Mixing Properties of Flows on Geometrically Finite Hyperbolic Manifolds

By Dale A. Winter

M.S., Brown University B.A., M.A., and M.A.St., University of Cambridge

A dissertation submitted in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy in the Mathematics Department at Brown University

PROVIDENCE, RHODE ISLAND

May 2015 c Copyright 2015 by Dale A. Winter This dissertation by Dale A. Winter is accepted in its present form by the Department of Mathematics as satisfying the dissertation requirement for the degree of Doctor of Philosophy.

Date Hee Oh, Advisor

Recommended to the Graduate Council

Date Jeﬀrey F. Brock, Reader

Date Richard E. Schwartz, Reader

Approved by the Graduate Council

Date Peter M. Weber, Dean of the Graduate School

iii Vita

Dale Winter was born on January 30th 1985 as the third and ﬁnal son of Alan and Marjorie Winter. He attended The Perse School before arriving at St. John’s College, Cambridge, where he gained his B.A. in mathematics in 2007. Over the subsequent three years he spent time as a Baden-W¨urttemberg Scholar at the University of Heidelberg, a Kennedy Scholar at the Massachusetts Institute of Technology, and acquired M.A. and M.A.St. degrees in mathematics from Cambridge. He came to Brown in 2010 to study for his Ph.D. under the direction of Professor Hee Oh.

iv Acknowledgements

First and foremost I would like to thank my advisor, Hee Oh, for her deep insight, support, and enthusiasm. She introduced me to a remarkably beautiful area of mathematics four years ago, and has guided me through many challenges since. The bulk of this thesis describes collaborative work by the two of us; without her vision in suggesting the problem, and her patience and thoroughness in seeing it through, this thesis would not have been possible.

Many other mathematicians have provided inspiration, knowledge, or advice. I would, however, like to single out a few. I’d like to thank the other members of my thesis committee, Rich Schwartz and Jeﬀ Brock, for their time, useful comments, attention and interest. Ralf Spatzier provided very helpful insight at a critical stage in an early project. Michael Magee provided useful comments on earlier versions of these arguments. Ilya Gekhtman, Wenyu Pan, and other dynamicists at Yale have gently and repeatedly disabused me of my greater misunderstandings and pushed me in more fruitful directions.

I beneﬁted greatly from the warmth and good humor of my colleagues in the mathematics department at Brown: some helped me through classes I might not otherwise have passed; others answered research questions; others deftly solved administrative problems that had seemed utterly insurmountable to me; many, many more made the department a happy and productive place to be.

I am grateful to the many institutions who have hosted me at one time or anther during my graduate studies: key amongst these have been Yale, who took me in for a year after my advisor moved departments; the Mathematical Sciences Research Institute in Berkeley, who are hosting me for parts of the current semester; and the Korea Institute for Advanced Studies, where much of the work described herein was done. My studies would have been

v vi much less rich and much less productive without their hospitality. There are also many outside of mathematics, whose contributions to this thesis are no less profound for being indirect. Once again the full list is too long for inclusion, and once again there are a few names that I really cannot omit. The burrowfolk, who became home for me when I came to America: Alan M., Kirby, Lusann, Tom. My family, who made me and have supported me in wisdom and in folly: Alan W., Marjorie, Roger, William. And Emily, whose contributions I cannot begin to describe. Contents

1 Introduction 1

1.1 Summary ...... 1

1.2 Outline ...... 2

2 Background 4

2.1 Mixing, counting, and equidistribution for lattices ...... 4

2.2 Selberg’s theorem, uniform spectral gap, and sieve...... 7

2.3 Thin groups and appropriate measures ...... 10

3 Statements of results and areas of application 16

3.1 Selberg’s theorem for thin subgroups of SL2pZq ...... 16 3.2 Mixing properties of frame ﬂow on geometrically ﬁnite manifolds ...... 21

4 Selberg’s theorem for thin subgroups 23

4.1 Congruence transfer operators ...... 23

4.2 Dolgopyat operators and vector valued functions ...... 34

4.3 The expansion machinery ...... 48

4.4 Uniform mixing of the BMS measure and the Haar measure ...... 64

4.5 Zero-free regions for the Selberg zeta functions ...... 75

5 Mixing properties of frame ﬂow 76

5.1 Further properties of rank one groups ...... 76

5.2 Algebraic properties of the Bruhat decomposition...... 77

5.3 Invariant functions for the horospherical groups ...... 92

vii viii

5.4 Triviality of Σ˜ and mixing of the frame ﬂow ...... 94 Chapter 1

Introduction

1.1 Summary

3 The main aim of this thesis is to discuss generalizations of Selberg’s 16 theorem to hyperbolic surfaces of inﬁnite area. These Selberg type theorems are deeply related to orbit counting problems, sieving questions for almost prime values of polynomial functions, and to the best known progress towards Zaremba’s conjecture.

The classical version of Selberg’s celebrated theorem asserts uniform spectral gap for an important family of finite area hyperbolic surfaces. If we write Γpqq ă SL2pZq for the 2 principal congruence subgroups and set Xpqq “ ΓpqqzH , then the two least eigenvalues 3 λ0pqq ď λ1pqq of the hyperbolic Laplacian on Xpqq satisfy λ0pqq “ 0 and λ1pqq ě 16 . ˜ In the present work we consider the situation where Γ ă SL2pZq is finitely generated but 2 infinite index, and Γ˜pqq :“ Γpqq X Γ.˜ In this case the base X˜ :“ Γ˜zH and its congruence 2 covers X˜pqq :“ Γ˜pqqzH are hyperbolic surfaces of infinite area. We recover both a uniform exponential mixing result for geodesic flows on the unit tangent bundles T1X˜pqq and a uniform spectral gap statement, which we phrase in terms of resonance free regions for the

resolvents of the hyperbolic Laplacians on X˜pqq. The uniform exponential mixing theorem may be regarded as a dynamical reformulation of Selberg’s theorem on the inﬁnite area

surfaces X˜pqq.

Our approach builds on breakthrough work of Bourgain-Gamburd-Sarnak, who studied

these questions using expander graph properties of Γ˜pqqzΓ.˜ Their results are highly satis-

1 2

1 1 factory in the case that the critical exponent δΓ˜ ą 2 . In the case that δΓ˜ P p0, 2 s, however, we are able to strengthen their results by combining their approach with the Dolgopyat operator technique from hyperbolic dynamics. Our main results on uniform exponential mixing and uniform spectral gap are summarized as Theorems 3.1.2, 3.1.4. and 3.1.6.

We also include some comments on mixing properties of frame flow for infinite area hyperbolic manifolds. These are intended to clarify the applicability of the Brin-Pesin argument in the infinite volume setting. We will see that the frame flow on “geometrically finite” hyperbolic manifolds is mixing for its natural measure. This allows refinements to work of Roblin on measure classification for horospherical group actions and on decay estimates for certain group representations. The principal outcomes of our discussion of mixing of frame flow are summarized as Theorems 3.2.1, 3.2.2, and 3.2.3.

The mathematics and much of the text of this thesis have already appeared in the preprints [41, 59]. The former was the result of joint work with Hee Oh and the latter of an independent project. Recent joint work with Magee-Oh [32] may also be of interest; this provides a readable account of the ideas of [41] in a diﬀerent setting and with diﬀerent goals in mind.

1.2 Outline

1.2.1. Background. We begin in Chapter 2 by brieﬂy recalling some tools of homogenous dynamics. The ﬁrst element will be the mixing, equidistribution, and counting techniques initiated by Margulis [33] and later developed by many authors [18, 19, 50]. We recall these in a simple setting, both as a reminder of the links between dynamical and geometric questions, and as a model for some later arguments (the energetic reader might usefully compare Theorems 2.1.3, 2.1.5, and 2.1.6 with Theorems 3.1.4, 4.4.18 and 4.5.1, for instance).

Most of our arguments will be concerned with more reﬁned statements than mixing. The topic of Section 2.2 will therefore be uniform exponential mixing results or, equivalently,

3 uniform spectral gap. The seminal result in this direction is Selberg’s 16 Theorem which we recall in both its spectral and dynamical formulations; see Theorems 2.2.1 and 2.2.2. We also recall links between these uniform spectral gap statements and sieve questions; see 3

Fact 2.2.3 and Theorem 3.1.7. With the classical theory on finite area hyperbolic manifolds and orbit counting questions in place, we next turn to infinite area manifolds in Section 2.3. The first tasks are to recall the notion of a rank one locally symmetric space, and to construct the measures appropriate to the study of flows in infinite volume. Key notions at this stage are of limit set, critical exponent, Patterson-Sullivan density, and generalized Bowen-Margulis-Sullivan measure.

1.2.2. Results. In Chapter 3 we state the principal results of the present work together with some current and potential applications. The main results are Theorems 3.2.1, 3.2.2, 3.2.3, 3.1.2, 3.1.4, and 3.1.6. We include one sample consequence of mixing of frame ﬂow in inﬁnite volume as Theorem 3.2.4; this is a result by Margulis-Mohammadi-Oh on equidistribution of holonomies. We also mention groundbreaking work of Bourgain and Kontorovich on Zaremba’s conjecture, which provides one of the most beautiful applications of Selberg type theorems for thin groups (or rather semigroups); it is hoped that the ideas in this thesis, as reformulated in [32], will lead to improved error bounds in Theorem 3.1.9.

1.2.3. Proofs of the main results. We finish in Chapters 4 and 5, which contain the proofs of the main results. Chapter 4 concerns generalizations of Selberg’s theorem to infinite area hyperbolic surfaces. The proof proceeds via symbolic dynamics using arguments of Bourgain-Gamburd-Sarnak and Dolgopyat. Chapter 5 describes the arguments for mixing of frame flow in infinite area manifolds; this uses Brin-Pesin moves and the observation that the limit set of a Zariski dense group cannot be contained in smooth submanifolds of the boundary. Chapter 2

Background

This chapter relates necessary background and notation. Our ﬁrst focus will be on relations between mixing, equidistribution and counting questions for ﬁnite area hyperbolic manifolds. These ideas underlie many of the applications of our main results.

3 Section 2.2 will address classical versions of Selberg’s 16 theorem in both spectral and dynamical formulations. We will also mention connections to sieving for almost prime values of polynomial functions on a group orbit, and recall some of the tools used to prove exponential mixing. Section 2.3 will provide the needed background on inﬁnite volume manifolds. Mostly this will involve a very brief introduction to Patterson-Sullivan theory and to generalized Bowen-Margulis-Sullivan measures.

2.1 Mixing, counting, and equidistribution for lattices

2.1.1. Setup. This thesis will be heavily concerned with mixing and equidistribution phe- nomena, as well as their connections with orbit counting and related questions. The fundamental relations among these concepts are due to Margulis [33] and have since been reﬁned by many authors [18, 19]. We begin by sketching some of this theory in a simple setting.

2 2 2 Let Qpx0 . . . xnq “ x0 ` ... ` xn´1 ´ xn be the standard quadratic form of signature n 1 pn, 1q on R ` . Real hyperbolic n-space can be realized as the subset

n n`1 H :“ tx P R : Qpxq “ ´1 and xn ą 0u.

4 5

n The metric on H induced by Q has constant sectional curvature ´1 and has been heavily studied.

We will denote by SOpn, 1q the group of determinant one pn ` 1q ˆ pn ` 1q real matrices preserving form Q, and write SOpn, 1q` for the identity component. Note that SOpn, 1q` n acts transitively and isometrically on H . The stabilizer of the standard base point o “ n p0,..., 0, 1q P H is then the subgroup B 0 K “ : B P SOpnq . $¨ ˛ , & 0 1 . n ˝ ‚ We recover H “ SOpn, 1q`{K. Another% important subgroup- of SOpn, 1q` is a choice of one-parameter diagonalizable subgroup; we will work with

Idn´1 0 0 $ ¨ ˛ , A “ at :“ 0 cosh t sinh t : t P R . (2.1.1) ’ / ’ ˚ ‹ / & ˚ ‹ . ˚ 0 sinh t cosh t ‹ ’ ˚ ‹ / Let Γ SO n, 1 denote’ a discrete˝ subgroup. Recall‚ that SO/ n, 1 is a unimodular ă p q` %’ -/p q group, so has an essentially unique bi-invariant Haar measure; uniqueness in this case is up to multiplication by constants.

Deﬁnition We say that Γ is a lattice in SOpn, 1q if the Haar measure of the quotient

Γz SOpn, 1q` is ﬁnite. We say that Γ ă SOpn, 1q` is thin if it is Zariski dense but not a lattice.

One basic dichotomy of this ﬁeld is between lattices and thin subgroups; lattices tend to be well understood while thin subgroups mostly are not. A central challenge of this thesis, as well as much other work [2, 50, 39], is to generalize techniques that are well developed for lattices and make them work for thin subgroups as well. For now we concentrate on the simpler lattice theory.

8 Notation 2.1.2. For a manifold X we write C0pXq or C0 pXq for the spaces of continuous or smooth functions on X with compact support.

One natural question in this setting is the orbit counting question for Γ, that is, understanding asymptotics (for T large) of the quantity

NT :“ #tγ P Γ: dHn po, γoq ă T u. 6

There is a well developed machine for estimating such quantities, which is succinctly described in the paper of Eskin-McMullen [19]. The details of that argument are not essential for us at this stage, but we will recall the outline, which serves as a useful model for arguments later in the thesis.

2.1.2. Mixing. The ﬁrst step in the process is to prove a mixing theorem. In the lattice case the mixing statement is an easy consequence of Howe-Moore’s theorem on the decay

2 of matrix coeﬃcients applied to L pΓz SOpn, 1q`q.

Theorem 2.1.3 ([25]). Let Γ ă SOpn, 1q` be a lattice. The A action on Γz SOpn, 1q` is mixing; for any functions φ1, φ2 P C0pΓz SOpn, 1q`q we have

φ1pgatqφ2pgqdg Ñ φ1pgqdg φ2pgqdg żΓz SOpn,1q` ˜żΓz SOpn,1q` ¸ ˜żΓz SOpn,1q` ¸ as t Ñ 8; all integrals are taken with respect to the probability Haar measure on Γz SOpn, 1q`.

Notation 2.1.4. We will refer to results of this form, where the integral of a twisted product converges to the product of the integrals, as a mixing result. This is a fundamental notion from ergodic theory.

The orbit counting statement currently under discussion can be deduced directly from here as described in [39]. We shall instead pursue a more general strategy via equidistribution.

2.1.3. Equidistribution. The principal insights of the equidistribution strategy are due to Duke-Rudnik-Sarnak and were greatly simpliﬁed by Eskin-McMullen. One aims to prove a theorem of the following form.

Theorem 2.1.5 ([18, 19]). Let Γ ă SOpn, 1q` be a lattice. Then right translates of K by

A equidistribute in Γz SOpn, 1q`. More precisely, let φ P C0pΓz SOpn, 1q`q. Then

φpkatqdk Ñ φpgqdg as t Ñ 8 żK żΓz SOpn,1q` where dk represents the probability Haar measure on K and dg represents the probability

Haar measure on Γz SOpn, 1q`. 7

To pass from a mixing theorem to an equidistribution theorem one must run a thickening argument: ﬁrst consider an -neighbourhood K of K; next apply a mixing theorem (like

Theorem 2.1.3) to the indicator function of Kat; ﬁnally let go to zero and hope to recover Theorem 2.1.5. The behavior as goes to zero is usually understood by means of a “wavefront Lemma”. The whole procedure is described cleanly in [19], and actually works in much greater generality; one can replace K by any “symmetric subgroup” of G.

2.1.4. Orbit counting. The passage from an equidistribution result to an orbit counting statement is completed by a smoothing argument (see [19] again). For the case under discussion one recovers the following asymptotics.

