Benjamini-Schramm convergence of periodic orbits

We prove a criterion for Benjamini-Schramm convergence of periodic orbits of Lie groups. This general observation is then applied to homogeneous spaces and the space of translation surfaces.

Let r (R) denote 1/2 the supremum of all such r , then B H Rie (e, r (R)) ⊂ B H (e, R) and r (R) → ∞ as R → ∞; indeed it is not difficult to see that r (R) ≥ C log R where C > 0 depends on the embedding H ⊂ SL N (R).
Let ⊂ H be a discrete subgroup. The injectivity radius of y ∈ H / is define as the supremum over all r > 0 so that the map h → hy is injective on B H Rie (e, r ). Let n ⊂ H be a sequence of lattices in H . The sequence {H / n : n ∈ N} Benjamini-Schramm converges to H if for every r > 0 we have μ n {y ∈ H / n : injectivity radius of y < r } → 0 as n → ∞ where μ n denote the H -invariant probability measure on H / n for every n.
Throughout, we assume that H acts continuously on X preserving the measure μ; also assume that Stab H (x) is discrete for every x ∈ X .
An This and the assumption (2) in the proposition imply that If f N R 2 (y) > 0, then the injectivity radius of h is at least r (R); (1.2) This and the fact that N δ (K ε ) ⊂ {y ∈ H x n : f N R 2 (y) > 0} imply: there exists some n 0 so that In consequence, using (1.2) we deduce that for all n > n 0 . Since r (R) → ∞ as R → ∞, the claim follows.
In subsequent sections, we discuss two settings where Proposition 1.1 is applicable: the homogeneous setting is discussed in §2 and the space of Abelian differentials in §3; see in particular Theorems 2.2 and 3.1.

Homogeneous spaces
Let G be a connencted algebraic group defined over R, and let G = G(R) • be the connected component of the identity in the Lie group G(R).
Let ⊂ G be a lattice. Throughout this section, we assume that is torsion free. Let X = G/ , and let μ X denote the G-invariant probability measure on X . Proof We prove the result for the case real hyperbolic manifold, the complex case is similar.
Let G = SO(d, 1) • , = π 1 (M), and H = SO(d − 1, 1) • . Then V n lifts to a closed orbit H x n in X = G/ for every n.
Note that H ⊂ G is a maximal connected subgroup which is not a parabolic subgroup of G. Therefore, the assumptions in Theorem 2.1 are satisfied for G, H , and the orbits {H x n : n ∈ N}. The claim thus follows from Theorem 2.1.
We note that when is arithmetic Theorem 2.1 can be proved using the results in [1, §5]. This condition holds if is an irreducible lattice and the real rank of G is at least two by Margulis' arithmeticity theorem [9]. Moreover, it was proved by Corlette and Gromov-Shoen [4,8] that lattices in SP(n, 1) and F −20 4 are arithmetic. While non-arithmetic lattices in SO(n, 1), for all n, and SU(n, 1), for n = 2, 3, exist, recent developments, [2,3,10], show that the presence of infinitely many totally geodesic hyperplanes 1 in real and complex hyperbolic manifolds of finite volume imply arithmeticity of their fundamental group. Therefore, in all of the interesting cases, the assertion of Theorem 2.1 can be obtained by combining rather deep existing results in the literature. However, the proof we provide here is different and is arguably simpler. In particular, our proof does not rely on the arithmeticity of and further property of congruences lattices; instead, our proof relies only on a special case of an equidistribution theorem of Mozes and Shah [14]. Lemma 2.3 Let the notation and the assumptions be as in Theorem 2.1. Then for μ X -a.e.
x ∈ X we have Proof Let H denote the Zariski closure of H in G. Since H is a connected semisimple Lie group, it has finite index in the group H := H(R) ∩ G.
By Chevalley's theorem, there exists a finite dimensional (real) representation (ρ, W ) of G and a vector w ∈ W so that H = {g ∈ G : gw = w}. In particular, we conclude that (2.2) Let now x = g 0 . Then Stab(x) = g 0 g −1 0 , and H ∩ g 0 g −1 0 is nontrivial if and only if there exists some e = γ ∈ so that γ ∈ g −1 0 Hg 0 . Since H ⊂ H , we conclude that For every γ ∈ , the set F γ is an algebraic variety defined over R. Moreover, G = G(R) • is Zariski dense in G. These and the fact that is countable imply that unless there exists some e = δ ∈ so that δgw = gw for all g ∈ G, the lemma holds -indeed in that case G \ ∪ γ ∈ F γ is a conull subset of G, and for every g in this set we have Stab H (g ) = {e}.
Assume now to the contrary that G = {g ∈ G : δgw = gw} for some nontrivial δ ∈ . Then by (2.2) we have δ ∈ g H g −1 for all g ∈ G, hence, Since [H : H ] < ∞, there exists some n so that δ n ∈ g Hg −1 for all g ∈ G. That is, δ n ∈ ∩ g∈G g Hg −1 . However, is torsion free and ∩ g∈G g Hg −1 is a finite group. This contradiction completes the proof.
Proof of Theorem 2. 1 We may and will assume that G ⊂ SL N (R) for some N . As before, for all subgroups L ⊂ G and all R > 0, let denotes the maximum norm on SL N (R) with respect to the standard basis. Recall that μ X denotes the G-invariant probability measure on X . First note that by a theorem of Mozes and Shah [14] and our assumptions (1) and (2) in the theorem, we have Let dist denote the right invariant Riemannian metric on G induced using the killing form. Let R > 1, and put Stab( Altogether, we deduce that f R satisfies the conditions in Proposition 1.1. The theorem thus follows from Proposition 1.1 in view of (2.3).

