that Ω is compact while ∩ p∈P \{q} H(p) is non-compact for any q ∈ P . Now so it suffices to show for each marked segmentŝ ∈ M F (X, ω)\P , that ∩ p∈P H(p) ⊂ H(ŝ).
We claim that for each p ∈ P , the ball centered at p of radius one contains no marked segments other than p, i.e.
B 1 (p) ∩ M F (X, ω)\{p} = ∅. Suppose on the contrary that the intersection is non-empty. As in the proof of Proposition 4.11, we can choose someû in the intersection with argument nearest that of p and find that [p,û] is a marked segment of length strictly less than one. Its corresponding saddle connection also has length strictly less than one, contrary to our assumptions, so the claim holds.
Next, we claim that the ball centered at the origin of radius √ 2 is contained in the union of balls of radius one centered at the various p ∈ P , B √ 2 (0) ⊂ p∈P B 1 (p). This is clear from elementary Euclidean geometry, restricting to each 2π-sector c 1 j ; see Figure 13. In particular, the previous two claims imply that (4) B √ 2 (0) ∩ M F (X, ω)\P = ∅.
With these two lemmas, we now prove the main result of this section.