Theorem 2.1.6. Let Γ ă SOpn, 1q` be a torsion free lattice. Then

volpBT poqq #tγ P Γ: dHn po, γoq ă T u “ p1 ` op1qq, vol pΓzHnq where all volumes are taken with respect to the hyperbolic metric.

Our observation at this stage is that even fairly simple dynamical data (mixing of a group action) can still provide interesting geometrical output. For subtler applications, however, particularly to number theory, we require much more precise dynamical information. This often takes the form of uniform spectral gap or, equivalently, uniform exponential mixing. These ideas, in their classical form, will be our next topic.

2.2 Selberg’s theorem, uniform spectral gap, and sieve.

Very often, crude dynamical information is insuﬃcient for a desired application. In such cases one tries to bring more reﬁned estimates to bear, the quintessential example of which

3 2 is Selberg’s celebrated 16 theorem. We now think of H “ H as the hyperbolic upper half plane, with isometry group PSL2pRq acting by M¨obiustransformations. The natural lattice at hand is Γ “ SL2pZq. We will be interested in the congruence subgroups; for each q P N deﬁne

Γpqq “ tγ P Γ: γ ” Id mod qu, which is a ﬁnite index normal subgroup of Γ. We can then form the congruence covers

3 Xpqq :“ ΓpqqzH of ΓzH. Selberg’s 16 theorem can be thought of either as giving uniform 8 spectral gap for the hyperbolic Laplacians on Xpqq, or as giving a uniform exponential mixing result for the associated geodesic ﬂows on the unit tangent bundles T1Xpqq. Denote by

0 “ λ0pqq ă λ1pqq ď λ2pqq ...

1 the eigenvalues of the hyperbolic Laplacians on Xpqq and by Gt ý T Xpqq the geodesic ﬂow.

Theorem 2.2.1 (Selberg). For all q P N we have

3 λ pqq ě . 1 16

3 1 Conjecturally the same statement holds with 16 replaced by 4 . Together with the theory of spherical functions this theorem admits the following dynamical reformulation.

8 1 Theorem 2.2.2. For any ą 0, q P N, and all smooth functions φ1, φ2 P C0 pT Xpqqq we have

T1Xpqq φ1pvqdv T1Xpqq φ1pvqdv 1 ´p 4 ´qt φ1pvqφ2pGtpvqqdv “ 2 ` Op||φq||S1 ||φ2||S1 e q, T1Xpqq AreapΓpqqzH q ż ş ş 2 where || ¨ ||S1 denotes the ﬁrst order L -Sobolev norm. The implied constant may be chosen to depend only on .

We refer to results of this form, a mixing statement with an exponential error term where the exponent can be taken uniformly over some class of examples, as a uniform exponential mixing statement. This is the model result that we would like to emulate in the case of inﬁnite area hyperbolic surfaces.

2.2.1. Uniform exponential mixing and sieve. Part of our interest in uniform exponential mixing results like Theorem 2.2.2 comes from their connections to sieve. We’ll start with a very simple sieving statement by way of example.

Deﬁnition For n, R P Z we say that n is R-almost prime if the prime factorization of n has at most R primes when counted with multiplicity. 9

˜ Fact 2.2.3. Let Γ ă SOZp2, 1q` be a lattice. The number of small Pythagorean triples px, y, zq P Γ˜ ¨ p3, 4, 5q with almost prime area satisﬁes

T #tpx, y, zq P Γ˜ ¨ p3, 4, 5q : x2 ` y2 ` z2 ď T and xy is 7-almost primeu ! plog T q4 and for any suﬃciently large R

T #tpx, y, zq P Γ˜ ¨ p3, 4, 5q : x2 ` y2 ` z2 ď T and xy is R-almost primeu " . plog T q4

The point is that one can solve orbit counting questions with some extra number theoretic conditions. The passage from uniform exponential mixing statements like Theorem 2.2.2 to sieve statements like Fact 2.2.3 will largely be regarded as a black box during this thesis; see [36] for a discussion of these ideas in a similar setting.

2.2.2. Techniques for proving exponential mixing. Roughly speaking there are two techniques used to prove exponential mixing results: the ﬁrst is representation theory/L2 spectral theory; the second is Dolgopyat’s argument.

The spectral/representation theory approach has been heavily used in homogeneous dynamics. One tries to understand which irreducible unitary representations of G appear as sub-representations of L2pΓzGq. If one can prove that all the irreducible representations that appear have some exponential decay of matrix coefficients then that statement can be translated into exponential mixing for the flows of interest. This technique works well in the case that Γ ă SOpn, 1q` is a lattice, and more generally when the critical exponent of Γ (see below) is sufficiently large. The approach is unworkable without making assumptions on the critical exponent, however.

Dolgopyat expounded a purely dynamical approach to exponential mixing in [17]. His approach is very powerful (for instance it proves exponential mixing of geodesic ﬂows on compact manifolds of variable negative curvature, a question that representation theory cannot address), but also rather technical. It is only fairly recently that this approach has been married with expander graph technology to give Selberg type results (see [6, 41, 32] and Chapter 4). 10

2.3 Thin groups and appropriate measures

As alluded above we have a very good understanding of mixing, equidistribution, and counting statements for the action of lattices on hyperbolic spaces. Our primary interest for the rest of the thesis will be in “thin” subgroups of SOpn, 1q or related groups, that is, in discrete subgroups that have infinite covolume for the Haar measure yet are Zariski dense. These groups arise naturally in a number of settings, perhaps most famously in the study of Apollonian circle packings. The immediate difficulty in the infinite volume case is that the natural flows are no longer even ergodic for the volume measure. This difficulty is overcome by changing the reference measures. The process, however, requires some work. Key ideas in this area are due to Patterson, Sullivan, Burger, Roblin and others [44, 45, 55, 56, 50].

2.3.1. Rank one locally symmetric spaces. We begin by ﬁxing some notation, which will be used without comment throughout the rest of the text. Let G be one of the following

´20 groups: PSL2pRq, SUpn, 1q, Sppn, 1q,F4 , or the identity component SOpn, 1q` with n ě 3. Let K ă G be a maximal compact subgroup. The quotient space X˜ :“ G{K carries a left G-invariant Riemannian metric d, which we normalize so that the maximum sectional curvature of pX,˜ dq is minus one. We denote by o “ rKs the natural base point of X˜, and 1 ˜ ﬁx a unit tangent vector v0 P ToX. The Lie group G also carries a left G- and right K-invariant metric d1. We write g for the real Lie algebra TeG, and note that g splits as the direct sum of k “ TeK and its orthocompliment p:

g “ k ‘ p.

Write π : G Ñ X˜ for the natural projection and observe that k is exactly the kernel of dπ : g Ñ TepX˜q. It follows that we can lift v0 to a unique unit vectorv ˆ0 P p. The one parameter subgroup

A “ tat :“ expptvˆ0q : t P Ru ă G

projects to a geodesic ato in pX,˜ dq. This subgroup plays the same role as the explicit choice

we made in equation (2.1.1). We will write M for the stabilizer stabK pv0q of v0 in K. The group M is well known. 11

Lemma 2.3.1. The centralizer ZK pAq is equal to M.

We denote by N ` (respectively N ´) the expanding (contracting) horospherical groups:

` N “ th P G : a´that Ñ e as t Ñ ´8u;

´ N “ tn P G : a´tnat Ñ e as t Ñ `8u.

We will occasionally write N “ N ` to save notation.

2.3.2. The boundary at inﬁnity of X˜. The geometric boundary BX˜ of X˜ may be identiﬁed as the set of unit speed geodesic rays xptq in X˜ under the equivalence relation x1ptq „ x2ptq if dpx1ptq, x2ptqq is bounded for t ě 0. We recall that the isometric action of G extends continuously to the boundary. For any tangent vector v P T1X˜ we write v` for the forward end point of the associated geodesic vptq in BX˜, and v´ for the backward

` ´ end point. Of particular importance are the forward and backward end points, v0 and v0 , of our ﬁxed tangent vector v0. It will also be convenient to talk about the forward and backward end points of group elements, by which we mean

` ` ´ ´ g :“ pgv0q and g :“ pgv0q .

` ` ´ ´ Note that g “ gpv0 q and that g “ gpv0 q.

2.3.3. Discrete subgroups of G We will be interested in discrete subgroups Γ ă G.

Deﬁnition We will write ΛpΓq for the limit set of Γ in BX˜, that is, the collection of accumulation points in BX˜ of the orbit Γo Ă X˜. We say that Γ is elementary if ΛpΓq is ﬁnite, and that Γ is non-elementary else.

We write

´sdpo,γoq δΓ :“ infts P R : e convergesu (2.3.1) Γ ÿ for the critical exponent of Γ. We will sometimes write δ instead of δΓ in places where it seems unlikely to cause confusion. In many interesting cases the critical exponent coincides with the Hausdorff dimension of the limit set ΛpΓq; this happens in particular whenever Γ is geometrically finite (see subsection 2.3.6 for definitions and [15] for the proof). Direct 12 geometric arguments imply that δΓ is bounded above by the volume entropy of X˜ (see subsection 2.3.5). The group Γ is said to be of divergence type if the sum in equation (2.3.1) diverges

at s “ δΓ, and of convergence type if that sum converges. Geometrically ﬁnite (or convex cocompact) groups are automatically of divergence type [15]. By and large groups of divergence type are the more tractable, at least to the Patterson-Sullivan theory approach we describe. The running assumption of this text is that

Γ ă G is discrete, torsion free, non-elementary, and of divergence type. (2.3.2)

2.3.4. Bowen-Margulis-Sullivan measures. Next we recall the construction of the Bowen- Margulis-Sullivan measures. This is essentially a synopsis of the necessary material from

Chapter I of [50]. We consider the unit tangent bundle G{M “ T1X˜. We will write

B2X˜ :“ pBX˜q2 ´ tdiagonalu.

We denote the Busemann function by

β : pBX˜q ˆ X˜ ˆ X˜ Ñ R

βξpx1, x2q “ lim rdpx1, ξptqq ´ dpx2, ξptqqs, tÑ`8

where ξptq is any geodesic whose forward end point is ξ PBX˜. Using the projection maps T1X˜ Ñ X˜ and G Ñ X˜ we may also evaluate the Busemann function on tangent vectors or on group elements. Our choice of base point o P X˜ gives an identiﬁcation

1 2 T X˜ ÑB X˜ ˆ R

` ´ v ÞÑ pv , v , βv´ pv, oqq.

A conformal density µ of dimension δµ on X˜ is a family of ﬁnite measures tµx : x P X˜u on BX˜ which are mutually absolutely continuous and satisfy

dµx 1 pξq “ eδµβξpx2,x1q dµx2 for all pairs xi P X˜. The conformal density is said to be invariant for a subgroup Γ ă G if

γ˚µx “ µγx 13 for all γ P Γ and all x P X˜.

Conformal densities allow us to construct measures on T1X˜. For Γ-invariant conformal

1 densities ν, µ of dimensions δν, δµ we can deﬁne a measure on T X˜ by

ν,µ δν β ´ po,vq`δµβ ` po,vq ´ ` dm˜ pvq “ e v v dνopv qdµopv qds.

Direct calculation shows that this measure is independent of the choice of base point and

ν,µ left Γ-invariant. It is invariant for geodesic ﬂow whenever δµ “ δν. We now liftm ˜ from T1X˜ to give an M-invariant measure on G, which we will also denote bym ˜ ν,µ. Sincem ˜ ν,µ is Γ-invariant, it then descends to give a measure mν,µ on ΓzG, which we will call the generalized Bowen-Margulis-Sullivan measure associated to the pair ν, µ.

Certain conformal densities lead to particularly interesting measures. One of these is the

Patterson-Sullivan density; if Γ ă G is of divergence type (for example, if Γ is geometrically finite), with critical exponent δ, then there is a unique Γ-invariant conformal density of dimension δ supported on ΛpΓq, called the Patterson-Sullivan density σ (see, for example, [50]). When we talk about the BMS measure without qualification we mean mBMS :“ mσ,σ; it is an MA-invariant measure on ΓzG, and our principal object of study. The support of the BMS measure will play a significant role in what follows, so observe that

supppmBMSq “ tg P G : g` P ΛpΓq and g´ P ΛpΓqu.

2.3.5. Burger-Roblin and Haar measures in the BMS framework. There is also a unique G-invariant conformal density whose dimension is given by the volume entropy

D “ DpX˜q (see the next paragraph); we will denote it by λ. We can express λo explicitly as the unique K-invariant probability measure on BX˜. This allows us to construct the

BR σ,λ BR λ,σ Burger-Roblin measure m :“ m (respectively m˚ :“ m ). Direct calculation shows BR BR ` ´ that m (resp. m˚ ) is invariant under the right translation action of N (resp. N ) on ΓzG. These are the most interesting examples of measures invariant under the expanding (respectively contracting) horospherical groups. We can also recover the Haar measure on

G as a generalized BMS measure: it is given by mHaar “ mλ,λ.

We now recall some well known properties of volume entropy. The volume entropy of 14

X˜ is deﬁned in terms of the volume of large metric balls;

logpvolpBpR, oqqq DpX˜q :“ lim . RÑ`8 R

The limit exists and is independent of base point; it coincides with the topological entropy of geodesic ﬂow on compact quotients of X˜. For the cases we are interested in we have explicit values of DpX˜q as follows.

Group Volume entropy

SOpn, 1q` n ´ 1 SUpn, 1q 2n Sppn, 1q 4n ` 2

´20 F4 22

2.3.6. Geometrically finite groups. General discrete subgroups Γ ă G can be very difficult to understand, so it is often useful to restrict our attention to certain subclasses whose behavior is simpler. For our purposes the most useful class is of geometrically finite groups. Let CHpΛpΓqq Ă X˜ be the convex hull of ΛpΓq (that is, the minimal convex subset containing all geodesics joining two elements of ΛpΓq). Denote by CCpΓq the convex core of Γ, i.e. the projection of CHpΛpΓqq to ΓzX˜.

Definition We say that the discrete group Γ ă G is geometrically finite if every neighborhood of CCpΓq has finite (Riemannian) volume in ΓzX˜. We say that Γ is convex cocompact if CCpΓq is compact.

Note that convex cocompact implies geometrically finite, but that the converse is false as geometrically finite groups may have cusps. The significance of this definition for our purposes is that it ensures finiteness and ergodicity of the BMS measure on ΓzT1X˜.

Theorem 2.3.3 (Theorem 4.1 and Corollary 5.5 of [15]). If Γ ă G is geometrically ﬁnite, then |mBMS| ă 8 and the A action on pΓzG{M, mBMSq is ergodic.

Geometrical ﬁniteness of Γ also gives a decomposition of ΛpΓq into the radial limit set and the parabolic limit set. We say that a point ξ P ΛpΓq is parabolic if the stabilizer 15

Γξ :“ stabΓpξq is parabolic (i.e. if the ﬁxed point set of Γξ in BX˜ is exactly tξu). We say that ξ is bounded parabolic if it is parabolic and the action of Γξ on ΛpΓq´tξu is cocompact.

We write ΛppΓq for the set of bounded parabolic limit points of Γ. We say that ξ P ΛpΓq is a radial limit point if every geodesic ray xptq in X˜ ending at ξ has some r neighborhood which meets Γo in an inﬁnite number of points. We write ΛrpΓq for the radial limit set.

Theorem 2.3.4 (See [11]). If Γ ă G is geometrically ﬁnite, then the limit set is the disjoint union of the radial and bounded parabolic limit sets. Moreover there are ﬁnitely many Γ orbits of parabolic limit points.