The space of Abelian differentials
Let g ≥ 2, and let T g denote the Teichmüller space of complex structure on a compact Riemann surface of genus g. We denote by M g the corresponding moduli space, i.e., the quotient of T g by the mapping class group, Mod g . As it is well-known, Mod g is not torsion free, however, it has subgroups of finite index which are torsion free -indeed the kernel of the natural map from Mod g to Sp 2g (Z/3Z) is torsion free.
We fix, once and for all, a covering map which corresponds to a torsion free finite index subgroup of Mod g . Let f : H 2 → M g be an isometric immersion for the Teichmüller metric. Typically, f (H 2 ) is dense in M g , however, there are situations where f (H 2 ) is an algebraic curve in M g . In the latter case, the stabilizer of f is a lattice in Isom(H 2 ), and we obtain a Teichmüller curve For every g ≥ 2, the moduli space M g contains a dense family of Teichmüller curves which arise as branched cover of flat tori. There are also examples of infinite families of primitive Teichmüller curves, i.e., Teichmüller curves which do not arise as a branched cover of flat tori, in M g when g = 2, 3, 4, [12,13]. C. Leininger and A. Wright (independently) have supplied an alternative (and arguably softer) proof of Theorem 3.1. This argument relies on the fact that the length of shortest geodesic on Teichmüller curves tends to infinity, see Proposition 3.2, and is independent of measure classification theorems. We also thank T. Gelander for helpful communications regarding IRSs.
Here, we present a proof based on Proposition 1.1 and [6] to highlight a unifying theme between the homogeneous setting and the setting at hand.
For every M ∈ T g , let (M) be the g-dimensional space of holomorphic 1-forms on M. By integrating a non-zero form ω ∈ (M) we obtain, away from the zeros of ω, a flat metric |ω| on M and local charts whose transition functions are translations.
Form a vector bundle over the Teichmüller space T g where the fiber over each point is (M). Let T g → T g be the complement of the zero section of this vector bundle.
There is a natural action of GL + 2 (R) (and hence of SL 2 (R)) on T g : given a holomorphic We let M g → M g denote the quotient of T g by action of the mapping class group of S g .  H 1 (M, , C) as a local coordinate on the stratum H(α). We will call these coordinates period coordinates.
The area of a translation surface is given by We let 1 M g and H 1 (α) denote the locus of unit area 1-forms in M g and H(α), respecitively.

The SL 2 (R)-action and the Kontsevich-Zorich cocycle
The action in (3.1) descends to an action of SL 2 (R) on H 1 (α). Indeed, write (M, ω) as a 2 × d matrix x. The action of SL 2 (R) in these coordinates is linear. Let Mod(M, ) be the mapping class group of M fixing each zero of ω. We choose a fundamental domain for the action of Mod(M, ), and think of the dynamics on the fundamental domain. Then, the SL 2 (R) action becomes where A(h, x) ∈ Sp 2g (Z) Z m−1 is the Kontsevich-Zorich cocycle.

Thus, A(h, x)
is the change of basis one needs to perform to return the point hx to the fundamental domain. It can be interpreted as the monodromy of the Gauss-Manin connection (restricted to the orbit of SL 2 (R)).

Affine measures and manifolds
For a subset E ⊂ H 1 (α) we write

A suborbifold M for which there exists a measure ν such that the pair (M, ν) satisfies (i) and (ii) is said to be affine invariant submanifold.
We sometimes write ν M to indicate the affine invariant measure ν on affine invariant submanifold M.
Note that in particular, any affine invariant submanifold is a closed subset of H 1 (α) which is invariant under the action of SL 2 (R), and which in period coordinates is an affine subspace. We also consider the entire stratum H 1 (α) to be an (improper) affine invariant submanifold.

Typical affine stabilizer is trivial
In this section, we prove the following statement:

Proposition 3.2 Let (M, ν) ⊂ H 1 (α) ⊂ M g,n be an affine invariant submanifold. Assume that M is not a Teichmüller curve. Then for ν-a.e. x ∈ M,
Recall that the set of self-intersections M of M is a proper closed invariant submanifold of M, hence, dim M < dim M, see [6]; in particular, ν(M ) = 0. Therefore, it suffices to prove the proposition for ν-a.e. x ∈ M \ M . Let M denote the lift of M \ M to T g .
Fix φ ∈ Mod(S g ) (that is, φ is not torsion). Define We will show, for every φ ∈ Mod(S), P(φ) is a ν-measure zero subset of M. Note that, by assumption, dim( M) > 3.
Consider x ∈ P(φ) and let E x be the GL + (2, R) orbit or x. Then E x can be considered as (an open subset of) the tangent space of the Teichmüller disk H x associated to x (the projection of E x to Teichmüller space). The restriction of Teichmüller metric to H x equips H x with the hyperbolic metric (up to a factor 2). We observe that φ stabilizes H x acting on H x by an isometry. In fact, we have either (see, for example, [11,Lemma 5.6
• φ acts parabolically on H x and φ is a multi-curve.
• φ acts elliptically on H x and φ has finite order in Mod(S).
Note that the third case is excluded since we are assuming φ is not torsion. We argue each case separately showing that P(φ) ∩ M is a ν-measure zero subset of M. Remark 3. 3 We are in fact proving more that what is stated in Proposition 3.2. Recall that, for x ∈ M, the Veech group of x is the subgroup of PSL 2 (R) which stabilizes H x setwise. Hence, the proof actually gives that for ν-a.e. x ∈ M, the Veech group of x is finite.

is pseudo-Anosov element
A pseudo-Anosov map φ stabilizes only one Teichmüller disk, the one where ∂H x contains F + (φ) and F − (φ); the stable and the unstable foliation associated to φ. Therefore, P(φ) = T 1 H x , the unit tangent bundle over H x . Since M is a not a Teichmüller curve, it has a dimension larger than 3. Hence P(φ) ∩ M is a ν-measure zero subset of M.

is a multi-twist
Let φ be a multi-twist around γ , namely Let RP(φ) be the subset of H(α) obtained from points in P(φ) after scaling. Then, for any x ∈ RP(φ), a measured foliation that is topologically equivalent to γ = {γ 1 , . . . , γ k ) has to appear in the boundary of H x . That is, after a rotation, we can assume x = (F − , F + ) and F + = c k γ k . Furthermore, x has a cylinder decomposition where the modulus of these cylinders are rationally multiples of each other ([11, Lemma 5.6]). That is, there are r i ∈ Q such that We also have That is, given γ , r i , F − and area(x), we can calculate the values of c i . Hence, F + and subsequently x are uniquely determined by γ , r i , F − and area(x). There are countably many choices for the values r i and the multi-curve γ . We now show that the dimension of the space of possible measured foliations F − is half the dimension of R M where R M is the subset of H(α) obtained from point in M after scaling. For a filling bi-recurrent train-track τ (see [15] for definition and discussion) any admissible weight on τ defines a measured foliation. We then say this measured foliation is carried by τ . The complementary regions of a filling train tracks are n-gons or punctured n-gons. A foliation carried by τ has a singular point associated to each complementary region of τ . We say τ is of type α = (α 1 , . . . , α m ) if τ has m complementary components that are punctured α i -gons, i = 1 . . . m. We denote the space of admissible weights in τ by W (τ ).