Note that, if Γ is geometrically ﬁnite, then any parabolic limit point of Γ is bounded parabolic (since it can’t be radial). Assuming that Γ is geometrically ﬁnite also allows us to choose useful cusp neighborhoods. By a horoball based at η PBX˜ we mean a set of the form tx P X˜ : βηpx, oq ă T u.

Lemma 2.3.2 (See, for example, Lemma 1 of Proposition 1.10 in [50]). Suppose that Γ ă G is geometrically ﬁnite and that ξ PBX˜ is a parabolic ﬁxed point. There is a collection of horoballs tBη : η P Γξu such that

• Bη is based at η,

• the sets Bη are pairwise disjoint, and

• γBη “ Bγη for all γ P Γ and η P Γξ.

We take this opportunity to fix one further piece of notation: we say that an element g P G is loxodromic if its fixed point set (in X˜ YBX˜) consists of exactly two boundary points. One of those fixed points will be attracting, and one repelling, and g acts as a translation along the geodesic joining the two. If Γ is non elementary, then the fixed points of loxodromic elements are dense in the limit set (see, for example, the review [48]). Chapter 3

Statements of results and areas of application

This chapter contains statements of results that will be proved in the current work, together with some current and hoped for applications. Section 3.1 describes generalizations of

3 Selberg’s 16 theorem to geometrically finite (but infinite area) hyperbolic surfaces. Section 3.2 describes some mixing properties of frame flow on infinite area rank one manifolds.

3.1 Selberg’s theorem for thin subgroups of SL2pZq

Our next target is to make sense of the Selberg theorem for geometrically ﬁnite subgroups

of SL2pZq. This setting is particularly nice because of the following well known fact.

Fact 3.1.1. A subgroup Γ ă SL2pZq is geometrically ﬁnite if and only if Γ is ﬁnitely generated.

Throughout this Section we will use the notation G “ PSL2pRq, and will assume that

Γ is a ﬁnitely generated non-elementary and torsion free subgroup of PSL2pZq. We write Γpqq ă Γ for the congruence subgroups of Γ, that is for the intersection of Γ with the

usual congruence subgroups of PSL2pZq. We can choose the maximal compact subgroup

K “ SORp2q{ ˘ I ă G and et{2 0 A “ at :“ : t P R . $ ¨ ´t{2 ˛ , & 0 e . ˝ ‚ % 16 - 17

The breakthrough in this ﬁeld is work of Bourgain-Gamburd-Sarnak [6] who understood connections between Cayley graphs of the ﬁnite quotients ΓpqqzΓ and the spectral or dy-

2 namical properties of the surfaces ΓpqqzH . Their theory is highly satisfactory in the case 1 1 when δΓ ą 2 . In the case when δΓ P p0, 2 s one can refine their work by using a key observation of Dolgopyat to retrieve a strengthened theory, which we shall describe. For now we confine ourselves to the following statement, which summarizes both their breakthrough work and our refinements.

Theorem 3.1.2 ([6, 9, 36, 41]). Suppose that Γ ă SL2pZq is ﬁnitely generated. There exist

η ą 0,C ě 3, and q0 ą 0 such that, for all square free q P N coprime to q0, and for any 1 ψ1, ψ2 P C0 pΓpqqzGq, we have

BMS ψ1pgatqψ2pgq dmq pgq “ żΓpqqzG BMS BMS C ´ηt mq pψ1q ¨ mq pψ2q ` Op||ψ1||C1 ||ψ2||C1 ¨ q ¨ e q (3.1.3)

BMS as t Ñ `8, with the implied constant depending only on Γ. Here mq denotes the probability Bowen-Margullis-Sullivan measure on ΓpqqzG

This can immediately be restated in terms of the Haar measure matrix coeﬃcients on

ΓzG.

Theorem 3.1.4 ([41]). Let Γ ă SL2pZq be a ﬁnitely generated subgroup. Then there exist

η ą 0,C ě 3 and q0 ą 1 such that for any square free q with pq, q0q “ 1 and any ψ1, ψ2 P

1 C0 pΓpqqzGq, we have

p1´δqt e ψ1pgatqψ2pgq dg “ żΓpqqzG 1 BR BR˚ C ´ηt BMS mq pψ1qmq pψ2q ` Op||ψ1||C1 ||ψ2||C1 ¨ q ¨ e q (3.1.5) mq pΓpqqzGq

BR BR˚ as t Ñ `8; here mq and mq denote respectively the unstable Burger-Roblin measure and the stable Burger-Roblin measure on ΓpqqzG which are chosen compatibly with the choice

BMS of dg, mq . 1 The implied constant can be chosen uniformly for all C -functions ψ1, ψ2 whose supports project to a ﬁxed compact subset of ΓzG. 18

The essential dichotomy in both Theorem 3.1.2 and Theorem 3.1.4 is between large

1 and small critical exponent. Where δ ą 2 the argument of Bourgain-Gamburd-Sarnak [6] provides uniform spectral gap for the hyperbolic Laplacian, which Bourgain-Kontorovich- Sarnak [9] and Mohammadi-Oh [36] then use to get eﬀective dynamical statements. When

1 δ ď 2 , on the other hand, the spectral arguments are unavailable, and Theorems 3.1.2 and 3.1.4 follow from the work of Oh-Winter [41] using ideas of Bourgain-Gamburd-Sarnak

1 and Dolgopyat. One useful observation is that any ﬁnitely generated group with δΓ ď 2 is automatically convex cocompact. It is for this reason that we focus on convex cocompact groups in Chapter 4. As in the lattice case, these mixing results lead to precise orbit counting asymptotics

for the action of Γpqq with exponential error terms independent of q. One application of this is to provide spectral data for the hyperbolic Laplacian ∆ on Xpqq in this setting.

1 2 When δ ď 2 , the L -spectrum of ∆ is known to be purely continuous [30], and the relevant spectral quantities are the resonances. The resolvent of the Laplacian

´1 8 2 8 2 RΓpqqpsq :“ p∆ ´ sp1 ´ sqq : C0 pΓpqqzH q Ñ C pΓpqqzH q

1 is holomorphic in the half plane

Theorem 3.1.6 ([6, 41]). Suppose that Γ is convex cocompact. There exist ą 0 and q0 ą 1 such that for all square free q P N with pq, q0q “ 1,

~~is a resonance free region for the resolvent RΓpqq except for a simple pole at s “ δ. 19~~

~~1 Together with the work of Bourgain-Gamburd-Sarnak for δ ą 2 this provides a reasonable analogue of Selberg’s theorem in the thin group case.~~

3.1.1. Applications of Selberg’s theorem for thin groups; Sieve and Zaremba Another application of uniform exponential mixing results such as Theorem 3.1.2 is to sieve; we use the mixing to get very precise counting results for Γpqq orbits. Feeding that data into some form of sieve one can recover number theoretic applications modeled on Fact 2.2.3. This approach was pushed through to a natural conclusion by Mohammadi-Oh in [36] who established very general sieving results for thin orbits using only uniform exponential mixing. By establishing the ﬁrst representation-theory-free approach to uniform exponential mixing, Theorem 3.1.2 expands the scope of their approach, and leaves the way open for further extensions in the future, as soon as the techniques addressed here can be generalized. We’ll state one example, which follows from the uniform exponential mixing theorem for thin groups and the counting and sieve machinery developed by Mohammadi-Oh [36]; see also earlier work on this question by Kontorovich-Oh [29] and Hong-Kontorovich [24], who obtained stronger results under the additional assumption that the critical exponent δΓ is very close to one.

~~Theorem 3.1.7 ([36, 41]). Let Γ ă SOZp2, 1q` be ﬁnitely generated and non-elementary. The number of small Pythagorean triples px, y, zq P Γ ¨ p3, 4, 5q with almost prime area satisﬁes~~

~~T δΓ #tpx, y, zq P Γ ¨ p3, 4, 5q : x2 ` y2 ` z2 ď T and xy is 7-almost primeu ! plog T q4 and for any suﬃciently large R~~

~~T δΓ #tpx, y, zq P Γ ¨ p3, 4, 5q : x2 ` y2 ` z2 ď T and xy is R-almost primeu " . plog T q4~~

Another application of Theorem 3.1.2 is to Zaremba’s conjecture. This was carried through by Bourgain-Kontorovich [8] using the initial statement of the thin Selberg theorem due to Bourgain-Gamburd-Sarnak [6]. It is hoped that the proof of Theorem 3.1.2 will allow an improved bound on the size of the exceptional set, and indeed a large part of the work necessary to establish this has already been completed [32]. We content ourselves at this 20 stage with a brief statement of the conjecture and the best known current bounds towards it.

For a ﬁnite sequence a1 . . . ak P N we recall the continued fraction 1 p ra1 . . . aks :“ 1 “ P Q a1 ` 1 q a2`... ak with p, q coprime. One natural question is this: if we make some restriction on the ai what

~~does that tell us about p and q? For instance, if we require that the ai be not too large does that imply any restriction on q? One precise version of this question is the following conjecture.~~

~~Conjecture 3.1.8 (Zaremba’s Conjecture). Let p D “ tq P : Dp P , a . . . a ď 5 such that “ ra . . . a su. 5 N N 1 k q 1 k~~

~~Then D5 “ N.~~

~~The best progress in this direction for which I am aware of full proofs in the literature is the groundbreaking work of Bourgain-Kontorovich, later reﬁned by Huang, which gives~~

Theorem 3.1.9 ([8, 26]). For some ą 0 we have N N ? #pD X r ,Nsq “ p1 ` Ope´ log N qq. 5 2 2 It is strongly expected by experts that the same will hold with an error term N ´, and indeed that statement (with only sketch proof) does appear in the literature [4]. The expectation is that the ideas of the proof of Theorem 3.1.2 will lead to an alternative proof of the power savings error term in Theorem 3.1.9. We ﬁnish the section with some comments on the strategy for proving bounds like

Theorem 3.1.9. The allowable alphabet (in our case t1, 2, 3, 4, 5u) leads to a free semigroup ˆ Γ Ă SL2pZq. There is then a very clever circle-method type argument to achieve the bounds in Theorem 3.1.9, which is the topic of [8]. However, no-one knows how to run this circle method argument with the full semigroup Γ;ˆ instead one is forced to restrict attention to a large subset Γ˜ Ă Γ,ˆ which is chosen to have good hyperbolic and congruence properties. It is in the choice of Γ˜ that orbit counting results for thin semigroups are needed. By improving the orbit counting results it is reasonable to hope for a “better” choice of Γ˜ and a better bound on the exceptional set. 21

~~3.2 Mixing properties of frame ﬂow on geometrically ﬁnite manifolds~~

~~Throughout this section we assume that the group G is either one of the following groups,~~

´20 or one of their ﬁnite covers: SUpn, 1q, Sppn, 1q,F4 , or the identity component SOpn, 1q` with n ě 3. We assume that Γ ă G is discrete, non-elementary, Zariski dense, and of divergence type. The subgroups K, A, M, N `, and N ´ are chosen as in Section 2.3. We

will mostly be interested in the case that Γ ă G has infinite covolume. In order to apply the mixing-equidistribution-counting machinery outlined in Section 2.1 in the infinite volume/thin group setting we need to establish mixing of the frame flow for

~~BMS the BMS measure. The ﬁrst step is to establish that the geodesic ﬂow pΓzG{M, m , atq is mixing. This was carried out by Martine Babillot using the Hopf argument [2]. One then~~

~~BMS seeks to lift mixing of the geodesic ﬂow up to mixing of the frame ﬂow pΓzG, m , atq. The strategy here is to use the Brin-Pesin argument [12, 20, 59].~~

~~Theorem 3.2.1 (Mixing of frame ﬂow, [2, 59]). The right translation action of A on~~

~~BMS pΓzG, m q is mixing: for any ψ1, ψ2 P C0pΓzGq we have~~

~~BMS BMS BMS m pψ1qm pψ2q lim ψ1pgatqψ2pgqdm pgq “ . tÑ8 |mBMS| żΓzG Very roughly speaking there are two ways for a compact M-extension of a mixing ﬂow~~

~~1 to be non-mixing. Consider an S extension of a mixing system pY, φt, µq. The pathologies we must exclude are demonstrated in the examples~~

1 iθ ips0t`θq pY ˆ S , φˆtpx, e q :“ pφtpxq, e q, µ ˆ dθq for s0 P R ﬁxed. If s0 is zero then the above extension is non-ergodic. If s0 is non-zero then the above extension is ergodic but not weak mixing. To prove Theorem 3.2.1 one must establish that neither of these problems arises in the cases of interest.

The s0 “ 0 pathology is easily excluded using Zariski density and the Bruhat decomposition for G. The second pathology requires more work. The key observation is that if the pathology occurs then, using Brin-Pesin moves, we can recover a restriction on the limit set ΛpΓq Ă BX˜; 22 roughly one shows that the limit set would need to be contained in a smooth submanifold of the boundary. By an expansion trick one upgrades this smooth obstruction to an algebraic one, and so obtains a contradiction with Zariski density of Γ. In fact the mixing property of mBMS is equivalent to the N `-ergodicity of the Burger-

~~Roblin measure on ΓzG. More precisely, work of Thomas Roblin, together with the remarks above, gives the following measure classﬁcation.~~

~~Theorem 3.2.2 ([50, 59]). Suppose that Γ ă G is geometrically ﬁnite. The Burger-Roblin measure is essentially the only N `-ergodic and -invariant Radon measure on ΓzG.~~

“Essentially” here means neglecting rescaling and measures supported on a single closed MN ` orbit. Roblin’s arguments also relate the mixing of the BMS measure to asymptotics of Haar measure matrix coeﬃcients; the following follows from Theorem 3.2.1 by the arguments of his thesis [50].

BR Theorem 3.2.3 ([50, 59]). Denote by m˚ the BR measure for the contracting horospher- Haar ical subgroup, and by m the Haar measure on ΓzG. For any pair ψ1, ψ2 P C0pΓzGq, we have BR BR pD´δqt Haar m pψ1qm˚ pψ2q lim e ψ1pgatqψ2pgqdm pgq “ tÑ`8 |mBMS| żΓzG These results are suﬃcient to run the mixing-equidistribution-counting machinery developed for the lattice case. As a sample application of this approach we mention a simple special case of work of Margulis-Mohammadi-Oh. For a geometrically ﬁnite hyperbolic

~~3 manifold X “ ΓzH of dimension 3 and a closed geodesic α : r0, 1s Ñ X of length lpαq we 1 have a holonomy taking values in the circle group θα P S .~~

~~Theorem 3.2.4 ([34]). The holonomies of closed geodesics of length less than T equidis-~~

~~1 1 tribute in S : for any continuous function φ : S Ñ R we have~~

~~eδΓT φpθαq „ φpθqdθ, δΓT 1 l α T S p ÿqă ż where the integral is taken with respect to the probability Lebesgue measure. Chapter 4~~

~~Selberg’s theorem for thin subgroups~~

This chapter relates a proof of Theorem 3.1.2. The key techniques here are from theory of transfer operators and the thermodynamic formalism. The key point is to prove that the Laplace transform of the correlation function has analytic extension to some appropriate half plane in C. One does this by relating the Laplace transform to a family of transfer operators La`ib with a, b P R. The required analytic extension will follow if we can establish spectral bounds on La`ib that are uniform in |a| small and b P R. In the region |b| ! 1 the required bounds follow the expander machinery and the arguments of Bourgain-Gamburd-

~~Sarnak. In the region where |b| is large one uses the Dolgopyat argument.~~

The outline of this chapter is as follows; Section 4.1 provides an overview of the necessary background on symbolic dynamics and Gibbs measures. Section 4.2 describes the Dolgopyat argument, and Section 4.3 the expander argument. We conclude in Section 4.4 by describing the relationships between transfer operators, correlation functions, and Laplace transforms.

~~4.1 Congruence transfer operators~~

In the whole chapter, let G “ SL2pRq and let Γ ă G be a non-elementary, convex cocompact subgroup containing the negative identity. We assume that ´e is the only torsion element of Γ. If p : SL2pRq Ñ PSL2pRq is the canonical projection, then ppΓq is a convex cocompact

~~23 24 torsion-free subgroup of PSL2pRq and we have Γz SL2pRq “ ppΓqz PSL2pRq. Since our results concern the quotient space ΓzG, we will henceforth abuse notation so that sometimes~~

~~G “ PSL2pRq and our Γ is considered as a torsion-free subgroup of PSL2pRq. We recall that the limit set ΛpΓq is a minimal non-empty closed Γ-invariant subset of~~

~~2 the boundary BH , and has Hausdorﬀ dimension δ “ δΓ is equal to the critical exponent of Γ (see [44]).~~

PS 2 We recall the Patterson-Sullivan densities tµx “ µx : x P H u for Γ and the Busemann function βξpy, xq “ limtÑ8 dpξt, yq ´ dpξt, xq where ξt is a geodesic ray tending to ξ as t Ñ 8. Since Γ is convex cocompact, µx is simply the δ-dimensional Hausdorﬀ measure on ΛpΓq with respect to a spherical metric viewed from x (up to a scaling).

2 We fix o P H in the convex hull of the limit set, and construct the Bowen-Margulis- BMS BMS 1 2 t{2 ´t{2 Sullivan measurem ˜ “ m˜ Γ on T pH q. We write A “ tat “ diagpe , e q : t P Ru, 1 2 and recall that the right at action on G corresponds to the geodesic flow on T H . Using the 1 2 BMS usual identification of T pH q and PSL2pRq we think ofm ˜ as an A-invariant measure on G. We denote by mBMS the measure on ΓzG induced bym ˜ BMS; it is called the Bowen- Margulis-Sullivan measure on ΓzG, or the BMS measure for short. It is a finite, A-invariant and A-mixing measure. We choose the left G- and right SO2pRq-invariant metric d on G such that dpe, atq “ t.

~~` ´ Let N and N be the expanding and contracting horocyclic subgroups for at:~~

1 0 1 s ` ` ´ ´ N “ tns :“ : s P Ru and N “ tns :“ : s P Ru. (4.1.1) ¨s 1˛ ¨0 1˛ ˝ ‚ ˝ ‚ ˘ For ą 0, we will denote by N the intersection of the ball around the identity, Bpeq, with N ˘. Write Ω for the support of the BMS measure. Since Γ is convex cocompact, Ω is compact and there is a uniform positive lower bound for the injectivity radii for points on ΓzG, which we will simply call the injectivity radius of Γ.

~~4.1.1. Markov sections We refer to [23] for basic facts about Markov sections. Let~~

~~α ą 0 be a small number. Consider a ﬁnite set z1, . . . , zk in Ω and choose small compact~~

~~` ´ neighborhoods Ui and Si of zi in ziNα XΩ and ziNα XΩ respectively of diameter at most α{2. 25~~

u ` s We write int pUiq for the interior of Ui in the set ziNα X Ω and deﬁne int pSiq similarly. u We will assume that Ui (respectively Si) are proper, that is to say, that Ui “ int pUiq s (respectively Si “ int pSiq). For x P Ui and y P Si, we write rx, ys for the unique local intersection of xN ´ and yN `A. We write the rectangles as

~~Ri “ rUi,Sis :“ trx, ys : x P Ui, y P Siu and denote their interiors by~~

~~u s intpRiq “ rint pUiq, int pSiqs.~~

~~Note that Ui “ rUi, zis Ă Ri. The family R “ tR1,...Rku is called a complete family of size α ą 0 if~~

~~k 1.Ω “Y1Riar0,αs~~

~~2. the diameter of each Ri is at most α, and~~

~~3. for any i ‰ j, at least one of the sets Ri X Rjar0,αs or Rj X Riar0,αs is empty.~~

~~Set R “ i Ri. Let τ : R Ñ R denote the ﬁrst return time and P : R ÞÑ R the ﬁrst return map: š~~

~~τpxq :“ inftt ą 0 : xat P Ru and Ppxq :“ xaτpxq.~~

~~Definition 4.1.2 (Markov section). A complete family R :“ tR1 ...Rku of size α is called a Markov section for the flow at if the following the Markov property is satisfied:~~

~~u u s s Pprint Ui, xsq Ą rInt Uj, Ppxqs and Pprx, Int Sisqq Ă rPpxq, Int Sjs~~

~~´1 whenever x P intpRiq X P pintpRjqq.~~

~~We consider the k ˆ k matrix~~

´1 1 if intpRlq X P intpRmq ‰ H Tr “ lm $ & 0 otherwise, which we will refer to as the transition% matrix. If there exists a positive integer N such that all the entries of TrN are positive, then we say that Tr is topologically mixing. 26

Ratner [49] and Bowen [10] established the existence of Markov sections of arbitrarily small size with topologically mixing transition matrix Tr: we now ﬁx such an R “ tR1 “ rU1,S1s,...,Rk “ rUk,Sksu of size α, where α ą 0 satisﬁes

~~1 α ă 1000 ¨ Injectivity radius of ΓzG and for all |s| ă 4α,~~

~~` ` dpe, ns q ď |s| ď 2dpe, ns q. (4.1.3)~~

~~Write~~

~~U :“ Ui and intpRq “ intpRiq. i i ž ž The projection map along stable leaves~~

~~πS : R Ñ U, taking rx, ys ÞÑ x~~

~~will be very important for us at several stages of the argument. We will writeσ ˆ for the map~~

~~σˆ :“ πS ˝ P : U Ñ U.~~

~~Deﬁnition 4.1.4. We deﬁne the cores of R and U by~~

~~m Rˆ “ tx P R : P x P intpRq for all m P Zu, and~~

~~ˆ m u U “ tu P U :σ ˆ u P int pUq for all m P Zě0u.~~

~~Note that Rˆ is P-invariant, and that Uˆ isσ ˆ-invariant. The cores are residual sets (that is, their complements are countable unions of nowhere dense closed sets).~~

~~4.1.2. Symbolic dynamics We choose Σ to be the space of bi-inﬁnite sequences x P~~

~~Z t1, . . . , ku such that Trxlxl`1 “ 1 for all l. Such sequences will be said to be admissible. We denote by Σ` the space of one sided admissible sequences~~

~~` Σ “ tpxiqiě0 : Trxi,xi`1 “ 1 for all i ě 0u.~~

~~We will write σ :Σ Ñ Σ for the shift map pσxqi “ xi`1. By abuse of notation we will also allow the shift map to act on Σ`. 27~~

~~` Deﬁnition 4.1.5. For θ P p0, 1q, we can give a metric dθ on Σ (resp. on Σ ) by choosing~~

~~1 1 inft|j|:xj ‰x u dθpx, x q “ θ j .~~

~~For a ﬁnite admissible sequence i “ pi0, . . . , imq, we obtain a cylinder of length m:~~

~~ˆ j Cris :“ tu P Ui0 :σ ˆ puq P intpUij q for all 1 ď j ď mu. (4.1.6)~~

Note that cylinders are open subsets of Uˆ. By a closed cylinder, we mean the closure of some (open) cylinder. We also take this opportunity to introduce embeddings of the symbolic space into the analytic space.

Deﬁnition 4.1.7 (The map ζ :Σ Ñ Rˆ). For x P Rˆ, we obtain a sequence ω “ ωpxq P Σ k ˆ ˆ by requiring P x P Rωk for all k P Z. The set Σ:“ tωpxq : x P Ru is a residual set in Σ. 2 Using the fact that any distinct pair of geodesics in H diverge from one another (either in positive time or negative time), one can show that the map x ÞÑ ωpxq is injective. We now deﬁne a continuous function ζ :Σ Ñ Rˆ by choosing ζpωpxqq “ x on Σˆ and extending continuously to all of Σ.

~~The restriction ζ : Σˆ Ñ Rˆ is known to be bijective and to intertwine σ and P.~~

~~Deﬁnition 4.1.8 (The map ζ` : Σˆ ` Ñ Uˆ). For u P Uˆ, we obtain a sequence ω1puq P Σ`~~

k ` ` by requiring P x P R 1 for all k P 0. We obtain an embedding ζ :Σ Ñ U by sending ωk Zě ω1puq ÞÑ u1 where possible and extending continuously. We write Σˆ ` :“ pζ`q´1pUˆq. The restriction ζ` : Σˆ ` Ñ Uˆ is known to be bijective and to intertwine σ andσ ˆ.

~~For θ suﬃciently close to 1, the embeddings ζ, ζ` are Lipschitz. We ﬁx such a θ once~~

~~` ` and for all. The space CθpΣq (resp. CθpΣ q) of dθ-Lipschitz functions on Σ (resp. on Σ ) is a Banach space with the usual Lipschitz norm~~

~~|fpxq ´ fpyq| ||f||dθ “ sup |f| ` sup . x‰y dθpx, yq~~

~~Writingτ ˜ :“ τ ˝ ζ P CθpΣq, we form the suspension~~

~~τ Σ :“ Σ ˆ R{px, t ` τx˜ q „ pσx, tq. 28~~

~~τ τ We write Σˆ for the set pΣˆ ˆ R{ „q Ă Σ . The suspension embeds into the group quotient via the map~~

~~τ τ ζ : Σˆ Ñ ΓzG, px, sq Ñ ζpxqas~~

~~τ τ and has an obvious ﬂow Gt : px, sq ÞÑ px, t ` sq. The restriction ζ : Σˆ Ñ ΓzG intertwines~~

~~Gt and at.~~

~~4.1.3. Pressure and Gibbs measures.~~

~~Deﬁnition 4.1.9. For a real valued function f P CθpΣq, called the potential function, we deﬁne the pressure to be the supremum~~

~~P rσpfq :“ sup fdµ ` entropyµpσq µ ˆżΣ ˙ over all σ-invariant Borel probability measures µ on Σ; here entropyµpσq denotes the measure theoretic entropy of σ with respect to µ.~~

For a given real valued function f P CθpΣq, there is a unique σ-invariant probability measure on Σ that achieves the supremum above, called the equilibrium state for f. We will denote it νf . It satisﬁes νf pΣˆq “ 1.

τ τ To any σ-invariant measure µ on Σ, we can associate a Gt-invariant measure µ on Σ ; simply take the local product of µ and the Lebesgue measure on R. Our interest in these equilibrium states is justiﬁed in light of the following fact.

~~Notation 4.1.10. We will write ν for the ´δpτ ˝ ζq-equilibrium state on Σ. We remark that the pressure P rσp´δpτ ˝ ζqq is known to be zero.~~

~~Theorem 4.1.11. Up to a normalization, the measure mBMS on ΓzG coincides with the~~

~~τ τ pushforward ζ˚ ν .~~

~~BMS Proof. Sullivan [56] proved that m is the unique measure of maximal entropy for the at~~

~~τ τ action on ΓzG. On the other hand ζ˚ ν is also a measure of maximal entropy on pΓzG, atq by [23]. The result follows.~~

~~τ τ BMS In particular, this theorem implies that pΣ , Gt, ν q and pΓzG, at, m q are measurably isomorphic as dynamical systems via ζτ . One simple consequence is the following corollary. 29~~

´1 PS Corollary 4.1.12. The measures pπ ˝ vis q˚µo and pπS ˝ ζq˚ν are mutually absolutely continuous on each Ui with bounded Radon-Nikodym derivative. Here vis denotes the visual ˜ 2 map from a lift Ui to BpH q, and π is the projection G Ñ ΓzG.

~~By abuse of notation, we use the notation ν for the measure pπS ˝ ζq˚ν on U.~~

~~4.1.4. Transfer operators. The identiﬁcation of Στ and ΓzG above allows the use of symbolic dynamics in the study of the BMS measure. In particular we will use the theory of transfer operators.~~

~~` ` ` Deﬁnition 4.1.13. For f P CθpΣ q, we obtain a transfer operator Lf : CpΣ q Ñ CpΣ q by taking~~

fpu1q 1 Lf phqpuq :“ e hpu q. σ u1 u pÿq“ A straightforward calculation shows that Lf preserves CθpΣq. The following is a consequence of the Ruelle-Perron-Frobenius theorem together with the well-known theory of Gibbs measure (see [43], [54]):

~~` Theorem 4.1.14. For each real valued function f P CθpΣ q, there exist a positive function ˆ ` ` h P CθpΣ q, a probability measure νˆ on Σ , and ą 0, c ą 0 such that~~

~~ˆ P rσpfqˆ •L f phq “ e h;~~

~~˚ P rσpfq • the dual operator satisﬁes Lf νˆ “ e νˆ;~~

~~• for all n P N,~~

~~´nP rσpfq n ˆ n |e Lf pψqpxq ´ νˆpψqhpxq| ď cp1 ´ q ||ψ||Lippdθq;~~

~~with hˆ normalized so that νˆphˆq “ 1;~~

~~• the measure hˆνˆ is σ-invariant and is the projection of the f-equilibrium state to Σ`.~~

~~The constants c, and the Lipschitz norm of hˆ can be bounded in terms of the Lipschitz norm of f; see [54]~~

Remark Using the identiﬁcation of Σ` and Uˆ by ζ`, we can regard the transfer operators deﬁned above as operators on CpUˆq. We can also regard the metric dθ as a metric on Uˆ. We will do both of these freely without further comment. 30

~~We also deﬁne the normalized transfer operators. For a P R with |a| suﬃciently small,~~

P rσp´pδàqτq consider the transfer operator L´pδàqτ on the space Cdθ pUq. Let λa :“ e be ˚ the largest eigenvalue,ν â the probability measure such that L´pδàqτ νâ “ λaνâ and let ha be the associated positive eigenfunction, normalized so that hadνˆ a “ 1. It is known that

~~λ0 “ 1, and that λa and ha are Lipschitz in a for |a| small.ş It is also known that for |a| small, each ha is Lipschitz in the d-metric [54].~~

Notation 4.1.15. For functions f : Uˆ Ñ R and h :Σ` Ñ R, we will write n´1 n´1 i i fnpuq :“ fpσˆ uq and hnpωq “ hpσ ωq. i“0 i“0 ÿ ÿ It follows from the fourth part of Theorem 4.1.14 that there exist c1, c2 ą 0 such that for all x P Σ` and for all n P N,

~~´pδàqτnpxq ń ´pδàqτnpxq ń c1e λa ď νâpCrx0, ¨ ¨ ¨ , xnsq ď c2e λa ; (4.1.16) moreover c1, c2 can be taken uniformly uniformly for |a| ă a0 for a fixed a0 ą 0.~~

~~We consider~~

~~paq f :“ ´pδ ` aqτ ` log h0 ´ log h0 ˝ σ ´ log λa, (4.1.17)~~

~~ˆ and let Lab :“ Lf paq`ibτ be the associated transfer operator, i.e.,~~

~~1 p´δàíbqτpu1q 1 Lâbphqpuq :“ e ph0 ¨ hqpu q. λah0puq σ u1 u pÿq“ paq Note that Lâb preserves the spaces CdpUˆq. We remark that the pressure P rσpf q is zero; so~~

~~p0q the leading eigenvalue of Lˆa0 is 1, with an eigenfunction ha{h0. Since f is cohomologous to ´δτ, the corresponding equilibrium states coincide.~~

~~4.1.5. Congruence transfer operators and the cocycle c Let D be the intersection of~~

2 the Dirichlet domain for pΓ, oq in H and the convex hull of ΛpΓq. For each Rj Ă ΓzG, we ˜ ˜ ˜ ˜ 2 choose a lift Rj “ rUj, Sjs to G so that the projection of Rj to H intersects D non-trivially.