Lemma 3.4
For every x ∈ H(α) there are train tracks τ + and τ − of type α such that a neighborhood of H(α) around x is homeomorphic to U × V where U, V are open subsets of W (τ + ) and W (τ − ) respectively. In fact, the real part of the period coordinates for H(α) give coordinates for U and the imaginary part of the period coordinates, give coordinates for V .
Proof Let be a triangulation of x by saddle connections (for example, L ∞ -Delanay triangulations see [7, §3]). Pick a subset B of the edges of that give a basis for the homology of x relative to the zeros of x. Then the complex numbers { ω x} ω∈B give local coordinates for H (α). For every edge ω of , we have In fact, F − can be constructed, triangle by triangle, from the set of real numbers {i(ω, F − )} ω∈ . That is there is a train-track τ − dual to the triangulation (again, see [7, §3] for the construction of such train-tracks) such that {R( ω x)} ω∈B form an admissible weights on τ − . At any point y ∈ H(α) near x, the triangulation can still be represented by saddle connections and the set {R( ω y)} ω∈B form an admissible weights on τ − that is associated to the vertical foliation at y. That is, {R( ω y)} ω∈B , thought of as admissible weights on τ − give local cooridinates for the set of measured foliation that appear as a horizontal foliation of an element of H 1 (α) near x. The same also holds for τ + and the vertical foliations.
Since R M is an affine sub-manifold of H(α), it is locally defined by a set of affine equations on period coordinates, see e.g. §3.1 and [5]. That is, there are subspaces U ⊂ U and V ⊂ V , defined by the same set of affine equations, such that a neighborhood of x in R M is naturally homeomorphic to U × V . In particular, where U and V have half the dimension of R M.
Let W be the intersection of RP(φ) with this neighborhood. Recall that, fixing the multicurve γ , rational numbers r i and the area, every point in W is determined, up to rotation, by a point in U . Therefore, W is a countable union of set of dimension dim(U ) + 2. But where the last inequality follows from the assumption that dim(R M) > 4. That is, RP(φ) ∩ R M is a countable union of lower dimensional subset of R M and therefore, hasν-measure zero, see §3.1 for the definition ofν. Since, Stab SL 2 (R) (x) does not change after scaling, we have, P(φ) ∩ M has ν-measure zero in M.

Proof of Theorem 3.1
In this section we prove Theorem 3.1. The proof is based on the following proposition. Altogether, we deduce that f R satisfies the conditions in Proposition 1.1. This and (3.3) imply that Proposition 1.1 applies and yields: SL 2 (R)/Stab SL 2 (R) (x k ) Benjamini-Schramm converges to SL 2 (R).
The proof is complete.
Proof of Theorem 3.1 Let {V k : k ∈ N} ⊂ M g,n be a sequence of Teichmüller curves. We will show that for every subsequence {V k i }, there exists a further subsequence {V k i j } which Benjamini-Schramm converges to H the theorem follows from this.
Let {V k i } be a subsequence of {V k }. Passing to a further subsequence, which we continue to denote by {V k i }, we may assume that the corresponding SL 2 (R) orbits {E k i } lie in H 1 (α) ⊂ M g,n for a fixed α. Now by [6,Thm. 2.3], see also [6,Cor. 2.5], there exists a subsequence {E k i j } of {E k i }, and an affine invariant manifold (M, ν), so that μ k i j → ν where μ k i j denotes the SL 2 (R)invariant measure on E k i j . By Proposition 3.5, we have V k i j Benjamini-Schramm converges to H; as we wished to show.