~~We write R˜ :“YR˜i.~~

~~Definition 4.1.18 (Definition of the cocycle c : R Ñ Γ). For x P R with (unique) lift x˜ P R˜, we define the cocycle c by requiring that~~

~~˜ xa˜ τpxq P cpxqR. (4.1.19) 31~~

~~For n P N and x P U Ă R, we write~~

~~n´1 cnpxq :“ cpxqcpσˆpxqq ... cpσˆ xq.~~

~~1 ´1 1 Lemma 4.1.20. 1. If x, x P Rj X P Rl, then cpxq “ cpx q.~~

~~1 1 2. If x, x are both contained in some cylinder of length n ě 1, then cnpxq “ cnpx q.~~

~~´1 Proof. Let x1, x2 P Rj X P Rl. Ifx ˜1, x˜2 P R˜j with xj “ ΓzΓ˜xj, then fory ˜i :“ ´1 ˜ cpxiq x˜iaτpxiq P Rj, we have~~

~~dpcpx1qy˜1, cpx2qy˜1q ď dpx˜1, cpx2qy˜1q ` α~~

~~ď dpx˜2, cpx2qy˜1q ` 2α~~

~~ď dpcpx2qy˜2, cpx2qy˜1q ` 3α~~

~~ď 4α,~~

~~which is less than the injectivity radius of Γ. Thus cpx1q “ cpx2q as desired. The second~~

~~statement is now straightforward from the deﬁnition of cn.~~

~~Let Γpqq be a normal subgroup of Γ of ﬁnite index and denote by Fq the ﬁnite group ΓpqqzΓ. We would like a compatible family of Markov sections for the dynamical systems~~

~~BMS pΓpqqzG, at, m q. The lifts R˜l give a natural choice; for l P t1, . . . ku and γ P Fq, we take~~

~~q ˜ Rl,γ “ ΓpqqγRl Ă ΓpqqzG.~~

~~The collection~~

~~q q R :“ tRl,γ : l P t1 . . . ku and γ P Fqu is a Markov section of size α for pΓpqqzG, atq as expected. The ﬁrst return time τq and ﬁrst~~

~~q return map Pq associated to R are given rather simply in terms of the cocycle c and the corresponding data for R.~~

~~q Remark Let πq :ΓpqqzG Ñ ΓzG be the natural covering map. Ifx ˜ P Rl,γ and πqpx˜q P ´1 Rl X P Rm, then~~

~~τqpx˜q “ τpπqpx˜qq~~

~~q and Pqpxq is the lift of Ppπpxqq to Rm,γcpx˜q. 32~~

~~q ˜ Embedded inside each partition element Rl,γ, we have a piece of an unstable leaf. Let Ul be the lift of Ul contained in R˜l. Then the subsets~~

~~q ˜ ˆ q q ´1 ˆ Ul,γ :“ ΓpqqγUl Ă ΓpqqzG and Ul,γ :“ Ul,γ X πq pUq~~

q ˆ q ˆ q ˆ q ˆ q are contained in Rl,γ. We then write U :“ Ul,γ for the union andσ ˆq : U Ñ U for the natural extension ofσ ˆ. Just as the partitionš R gives rise to a symbolic model of the geodesic ﬂow on ΓzG, so Rq provides a model for ΓpqqzG. In particular we can identify Uˆ q with Uˆ ˆ Fq in a natural way; simply send pu, γq to the image γu˜ whereu ˜ is the lift of u to

~~R˜. Note then thatσ ˆq acts as the the map~~

~~σˆqpu, γq “ pσu,ˆ γcpuqq.~~

~~ˆ q ˆ q ˆ q For fq P CpU q, we may consider the following transfer operators Lfq,q : CpU q Ñ CpU q given by~~

1 1 fqpu ,γ q 1 1 pLfq,qhqpu, γq :“ e hpu , γ q σ u1,γ1 u,γ qp ÿq“p q 1 1 ´1 “ efqpu ,γcpu q qhpu1, γpcpu1qq´1q. σ u1 u pÿq“ It will very often be helpful to think of a function h P CpUˆ qq as a vector valued function

~~ˆ Fq U Ñ C . In the case where fqpω, γq “ fpωq doesn’t depend on the group element, we can~~

~~ˆ Fq ˆ Fq then recover the congruence transfer operator Mf,q : CpU, C q Ñ CpU, C q given by~~

~~fpu1q 1 1 pMf,qHqpuq “ e Hpu qcpu q; σˆ u1 u pÿq“~~

1 1 Fq where cpu q acts on Hpu q P C by the right regular action. We will often write pMf,qHqpu, γq to mean the γ component of pMf,qHqpuq . Most of this chapter is going to be devoted to a study of these congruence transfer operators. The key example for us will be the normalized

~~ˆ ˆ Fq ˆ Fq congruence transfer operator Mab,q :“ Mf paq`ibτ,q : CpU, C q Ñ CpU, C q:~~

~~1 ´pδ`a´ibqτpuq 1 1 pMˆ ab,qHqpuq “ e ph0Hqpu qcpu q. λah0puq σˆ u1 u pÿq“ We then have that for any n P N,~~

~~paq 1 ˆ n pfn `ibτnqpu q 1 ´1 1 Mab,qHpu, γq :“ e Hpu , γcn pu qq. σˆn u1 u pÿq“ 33~~

The key point will be to establish spectral properties of these congruence transfer operators. To do this we must ﬁrst establish norms and banach spaces appropriate to the task. We will write | ¨ | for the usual Euclidean inner product (in other words the l2 inner product)

~~F F on C q . For Lipschitz functions H : Uˆ Ñ C q , we deﬁne the norms~~

~~1 |Hpuq ´ Hpu1q| H : sup H u sup and (4.1.21) || ||1,b “ | p q| ` 1 uPUˆ maxp1, |b|q u‰u1 dpu, u q~~

~~1{2 2 ||H||2 :“ |Hpuq| dνpuq . (4.1.22) ˆż ˙ We will sometimes also write || ¨ ||Lippdq :“ || ¨ ||1,1 for the Lipschitz norm and denote by~~

~~ˆ Fq CLippdqpU, C q the space of Lipschitz functions for the norm || ¨ ||Lippdq. Consider the space of functions~~

ˆ Fq ˆ Fq ˆ WpU, C q “ tH P CLippdqpU, C q : Hpu, γq “ 0 for all u P Uu. (4.1.23) γPF ÿq 2 We will write L0pFqq for the space of complex valued functions on Fq that are orthogonal F to constants. We can then think of WpU,ˆ C q q as the space of Lipschitz functions from Uˆ 2 to L0pFqq. We’re now in a position to state the main technical result of our argument. Suppose that

Γ is a (non-elementary) convex cocompact subgroup of SL2pZq. We recall the congruence subgroups Γpqq of Γ. Since Γ is Zariski dense in SL2, it follows from the strong approximation theorem that there exists q0 ě 1 such that for all q P N with pq, q0q “ 1, we have

~~ΓpqqzΓ “ GpZ{qZq “ SL2pqq. (4.1.24)~~

~~1 Theorem 4.1.25. There exist ą 0, a0 ą 0,C ą 0, q0 ą 1 such that for all |a| ă a0, b P R, 1 and for all square free q P N with pq, q0q0q “ 1, we have~~

~~ˆ m m C ||Mab,qH||2 ď Cp1 ´ q q ||H||1,b~~

~~F for all m P N and all H P WpU,ˆ C q q.~~

~~The next two sections will be focused on the proof of this Theorem. In Section 3 we~~

prefer to work with the analytic space Uˆ and the associated function spaces CpUˆq, while in Section 4 the symbolic space Σˆ ` is preferred. For the most part we can unify these 34 viewpoints through the identiﬁcation ζ : Σˆ ` Ñ Uˆ; in particular for those parts of the argument where we consider the transfer operators acting on the L2pνq spaces there is no problem, as the measure theory does not see the precise geometry of the spaces Σˆ and Uˆ.

The one potential diﬃculty is where we want to use the d-Lipschitz properties of ha and f paq, which a priori do not follow from the usual statement of the RPF theorem 4.1.14. This is clariﬁed by [46], which ensures we can proceed as required.

~~4.2 Dolgopyat operators and vector valued functions~~

~~In this section we aim to prove that Theorem 4.1.25 holds whenever |b| is suﬃciently large:~~

~~Theorem 4.2.1. There exist ą 0, a0 ą 0, b0 ą 0,C ą 0 such that for all |a| ă a0, |b| ą b0, and for any normal subgroup Γpqq of Γ of ﬁnite index, we have~~

~~ˆ m m ||Mab,qH||2 ă Cp1 ´ q ||H||1,b~~

~~ˆ Fq for all m P N and all H P CLippdqpU, C q for Fq “ ΓpqqzΓ.~~

The strategy here is due to Dolgopyat [17], and uses the construction of so-called Dolgo- pyat operators. This construction was generalized to axiom A ﬂows by Stoyanov [53], and we will follow his argument. The remaining task is to relate these operators to our vector valued functions.

~~We begin by deﬁning a new metric on Uˆ: for u, u1 P Uˆ, set~~

~~Dpu, u1q “ inftdiampCq : C is a cylinder containing u and u1u (4.2.2)~~

~~where diampCq means the diameter of C in the metric d. Note that for all u, u1 P Uˆ,~~

~~dpu, u1q ď Dpu, u1q.~~

~~Deﬁnition 4.2.3. For E ą 0, we write KEpUˆq for the set of all positive functions h P CpUˆq satisfying~~

~~|hpuq ´ hpu1q| ď Ehpu1qDpu, u1q~~

~~1 for all u, u P Uˆ both contained in Uˆi for some i. 35~~

~~Theorem 4.2.1 follows from the following technical result as in the works of Dolgopyat and Stoyanov.~~

~~Theorem 4.2.4. There exist positive constants N P N,E ą 1, , a0, b0 such that for all a, b with |a| ă a0, |b| ą b0 there exist a ﬁnite set J pbq and a family of operators~~

~~NJ,a : CpUˆq Ñ CpUˆq for J P J pbq with the properties that:~~

~~ˆ 1. the operators NJ,a preserve KE|b|pUq;~~

~~|Hpuq| ď hpuq and |Hpuq ´ Hpu1q| ď E|b|hpuqDpu, u1q~~

~~for all u, u1 P Uˆ, then there exists J P J pbq such that~~

~~ˆ N • |Mab,qH| ď NJ,ah;~~

~~• for all u, u1 P Uˆ,~~

~~ˆ N ˆ N 1 1 |Mab,qHpuq ´ Mab,qHpu q| ď E|b|pNJ,ahqpuqDpu, u q.~~

~~The operators NJ,a are called Dolgopyat operators. Before moving on we indicate how to deduce Theorem 4.2.1 from Theorem 4.2.4.~~

~~Proof that Theorem 4.2.4 implies Theorem 4.2.1. Choose N P N, , |a| ă a0, |b| ą b0,E,~~

~~and H as in Theorem 4.2.4 and set h0 to be the constant function ||H||1,b. Theorem 4.2.4 ˆ allows us to inductively construct sequences Jl P J pbq, and hl P KE|b|pUq such that~~

~~1. hl`1 “ NJl,ahl,~~

~~ˆ lN 2. |Mab,qHpuq| ď hlpuq pointwise, and~~

~~ˆ lN l 3. ||Mab,qH||2 ď ||hl||2 ď p1 ´ q ||H||1,b. 36~~

~~Now choose 1 ą 0 such that p1 ´ 1qN “ p1 ´ q. There is a uniform upper bound, say~~

~~2 R0 ą 1, on the L pνq operator norm of Mˆ ab,q, valid for all b and all |a| ă a0. For any m “ lN ` r, with r ă N, we have~~

~~1{2 ˆ m ˆ r ˆ lN 2 ||Mab,qH||2 “ |Mab,qMab,qHpuq| dν ˆ ˆżU ˙ 1{2 r ˆ lN 2 ď R0 |Mab,qH| dν ˆ ˆżU ˙ r l ď R0p1 ´ q ||H||1,b~~

~~r 1 lN ď R0p1 ´ q ||H||1,b~~

~~N 1 m´N ď R0 p1 ´ q ||H||1,b.~~

~~This proves the claim.~~

~~4.2.1. Notation and constants We ﬁx notations and constants that will be needed later on. From hyperbolicity properties of the mapσ ˆ, we obtain constants c0 P p0, 1q, κ1 ą κ ą 1,~~

~~such that for all n P N,~~

~~n 1 n n 1 ´1 n 1 c0κ dpu, u q ď dpσˆ u, σˆ u q ď c0 κ1 dpu, u q (4.2.5)~~

~~1 for all u, u P Uˆi both contained in some cylinder of length n. Note that this implies a similar estimate for D:~~

~~n 1 n n 1 ´1 n 1 c0κ Dpu, u q ď Dpσˆ u, σˆ u q ď c0 κ1 Dpu, u q (4.2.6)~~

~~1 ˆ 1 for all u, u P Ui both contained in some cylinder of length n. Fix 0 ă a0 ă 0.1. The paq functions τ and h0, and hence f , are not d-Lipschitz globally, but they are essentially~~

~~d-Lipschitz in the following sense; there exists 0 ă T0 ă 8 such that~~

~~paq T0 ě max ||f || ` ||τ|| , (4.2.7) 1 8 8 |a|ďa0 ! ) and |f paqpuq ´ f paqpu1q| ` |τpuq ´ τpu1q| T0 ě (4.2.8) dpu, u1q~~

~~1 1 for all |a| ă a0 and all u, u both contained in the same cylinder of length 1. The following lemma follows from the Markov property. 37~~

~~n Lemma 4.2.9. Suppose that Cri0, . . . , iN s is a non-empty cylinder. The map σˆ : Cri0, . . . , iN s Ñ~~

n Crin, . . . , iN s is a bi-Lipschitz homeomorphsim. Moreover any section v of σˆ whose image contains Cri0, . . . , iN s restricts to a bi-Lipschitz homeomorphsm Crin, . . . , iN s Ñ Cri0, . . . , iN s.

~~The proof is omitted for brevity. We choose a small r0 ą 0 and zi P Uˆi such that~~

~~` u 2r0 ă minipdiampUiqq and ziNr0 X Ω Ă int pUiq for each i (here again Ω denotes the~~

~~support of the BMS measure). We ﬁx C1 ą 0 and ρ1 ą 0 to satisfy the following lemma:~~

~~Lemma 4.2.10. [53, Lemma 3.2] There exist C1 ą 0 and ρ1 ą 0 such that, for any cylinder Cris of length m, we have~~

~~´m m c0r0κ1 ď diampCrisq ď C1ρ1 .~~

~~We also ﬁx p0 P N and ρ P p0, 1q to satisfy the following proposition:~~

~~Proposition 4.2.11. [53, Proposition 3.3] There exist p0 P N and ρ P p0, 1q such that, for any n, any cylinder Cris of length n and any sub-cylinders Cri1s, Cri2s of length pn ` 1q and~~

~~pn ` p0q respectively, we have~~

~~diampCri2sq ď ρ diampCrisq ď diampCri1sq.~~

~~Choose also p1 ą 1 such that~~

~~1{4 ď 1{2 ´ 2ρp1´1. (4.2.12)~~

~~Fact 4.2.13. It follows from a property of an equilibrium state and the fact that Prσp´δτq “~~

~~0 that there is a constant 0 ă c1 ă 1 such that for any m P N,~~

~~´δτmpyq ´1 ´δτmpyq c1e ď νpCrisq ď c1 e for any cylinder Cris of length m and any y P Cris.~~

~~Now we need to recall some consequences of non-joint-integrability of the N `,N ´ foli- ations.~~

~~Lemma 4.2.14 (Main Lemma of [53]). There exist n1 P N, δ0 P p0, 1q, a non-empty subset~~

~~U0 Ă U1 which is a ﬁnite union of cylinders of length n1 ě 1, and z0 P U0 such that, setting~~

~~n U “ σ 1 pU0q, U is dense in U and that for any N ą n1, 38~~

~~N 1. there exist Lipschitz sections v1, v2 : U Ñ U such that σ pvipxqq “ x for all x P U,~~

~~and vipUq is a ﬁnite union of open cylinders of length N;~~

~~2. v1pUq X v2pUq “ H;~~

~~` ` 3. for all s P R such that z0ns P U0, all 0 ă |t| ă δ0 with z0ns`t P U0 X Ω, we have~~

~~1 |pτ ˝ v ˝ σˆn1 ´ τ ˝ v ˝ σˆn1 qpz n` q´ t N 2 N 1 0 t`s δ0 pτ ˝ v ˝ σˆn1 ´ τ ˝ v ˝ σˆn1 qpz n`q| ě N 2 N 1 0 s 2~~

~~(see (4.1.1) for other notation).~~

~~The next step is to establish certain a priori bounds on the transfer operators. Fix notation as in the previous subsection and choose~~

~~T0 T ´1 c κ 1 0 A0 ą 2c0 e 0p ´ q max 1, . (4.2.15) κ ´ 1 " * 1 Lemma 4.2.16. For all a P R with |a| ă a0 as in (4.2.7) and all |b| ą 1, the following hold:~~

~~• if h P KBpUˆq for some B ą 0, then~~

~~ˆm ˆm 1 La0hpuq ´ La0hpu q B T0 1 ď A0 m ` Dpu, u q ˇ Lˆm hpu1q ˇ κ κ ´ 1 ˇ a0 ˇ „  ˇ ˇ ˇ 1 ˇ for all m ě 0 andˇ for all u, u P Uˆi forˇ some i;~~

~~F • if the functions 0 ă h P CpUˆq,H P CpU,ˆ C q q and the constant B ą 0 are such that~~

~~|Hpvq ´ Hpv1q| ď Bhpv1qDpv, v1q~~

~~1 ˆ whenever v, v P Ui for some i, then for any m P N and any |b| ą 1,~~

~~B |Mˆ m Hpuq ´ Mˆ m Hpu1q| ď A Lˆm hpu1q ` |b|pLˆm |H|pu1qq Dpu, u1q ab,q ab,q 0 κm a0 a0 „  1 whenever u, u P Uˆi for some i.~~

~~Proof. The ﬁrst part is essentially proved in [53]. We concentrate on the second claim. Let~~

~~1 m u, u P Uˆi for some i and let m ą 0 be an integer. Given v P Uˆ withσ ˆ v “ u, let Cri0, . . . , ims~~

~~m be the cylinder of length m containing v. Note that im “ i and thatσ ˆ Cri0, . . . , ims “ Uˆi by 39~~

~~m the Markov property. Moreover we know thatσ ˆ : Cri0, . . . , ims Ñ Uˆi is a homeomorphism, so there exists v1 “ v1pvq withσ ˆmv1 “ u1. We therefore have~~

~~1 d σˆjv1, σˆjv d u, u1 p q ď m´j p q c0κ and so~~

~~m´1 paq paq 1 paq j paq j 1 |fm pvq ´ fm pv q| ď |f pσˆ vq ´ f pσˆ v q| j“0 ÿ m´1 Dpu, u1q ď ||f paq|| Lippdq c κm´j j“0 0 ÿ T0 ď Dpu, u1q. c0pκ ´ 1q~~

~~1 A similar estimate holds for |τmpv pvqq ´ τmpvq| by a similar calculation. In particular~~

~~paq paq 1 fm pvq fm pv pvqq e ď c0A0e , (4.2.17)~~

~~and~~

~~paq paq 1 |epfm `ibτmqpvq´pfm `ibτmqpv pvqq ´ 1|~~

~~paq paq 1 |fm pvq´fm pv q| paq paq 1 ď e |pfm ` ibτmqpvq ´ pfm ` ibτmqpv pvqq|~~

~~1 ď |b|A0Dpu, u q. (4.2.18)~~

~~Remark This type of estimate will be used repeatedly for the rest of the chapter, often with little comment.~~

~~1 Recall that cmpv pvqq “ cmpvq by Lemma 4.1.20. Using the fact that the diameter of Uˆi is 40 bounded above by 1, we now compute~~

~~ˆ m ˆ m 1 |Mab,qHpuq ´ Mab,qHpu q|~~

paq paq 1 ď epfm íbτmqpvqHpvq ´ epfm íbτmqpv pvqqHpv1pvqq σˆmv“u ÿ ˇ ˇ ˇ paq ˇ ď eˇ fm pvq|Hpvq ´ Hpv1pvqq| ˇ σˆmv“u ÿ paq paq 1 ` epfm íbτmqpvq ´ epfm íbτmqpv pvqq ¨ |Hpv1pvqq| σˆmv“u ÿ ˇ ˇ ˇ paq ˇ ď ˇefm pvqBhpv1pvqqDpv, v1pvqq ˇ σˆmv“u ÿ paq 1 paq paq 1 ` efm pv pvqq epfm ìbτmqpvq´pfm ìbτmqpv pvqq ´ 1 ¨ |Hpv1pvqq| σˆmv“u ÿ ˇ ˇ ˇ paq 1 ˇ 1 fm pv pvqq 1 ď c0A0BDpv, v pvqqˇ e hpv pvqq ˇ σˆmv“u ÿ paq 1 1 fm pv pvqq 1 ` |b|A0Dpu, u q e |Hpv pvqq| σˆmv“u ÿ by (4.2.17) and (4.2.18). By definitions and (4.2.6) this then yields

~~ˆ m ˆ m 1 |Mab,qHpuq ´ Mab,qHpu q| 1 A0BDpu, u q ď Lˆ hpu1q ` |b|A Dpu, u1qLˆ |H|pu1q κm a0 0 a0 B ď A Lˆ hpu1q ` |b|Lˆ |H|pu1q Dpu, u1q 0 κm a0 a0 ˆ ˙ as expected.~~

4.2.2. Construction of Dolgopyat operators. We now recall the construction of Dol- gopyat operators. Their deﬁnitions rely on a number of constants, which we now ﬁx. The meanings of these constants will become clear throughout the rest of the section. Choose

2A0T0 E ą max , 4A0, 1 ; (4.2.19) κ ´ 1 " * n1 n1 E 512κ E 200κ A0 N ą n such that κN ą max , 6A , 1 , 1 ; (4.2.20) 1 4c 0 c2δ ρ c2 " 0 0 0 0 * 2 c pκ ´ 1q c0r0 δ0 ă min 0 , , ; (4.2.21) 1 16T κn1 κn1 2 " 0 1 1 * 2 p ,p `2 2 2 1 c ρ 0 1 1 c µ ă min , 0 , 2 1 ; (4.2.22) 4 4κN 256 ˆ 1 ˙ 41 where A0 is given in (4.2.15), and other constants are as in subsection 4.2.1. Moreover set

~~b0 “ 1.~~

~~For the rest of this subsection, we ﬁx |b| ą b0. Let~~

~~tCm :“ Cmpbqu be the family of maximal closed cylinders contained in U0 (see Lemma 4.2.14) with diampCmq ď~~

~~1{|b|. As a consequence of (4.2.21) and Lemma 4.2.10 we have:~~

~~Lemma 4.2.23. Each of the cylinders Cm has length at least n1 ` 1.~~

~~N Corollary 4.2.24. Let v1, v2 be the sections for σˆ constructed by Lemma 4.2.14. If~~

~~1 n n 1 u, u P Cm X Uˆ, then cN pvipσˆ 1 uqq “ cN pvipσˆ 1 u qq for i “ 1, 2.~~

~~1 Proof. Choose u, u P Cm X Uˆ. They are both contained in some cylinder of length~~

~~n n 1 n1 ` 1. Thusσ ˆ 1 u, σˆ 1 u are both contained in some cylinder of length 1. But then~~

~~n n 1 vipσˆ 1 uq, vipσˆ 1 u q are both contained in the same cylinder of length N by Lemma 4.2.14. The result then follows by Lemma 4.1.20.~~

~~pmq n1 Notation 4.2.25. We set ci “ cN pvipσˆ uqq P Γ for any u P Cm; this is well deﬁned by Corollary 4.2.24.~~

~~Let tDj :“ Djpbq : j “ 1, . . . , pu be the collection of sub cylinders of the Cm of~~

~~lengthpCmq ` p0p1. We will say that Dj,Dj1 are adjacent if they are both contained in~~

~~the same Cm. We set Ξpbq :“ t1, 2u ˆ t1, . . . , ppbqu,~~

~~n Dˆj :“ Dj X U,Zˆ j :“ σ 1 pDˆjq, Zˆj :“ Zj X Uˆ and~~

~~Xi,j :“ vipZˆjq, Xˆi,j :“ Xi,j X Uˆ~~

~~ˆ for each i P t1, 2u and j P t1, . . . , pu. For J Ă Ξpbq, we deﬁne βJ : CpUq Ñ R by~~

~~βJ “ 1 ´ µ wi,j i,j J p ÿqP where wij is the indicator function of Xi,j. We recall a number of consequences of the constructions above: 42~~

~~1. Each cylinder Cm is contained in some Un and has diameter at least ρ1{|b|; apply Lemma 4.2.23 and Proposition 4.2.11.~~

~~p0p1`1 1 p1 1 2. ρ |b| ď diampDjq ď ρ |b| ; this follows from the deﬁnition of Dj and Proposition 4.2.11.~~

~~3. The sections vi are d-Lipschitz on each Uˆi, with Lipschitz constant no larger than 1 N ; this follows from (4.2.5). c0κ~~

~~4. The sets Xˆi,j are pairwise disjoint cylinders with diameters~~

~~2 p0p1`1 n1 p1 n1 c01ρ κ ˆ 1ρ κ1 N ď diampXi,jq ď 2 N ; (4.2.26) κ1 |b| c0κ |b| apply the previous two comments and (4.2.5).~~

~~5. The function βJ is D-Lipschitz on Uˆ with Lipschitz constant~~

~~µκN |b| 1 ; (4.2.27) 2 p0p1`1 n1 c01ρ κ this follows from the previous comment and the deﬁnition (4.2.2) of the metric D.~~

~~1 n 6. If u, u P σˆ 1 pCmq X Uˆ for some m, then~~

~~n1 1 1κ1 Dpvipuq, vipu qq ď 2 N for all i P t1, 2u (4.2.28) c0|b|κ~~

~~see the deﬁnition of Cm and (4.2.5).~~

~~1 2 n 7. If u , u P σˆ 1 pCmq X Uˆ, then~~

~~1 1 2 2 1 |b| ¨ τN pv2pu qq ´ τN pv1pu qq ´ τN pv2pu qq ´ τN pv1pu qq ď 8 ; (4.2.29) ˇ` ˘ ` ˘ˇ ˇ ˇ this follows from the deﬁnition of Cm, the choice (4.2.21) of 1, and (4.2.5).~~

~~Our next lemma, a simple special case of [53, Lemma 5.9] encapsulates the essential output of non-integrability for our argument. It is deduced from Lemma 4.2.14.~~

~~Lemma 4.2.30. For any Cm, there exist Dj1 ,Dj2 Ă Cm such that~~

~~1 1 2 2 1δ0ρ |b| ¨ τN pv2pu qq ´ τN pv1pu qq ´ τN pv2pu qq ´ τN pv1pu qq ě 16 (4.2.31) ˇ` ˘ ` ˘ˇ 1 ˇˆ 2 ˆ ˇ for all u P Zj1 and u P Zj2 . 43~~

1 2 1 2 1 2 Proof. Fix m and choose v0, v0, j , j such that v0 P Dj1 Ă Cm, and v0 P Dj2 Ă Cm with 1 2 1 1 2 dpv0, v0q ą 2 diampCmq. For any v P Dj1 and v P Dj2 , we have ρ 1 2 1 2 1 1 p1´1 1ρ dpv , v q ě dpv , v q ´ diampD 1 q ´ diampD 2 q ě ´ 2ρ ě 0 0 j j |b| 2 4 ` ˘ by (4.2.12). Now we recall z0 as in Lemma 4.2.14 and choose s1, s2 P p´α, αq such that v1 z n` and v2 z n` . Thus s s 1ρ by (4.1.3). On the other hand “ 0 s1 “ 0 s2 | 1 ´ 2| ě 8

~~δ0 1 2 1 2 ě diampCmq ě dpv , v q ě 2 |s1 ´ s2| by (4.2.21) and (4.1.3); the result follows by Lemma 4.2.14 part 3.~~

~~We are ﬁnally in a position to give the deﬁnition of our Dolgopyat operators. For~~

~~1 ˆ ˆ |b| ą b0, |a| ă a0 and for each J Ă Ξpbq, we deﬁne an operator NJ,a : CpUq Ñ CpUq by~~

~~ˆN NJ,aphq :“ La0pβJ hq.~~

4.2.3. Vector valued transfer operators and Dolgopyat operators. We will now check that appropriate operators NJ,a satisfy the conditions of Theorem 4.2.4. First choose the subsets J P Ξ that will be of interest.

~~Deﬁnition 4.2.32. A subset J Ă Ξpbq will be called dense if for every Cm, there exists pi, jq P J with Dj Ă Cm. We write J pbq for the collection of all dense subsets of Ξpbq.~~

~~The following proves parts 1 and 2 of Theorem 4.2.4.~~

~~1 Lemma 4.2.33. There exist a0 P p0, a0q and ą 0 such that for any |a| ă a0 and |b| ą b0, the family of operators tNJ,a : J Ă J pbqu satisﬁes:~~

~~ˆ ˆ 1. NJ,ah P KE|b|pUq whenever h P KE|b|pUq;~~

~~2 2 ˆ 2. U |NJ,ah| dν ď p1 ´ q U |h| dν for all h P KE|b|pUq; ş ş ˆ Fq ˆ 3. if H P CDpU, C q and h P KE|b|pUq are such that |H| ď h and~~

~~|Hpvq ´ Hpv1q| ď E|b|hpv1qDpv, v1q,~~

~~then ˆ N ˆ N 1 1 1 |Mab,qHpvq ´ Mab,qHpv q| ď E|b|pNJ,ahqpv qDpv, v q~~

~~where N is given as in (4.2.20). 44~~

Proof. The second part is Lemma 5.8 of [53]; although that paper uses a diﬀerently normalized transfer operator, the error is at most a factor sup sup ha , which can be absorbed |a|ďa0 inf ha into the decay term for a0 suﬃciently small. The other parts are contained in the same paper for complex valued functions; we include the argument for completeness. Suppose

~~ˆ 1 ˆ that h P KE|b|pUq, and that u, u P U. We compute~~

~~1 1 1 1 hβJ puq ´ hβJ pu q ď |hpuq ´ hpu q| ` hpu q|βJ puq ´ βJ pu q|. ˇ ˇ ˇ ˇ Thus, recalling (4.2.27);~~

~~hβJ P K ρ pUˆq. pE` 4 q|b|{p1´µq~~

~~We may now apply Lemma 4.2.16 above to give~~

~~1 N N 1 NJ,ahpuq ´ NJ,ahpu q “ La0phβJ qpuq ´ La0phβJ qpu q~~

~~ˇ ˇ ˇ pE ` ρ{4q|b| T0 ˇ 1 N 1 ď A0 ` Dpu, u qL phβJ qpu q ˇ ˇ ˇ κN p1 ´ µq κ ´ 1 ˇ a0 ˆ ˙ E|b| T0 1 1 ď A0 2 ` Dpu, u qNJ,ahpu q κN κ ´ 1 ˆ ˙ 1 1 ď E|b|Dpu, u qNJ,ahpu q~~

~~as required. The ﬁnal part also follows as a direct calculation using Lemma 4.2.16~~

~~Our ﬁnal task for this section is to prove the following key proposition, which completes the proof of Theorem 4.2.4 by addressing part 3.~~

~~Proposition 4.2.34. There exists a0 ą 0 with the following property. For any h, H as in~~

~~Theorem 4.2.4, any |a| ă a0, and any |b| ą b0 there exists J P J pbq such that for all v P Uˆ,~~

~~ˆ N |Mab,qHpvq| ď NJ,ahpvq where N is given as in (4.2.20). 45~~

~~We proceed via a series of lemmas.~~

~~Lemma 4.2.35. For |b| ą b0, functions h, H as in Theorem 4.2.4, we have, for any pi, jq P Ξpbq,~~

~~1. 1 1 hpvipu qq 1 ď ď 2 for all u, u P Zˆj; 2 hpvipuqq~~

~~3 ˆ 1 ˆ 2. either |Hpvipuqq| ď 4 hpvipuqq for all u P Zj or |Hpvipuqq| ě 4 hpvipuqq for all u P Zj.~~

~~ˆ 1 ˆ Proof. For h P KE|b|pUq and u, u P Zj with Dj Ă Cm, we simply calculate~~

~~1 1 hpvipu qq ď hpvipuqq ` E|b|Dpvipu q, vipuqqhpvipuqq~~

~~ď hpvipuqq 1 ` E|b| diampXˆi,jq ´ ¯ ď 2hpvipuqq~~

~~by (4.2.26) and (4.2.20). The other bound follows by symmetry. The second part of the the observation follows by similar calculations, which we shall omit.~~

~~1 Deﬁnition 4.2.36. Let a P p0, a0q and choose |b| ą b0, h, H as in Theorem 4.2.4. For each~~

~~ﬁxed Cm “ Cmpbq, recall that~~

~~pmq n1 ci “ cN pvipσˆ puqqq for all u P Cm.~~

~~Deﬁne the the functions~~

~~paq paq pf `ibτN pv1puqq pmq pf `ibτN qpv2puqq pmq e N Hpv1puqqc1 ` e N Hpv2puqqc2 p1q χ rH, hspuq :“ a a ˇ f p qpv puqq f p qpv puqq ˇ ˇ p1 ´ µqe N 1 hpv1puqq ` e N 2 hpv2puqq ˇ ˇ ˇ and~~

paq paq pf `ibτN qpv1puqq pmq pf `ibτN qpv2puqq pmq e N Hpv1puqqc1 ` e N Hpv2puqqc2 p2q χ rH, hspuq :“ a a . ˇ f p qpv puqq f p qpv puqq ˇ ˇ e N 1 hpv1puqq ` p1 ´ µqe N 2 hpv2puqq ˇ ˇ ˇ We claim the following:

~~Lemma 4.2.37. For every Cm, there exist i P t1, 2u and j P t1 . . . pu such that Dj Ă Cm~~

~~piq and χ rH, hspuq ď 1 for all u P Zˆj. 46~~

~~1 2 ˆ ˆ Proof. Fix m and choose j , j as in Lemma 4.2.30. Consider Zj1 and Zj2 . If there exist~~

~~1 2 t P tj , j u and i P t1, 2u such that the ﬁrst alternative of Lemma 4.2.35 (2) holds for Zˆt,~~

piq then χ rH, hspuq ď 1 for all u P Zˆt. So from now on in this proof we assume the converse, 1 ˆ ˆ i.e., for each i, |Hpvipuqq| ě 4 hpvipuqq for all u P Zj1 Y Zj2 . 1 ˆ 2 ˆ Consider now u P Zj1 and u P Zj2 . Then the properties of h, H imply

~~1 2 1 1 2 |Hpvipu qq ´ Hpvipu qq| E|b|hpvipu qqDpvipu q, vipu qq 1 2 ď 1 2 mint|Hpvipu qq|, |Hpvipu qq|u mint|Hpvipu qq|, |Hpvipu qq|u 1 2 ď 4E|b|Dpvipu q, vipu qq~~

~~1δ0ρ ă by (4.2.28), 128~~

~~1 2 where we have assumed |Hpvipu qq| ď |Hpvipu qq| without loss of generality.~~

~~1 δ0ρ In particular this is less than 2 . We write c2 “ 16 . The sine of the angle θi between~~

~~1 2 c21 Hpvipu qq and Hpvipu qq is therefore at most sin θi ď 8 , so~~

~~c21 θ ď . (4.2.38) i 4~~

~~We need to use this to show that at least one of the angles~~

~~1 1 ibτN pv1pu q 1 pmq ibτN pv2pu q 1 pmq θpe Hpv1pu qqc1 , e Hpv2pu qqc2 q~~

~~2 2 ibτN pv1pu q 2 pmq ibτN pv2pu q 2 pmq θpe Hpv1pu qqc1 , e Hpv2pu qqc2 q~~

~~is greater than c21{4. Supposing that the ﬁrst term is less than c21{4, we will show that~~

~~the second term is bigger than c21{4. Write~~

~~φpwq :“ b ¨ pτN pv2pwqq ´ τN pv1pwqqq~~

~~and note that~~

~~1 2 1 c21 ď |φpu q ´ φpu q| ď 8~~

~~1 ˆ 2 ˆ for all u P Zj1 and all u P Zj2 by (4.2.29) and Lemma 4.2.30. 47~~

~~We compute~~

~~2 2 ibτN pv1pu qq 2 pmq ibτN pv2pu qq 2 pmq θpe Hpv1pu qqc1 , e Hpv2pu qqc2 q~~

~~´iφpu2q 2 pmq 2 pmq “ θpe Hpv1pu qqc1 ,Hpv2pu qqc2 q~~

~~´iφpu2q 2 pmq ´iφpu1q 2 pmq ě θpe Hpv1pu qqc1 , e Hpv1pu qqc1 q~~

~~´iφpu1q 2 pmq 2 pmq ´θpe Hpv1pu qqc1 ,Hpv2pu qqc2 q~~

~~1 2 ´iφpu1q 2 pmq 2 pmq ě |φpu q ´ φpu q| ´ θpe Hpv1pu qqc1 ,Hpv2pu qqc2 q~~

~~´iφpu1q 1 pmq 1 pmq ě c21 ´ c21{2 ´ θpe Hpv1pu qqc1 ,Hpv2pu qqc2 q~~

~~ě c21{4~~

~~by (4.2.38) and the assumption. Write~~

~~paq 2 pmq pfN `ibτN pv1pu qq 2 v “ e Hpv1pu qqc1 and~~

~~paq 2 pmq pfN `ibτN qpv2pu qq 2 w “ e Hpv2pu qqc2~~

~~so that |v ` w| is the numerator of χpiqrH, hspu2q. Without loss of generality, we assume p1q 2 2 ˆ that |v| ď |w|. We now claim that χ rH, hspu q ď 1 for all u P Zj2 . This now follows~~

~~from rather simple trigonometry. Since the angle θ˜ between v and w is at least c21{4, we have c22 1 ` 2 cos θ˜ ď 2 ` cos θ˜ ď 3 ´ 2 1 ď 3p1 ´ µq2 ď p1 ´ µq2 ` 2p1 ´ µq. 16~~

~~Thus~~

~~|v| ` 2|v| cos θ˜ ď p1 ´ µq2|v| ` 2p1 ´ µq|v|.~~

~~Now~~

~~|v| ` 2|w| cos θ˜ ď p1 ´ µq2|v| ` 2p1 ´ µq|w|,~~

~~p1q ˆ and so p1 ´ µq|v| ` |w| ě |v ` w|, and χ rH, hs ď 1 on Zj2 as expected.~~

~~Proof of Proposition 4.2.34. Choose h, H, |b| ą b0 as in the hypotheses of Theorem 4.2.4~~

~~1 and choose a0 P p0, a0q to satisfy Lemma 4.2.33. We choose a subset J P J pbq as follows. p1q First include in J all p1, jq P Ξ such that χ rH, hs ď 1 on Zˆj. Then for any j P t1 . . . pu,~~

p2q include p2, jq in J if p1, jq is not already in J and χ rH, hs ď 1 on Zˆj. By Lemma 4.2.37, 48 this subset J is dense (in the sense of Deﬁnition 4.2.32), so that J P J pbq. We will show that for all u P Uˆ ˆ N |Mab,qHpuq| ď NJ,ahpuq.

~~N Let u P Uˆ. Suppose ﬁrst that u R Zˆj for any pi, jq P J; then βJ pvq “ 1 wheneverσ ˆ pvq “ u, and the bound follows. Suppose instead that u P Zˆj Ă Cm with p1, jq P J. Then p2, jq R J~~

~~p1q and so βJ pv1puqq ě 1 ´ µ and βJ pv2puqq “ 1. We therefore have χ rH, hs ď 1 on Zˆj, so~~

~~a ˆ N f p qpvq |Mab,qHpuq| ď e N |Hpvq| N σˆ v“u,v‰ÿv1puq,v2puq paq pmq paq pmq pfN `ibτN pv1puqq pfN `ibτN qpv2puqq ` e Hpv1puqqc1 ` e Hpv2puqqc2~~

~~ˇ paq ˇ ˇ fN pvq ˇ ď ˇ e |hpvq| ˇ N σˆ v“u,v‰ÿv1puq,v2puq paq paq f pv1puqq f pv2puqq ` p1 ´ µqe N hpv1puqq ` e N hpv2puqq~~

~~ď NJ,ahpuq.~~

~~The case u P Zˆj with p2, jq P J is similar. This ﬁnishes the proof.~~

~~Together with Lemma 4.2.33, this completes the proof of Theorem 4.2.4.~~

~~4.3 The expansion machinery~~

4.3.1. Some reductions. In this section we assume that Γ is a convex cocompact subgroup in SL2pZq and that q0 is as in (4.1.24). Let b0 ą 0 be as in Theorem 4.2.1. The main aim of this Section is to prove the following theorem.

~~1 Theorem 4.3.1. There exist P p0, 1q, a0 ą 0,C ą 1, q0 ą 1 such that for all |a| ă a0, 1 |b| ď b0, and all square free q ě 1 with pq, q0q0q “ 1, we have~~

~~ˆ m m C ||Mab,qH||2 ă Cp1 ´ q q ||H||Lippdq~~

~~SL q for all m P N and all H P WpU,ˆ C 2p qq; see (4.1.23) for notation.~~

~~Since the Lippdq norm and the || ¨ ||1,b norm are equivalent for all |b| ď b0, this theorem and Theorem 4.2.1 imply Theorem 4.1.25. 49~~

The key ingredient of the proof of Theorem 4.3.1 is the expander technology, introduced in this context by Bourgain, Gamburd, and Sarnak [6], from which we draw heavily throughout this section. The idea of the expansion machinery is that random walks on the

~~Cayley graphs of SL2pqq have good spectral properties. We don’t have a random walk in the usual sense, but the randomness inherent in the Gibbs measure provides the same eﬀect.~~

~~We recall the sequence spaces Σ`, Σ, the shift map σ and the embedding ζ :Σ` Ñ Uˆ.~~

~~SL q Notation 4.3.2. For any function H P CpU,ˆ C 2p qq, we will denote H˜ “ H ˝ ζ :Σ` Ñ SL q C 2p q. Similarly τ˜ will denote τ ˝ ζ.~~

~~We recall the constant θ P p0, 1q chosen suﬃciently close to one (see subsection 4.1.2)~~

~~` and the metric dθ on Σ (resp. on Σ ). Write~~

~~˜ ˜ ||H||8 :“ sup |Hpωq| ωPΣˆ ` and |H˜ pωq ´ H˜ pω1q| Lip pH˜ q :“ sup dθ 1 ω‰ω1PΣˆ ` dθpω, ω q which is the minimal Lipschitz constant of H˜ . We also write~~

~~H˜ : H˜ Lip H˜ . || ||dθ “ || ||8 ` dθ p q~~

~~We ﬁx the following constant for later convenience~~

~~paq Lipd pτq ` sup|a|ă1 Lipd pf q η :“ θ θ . (4.3.3) θ 1 ´ θ~~

~~Rather than proving Theorem 4.3.1 directly, we will instead start by describing some~~

~~1 ˆq 2 reductions to a simpler form. For q |q, we deﬁne Eq1 Ă L0pSL2pqqq to be the space of functions invariant under the left action of Γpq1q. We may then write~~

~~q ˆq ˆq K Eq1 :“ Eq1 X ‘q1‰q2|q1 Eq2 . ´ ¯ q 1 ˜q We think of Eq1 as the space of new functions at the level q . We can then deﬁne Eq1 as~~

~~ˆ SL2pqq q the subspace of functions H in WpU, C q with Hpu, ¨q P Eq1 for all u. We recall the orthogonal decomposition~~

~~2 q L0pSL2pqqq “ ‘1‰q1|qEq1 50 and the induced direct sum decomposition~~

~~ˆ SL2pqq ˜q WpU, C q “ ‘1‰q1|qEq1 .~~

~~Write~~

~~ˆ SL2pqq ˜q eq,q1 : WpU, C q Ñ Eq1 for the projection operator, and note that eq,q1 is norm decreasing for both the || ¨ ||Lippdq norm and the || ¨ ||2 norm.~~

~~Remark The projection operators commute with the congruence transfer operators: we have ˆ ˆ eq,q1 ˝ Mab,q “ Mab,q ˝ eq,q1 for any q1|q.~~

~~q The ﬁrst reduction is that we only need to consider functions in E˜q .~~

~~Theorem 4.3.4. There exist P p0, 1q, a0 ą 0,C ą 1, q1 ą 1 such that for all |a| ă a0,~~

~~|b| ă b0 and q ě q1 square free with pq, q0q “ 1, we have~~

~~ˆ m m C ||Mab,qH||2 ă Cp1 ´ q q ||H||Lippdq (4.3.5)~~

~~˜q for all m P N and all H P Eq .~~

~~1 Proof that Theorem 4.3.4 implies Theorem 4.3.1. Set q0 to be the product of all primes less than or equal to q1. We will ﬁrst explain how to deduce Theorem 4.3.1 from Theorem 4.3.4.~~

~~1 Fix , a0, b0, C, q1, q0 as in Theorem 4.3.4. Fix also q square free such that pq, q0q0q “ 1. For q1|q, we consider the projection maps~~

~~q q1 projq,q1 : Eq1 Ñ Eq1~~

by choosing pprojq,q1 F qpγq “ F pγ˜q, whereγ ˜ is any pre-image of γ under the natural projec- 1 tion map SL2pqq Ñ SL2pq q. By abuse of notation we will also write projq,q1 for the induced 1 1 ˜q ˜q # SL2pq q maps E 1 Ñ E 1 . Write ♠ 1 :“ . We note that q q q,q # SL2pqq

~~||pprojq,q1 Hq||Lippdq “ ♠q,q1 ||H||Lippdq, a 51 that~~

~~||pprojq,q1 Hq||2 “ ♠q,q1 ||H||2, a and that ˆ ˆ Mab,q1 ˝ projq,q1 “ projq,q1 ˝ Mab,q.~~

~~ˆ SL2pqq 1 Now consider H P WpU, C q. We calculate, for |a| ă minpa0, a0q and |b| ď b0,~~

ˆ m 2 ˆ m 2 ||Mab,qH||2 “ ||eq,q1 Mab,qH||2 1 q1 q ‰ÿ| ˆ m 2 “ ||Mab,qpeq,q1 Hq||2 1 q1 q ‰ÿ| ˆ m 2 “ ♠q,q1 ||projq,q1 pMab,qpeq,q1 Hqq||2 1 q1 q ‰ÿ| ˆ m 2 “ ♠q,q1 ||Mab,q1 pprojq,q1 peq,q1 Hqq||2. 1 q1 q ‰ÿ| Applying Theorem 4.3.4, we obtain

ˆ m 2 2 2m 1 2C 2 ||Mab,qH||2 ď C p1 ´ q pq q ♠q,q1 ||projq,q1 peq,q1 Hq||Lippdq. 1 q1 q ‰ÿ| 2 2m 1 2C 2 ď C p1 ´ q pq q ||eq,q1 H||Lippdq 1 q1 q,q1 q ‰ |ÿ ě 1 2 2 2 2m 2C2`1 2 ď pC q p1 ´ q q ||H||Lippdq as expected.

~~The most convenient formulation to prove will be the following:~~

~~Theorem 4.3.6. There exist κ ą 0, a0 ą 0, q1 ą 0 such that~~

~~ˆ lnq ˜ ´lκ ˜ ||Mab,qH||2 ď q ||H||dθ~~

~~˜q for all |a| ă a0, |b| ď b0, l P N, all q ą q1 square free and coprime to q0, and all H P Eq ; here nq denotes the integer part of log q.~~

~~Proof that Theorem 4.3.6 implies Theorem 4.3.4. Choose a0 ą 0 small enough that | log λa| ď~~

~~ ď minpκ{2, 1q for all |a| ă a0. Set~~

~~C :“ max log sup ||Mˆ ab,q||2 , 0 . # ˜|a|ďa0,bPR ¸ + 52~~

~~ˆ r C Then for all 0 ď r ă nq, we have ||Mab,q||2 ď q .~~

~~For any m P N, we write m “ lnq ` r, with 0 ď r ă nq. Thus Theorem 4.3.6 yields~~

~~ˆ m ˜ ˆ r ˆ lnq ˜ ||Mab,qH||2 ď ||Mab,q||2 ¨ ||Mab,qH||2 C ´lκ ˜ ď q q ||H||dθ~~

~~C ´lnq ˜ ď q e ||H||dθ~~

~~C`1 ´m ˜ ď q e ||H||dθ ,~~

~~as desired.~~

4.3.2. The `2-ﬂattening lemma The rest of this section is devoted to a proof of Theorem 4.3.6. The key ingredient is a version of the `2-ﬂattening lemma 4.3.9 of Bourgain-Gamburd- Sarnak [6, Lemma 7.2]. For the rest of this section we will assume that

~~q is square free and coprime to q0 (as in (4.1.24)).~~

~~1 Deﬁnition 4.3.7. For a complex valued measure µ on SL2pqq and q |q, we deﬁne |||πq1 pµq|||8~~

~~1 to be the maximum weight of |µ| over all cosets of subgroups of SL2pq q that have proper projection in each divisor of q1.~~

~~Notation 4.3.8. For a function φ and a measure µ on SL2pqq, we denote the convolution by~~

~~µ ˚ φpgq “ µpγqφpgγ´1q. γ SL q P ÿ2p q Lemma 4.3.9 ([5], [6]). Given κ ą 0 there exist κ1 ą 0 and C ą 0 such that if µ satisﬁes~~

~~||µ||1 ď B and~~

~~´κ 1 1 1{10 |||πq1 pµq|||8 ă q B for all q |q, q ą q for some B ą 0,~~

~~q then for each q and φ P Eq ,~~

~~´κ1 ||µ ˚ φ||2 ď Cq B||φ||2.~~

4.3.3. Measure estimates on cylinders Before we can apply the expansion machinery we must ﬁrst establish certain a priori measure estimates on cylinders; that will be the topic of this subsection. We deﬁne: 53

~~` Notation 4.3.10. • For x “ px1, x2,...q P Σ and a sequence i1, . . . , in of symbols, we denote the concatenation by~~

~~pin, . . . , i1, xq “ pin, . . . , i1, x1, x2,...q;~~

~~• For a function f on Σ` and x P Σ`, we set~~

~~n´1 fnpxq :“ fpxq ` fpσpxqq ` ... ` fpσ pxqq;~~

~~` • For x “ pxiq P Σ , put cpxq “ cpζpxqq, and~~

~~n´1 cnpxq :“ cpζpxqqcpζpσxqq ... cpζpσ xqq P Γ.~~

~~` Lemma 4.3.11. For sequences x, y P Σˆ with xi “ yi for i “ 0 . . . k for some k ě 1, we~~

~~have ckpxq “ ckpyq.~~

~~Proof. This is a straightforward consequence of Lemma 4.1.20.~~

~~We may therefore write cpxq “ cpx0, x1q, and more generally, for n ě 2, cnpxq is the product cpx0, x1q ¨ ¨ ¨ cpxn´1, xnq.~~

~~Notation 4.3.12. In the rest of the section, the notation means the sum taken over i1,¨¨¨ ,i`~~

~~all sequences pi1, ¨ ¨ ¨ , i`q such that any concatenation followingř the sum sign is admissible.~~

~~` Lemma 4.3.13. There exist 0 ă a0 ă 1 and c ą 1 such that for all |a| ă a0, x P Σˆ and~~

~~for all n P N, paq efn pin,¨¨¨ ,i1,xq ď c. i ,¨¨¨ ,i nÿ 1 Proof. This follows easily from (4.1.16).~~

2 We recall that for x, y P H and r ą 0, the shadow Orpx, yq is deﬁned to be the set 2 of all points ξ P BpH q such that the geodesic ray from x to ξ intersects the ball Brpyq PS 2 non-trivially. We need the following: recall the Patterson-Sullivan density tµo x : x P H u for Γ.

~~2 Lemma 4.3.14 (Sullivan’s shadow lemma [?]). Let x P H . There exists r0 “ r0pxq ą 1 such that for all r ą r0, there exists c ą 1 such that for all γ P Γ,~~

~~´1 ´δdpx,γxq PS ´δdpx,γxq c e ď µx pOrpx, γxqq ď ce . 54~~

~~Lemma 4.3.15. There exists c1 ą 1 such that for any x P Σˆ `, γ P Γ, any m P N, and any~~

~~ﬁxed im`1, we have~~

~~p0q efm pim,...,i1,xq ď c1 ¨ e´δm infpτq. i ,...,i ,c i ,i ,...,i ,x γ 1 m m`1p ÿm`1 m 1 q“ Proof. Recalling the deﬁnition (4.1.17) of f p0q in terms of τ, we calculate~~

p0q efm pim,...,i1,xq i ,...,i ,c i ,i ,...,i ,x γ 1 m m`1p ÿm`1 m 1 q“ supph0q ď e´δτmpim,...,i1,xq infph0q i ,...,i ,c i ,i ,...,i ,x γ 1 m m`1p ÿm`1 m 1 q“ supph0q ď eδ suppτq e´δτm`1pim`1,...,i1,xq infph0q i ,...,i ,c i ,i ,...,i ,x γ 1 m m`1p ÿm`1 m 1 q“ 1 ď c1 νpCrim`1, im, . . . , i1sq (see Fact 4.2.13q i ,...,i ,c i ,i ,...,i ,x γ 1 m m`1p ÿm`1 m 1 q“

~~1 δ suppτq supph0q where c “ c1e . 1 infph0q Recall that D denotes the intersection of the Dirichlet domain for pΓ, oq with the convex~~

~~core of Γ, and the lifts U˜i of Ui chosen to intersect D. Recall also the projection map π from G to ΓzG. It is a consequence of the deﬁnition of c (4.1.19) that~~

~~˜ Crim`1, im, . . . , i1s Ă π tu˜ P Uim`1 : dpua˜ τm`1pu˜q, γoq ă R1u (4.3.16) i ,...,i :c i ,...,i ,x γ m 1 m`1ďp m`1 1 q“ ´ ¯~~

~~where R1 denotes thrice the size of the Markov section plus twice the diameter of D plus~~

~~the constant r0poq deﬁned in Lemma 4.3.14.~~

~~Case 1: If dpu,˜ γu˜q ă pm ` 1q infpτq ´ R1, then~~

~~˜ tu˜ P Uim`1 : uaτm`1pu˜q P BR1 pγu˜qu “ H,~~

~~and the claim follows.~~

~~Case 2: We now assume that dpu,˜ γu˜q ě pm ` 1q infpτq ´ R1. Then dpo, γoq ě pm `~~

~~1q infpτq ´ 2R1. A straightforward argument in hyperbolic geometry yields~~

~~˜ ˜ tu˜ P Uim`1 : dpua˜ τm`1pu˜q, γoq ă R1u Ă tu˜ P Uim`1 : vispu˜q P OR1 pu,˜ γoqu ˜ Ă tu˜ P Uim`1 : vispu˜q P O2R1 po, γoqu. 55~~

~~Applying Corollary 4.1.12 and Lemma 4.3.14, we obtain~~

e´δτmpim,...,i1,xq i ,...,i ,c i ,i ,...,i ,x γ 1 m m`1p ÿm`1 m 1 q“ 1 ď c1 νpCrim`1, im, . . . , i1sq i ,...,i ,c i ,i ,...,i ,x γ 1 m m`1p ÿm`1 m 1 q“ 1 ˜ ď c1ν tπu˜ :u ˜ P Uim`1 and vispu˜q P O2R1 po, γoqu

~~1 PS´ ¯ ď c2µo o pO2R1 po, γoqq~~

~~1 ´δ infpτqm ď c3e as required.~~

4.3.4. Decay estimates for convolutions. We will now use the cylinder estimates and the expansion machinery to provide technical estimates on the L2 norm of certain convolutions. This is the last preparatory step before we begin the proof of Theorem 4.3.6 in earnest.

~~We observe that the set~~

~~S :“ t˘cpxq, ˘cpxq´1 P Γ: x P Σ`u~~

~~is a ﬁnite symmetric subset of Γ.~~

~~Lemma 4.3.17. The set S generates Γ.~~

~~Proof. By our assumption, the projection ppΓq is a torsion-free convex cocompact subgroups~~

~~of PSL2pRq, and hence it is a classical Schottky group by [13]. Therefore the Dirichlet~~

~~domain D for pppΓq, oq is the common exterior of a ﬁnitely many disks Di, i “ 1, ¨ ¨ ¨ , 2`,~~

2 which meets BpH q perpendicularly and whose closures are pairwise disjoint. It is now clear from the deﬁnition of the cocycle that tcpxqu contains all γ P ppΓq such that D X γpDq is non-empty, and hence contains a generating set for ppΓq.

~~Notation 4.3.18. For m P N, we write Bmpeq Ă Γ for the ball of radius m around the identity e in the word metric deﬁned by S. 56~~

Notation 4.3.19. We write nq for the integer part of log q. There exists d0 ą 3 such that for any m ď 1 log q, the ball B peq injects to SL pq1q whenever q1|q, q1 ą q1{10. For each q d0 mq 2 q we ﬁx a choice nq nq ă mq ă 2d0 d0 and denote rq “ nq ´ mq, so that

~~pd0 ´ 1qnq pd0 ´ 1{2qnq ă rq ă . d0 d0 Notation 4.3.20. For each element i of the alphabet deﬁning Σ, we choose an element~~

~~ωpiq P Σˆ ` such that the concatenation pi, ωpiqq is admissible.~~

For γ P SL2pqq, we write δγ for the dirac measure at γ. Given real numbers a, b, an ˆ ` element x P Σ , and an admissible sequence pinq , ¨ ¨ ¨ , imq`1q, we deﬁne a complex valued a,b measure µ on SL2pqq by x,pinq ,¨¨¨ ,imq`1q

~~paq a,b pfnq `ibτnq qpinq ,...,i1,xq µ :“ e δc pi ,...,i ,xq. (4.3.21) x,pinq ,¨¨¨ ,imq`1q mq`1 mq`1 1 i ,...,i 1 ÿmq That is, for γ P SL2pqq,~~

µa,b pγq “ x,pinq ,¨¨¨ ,imq`1q paq pfnq `ibτnq qpinq ,...,i1,xq te : cpimq`1, imq q ¨ ¨ ¨ cpi2, i1qcpi1, x0q “ γu. i ,...,i 1 ÿmq Our ﬁrst goal in this subsection is to prove the following proposition, which is essential to ˆ ˜ prove bounds on the supremum norm }Mab,qH}8.

~~Proposition 4.3.22. Let~~

~~µ :“ µa,b x,pinq ,¨¨¨ ,imq`1q and~~

~~paq B “ Ba :“ cefrq pinq ,...,imq`1,ωqeηθ , inq ,...,imq`1~~

with c ą 1 the constant from Lemma 4.3.13, ω “ ωpimq`1q, and ηθ as in (4.3.3). There ` exist constants κ ą 0, a0 ą 0, q1 ą 1,C ą 1 such that for any x P Σˆ , for any q ą q1, q |a| ă a0, |b| ď b0, and for all φ P Eq ,

~~´κ ||µ ˚ φ||2 ď BCq ||φ||2.~~

~~The constants C, κ may be chosen independent of q, x, a, b, and inq , . . . , imq`1. 57~~

Proof. The idea is to apply the `2 ﬂattening lemma 4.3.9. For ease of notation, we will ﬁx n “ nq and r “ rq throughout this proof. We will assume that a0 is small enough that we may apply Lemma 4.3.13.

~~Claim 1: We have the following bound~~

~~||µ||1 ď B. (4.3.23)~~

~~We ﬁrst observe that~~

~~paq paq paq fn pin, . . . , i1, xq “ fr pin, . . . , i1, xq ` fm pim, . . . , i1, xq~~

~~and~~

paq paq |fr pin, . . . , i1, xq ´ fr pin, . . . , im`1, ωpim`1qq| r´1 paq paq ď |f pin´j, . . . , im`1, ωpim`1qq ´ f pin´j, . . . , i1, xq| j“0 ÿ r´1 θr´1´j Lip f paq η . ď dθ p q ď θ j“0 ÿ Using the triangle inequality, we deduce

~~paq fr pin, . . . , i1, xq~~

~~paq paq paq ď fr pin, . . . , im`1, ωpim`1qq ` |fr pin, . . . , i1, xq ´ fr pin, . . . , im`1, ωpim`1qq|~~

~~paq ď fr pin, . . . , im`1, ωpim`1qq ` ηθ.~~

~~We therefore have~~

~~paq fn pin,...,i1,xq ||µ||1 ď e i1,...,im ÿ paq paq ď efr pin,...,im`1,ωpim`1qqefm pim,...,i1,xqeηθ i ,...,i 1 ÿ m ď B.~~

~~Claim 2: For some κ1 ą 0,~~

~~´κ1 ||µ||8 ď q B. (4.3.24) 58~~

~~Using the bound on m ă n{d0, it suﬃces to bound~~

~~ď Bq´κ˜~~

~~so long as we choose a0 is small enough that~~

~~mp||τ|| |a|`| log λ |q 1 δ infpτq max e 8 a ď e 3 , |a|ăa0 and κ1 ą 0 is chosen such that (recalling m — log q{d0)~~

~~κ 1 δm infpτq q 1 ď e 3~~

~~1 1{3κ and q ą q1 ą pc q 1 . Claim 3: We have~~

~~1 ´κ |||πq1 pµq|||8 ă Bpq q (4.3.25) for all q1|q with q1 ą q1{10 for some κ ą 0.~~

~~1 Choose such a q , and let Γ0 ă Γ be a subgroup such that the projection πppΓ0q is a~~

~~1 proper subgroup of SL2ppq for each divisor p|q . As in [6] (see also Lemma 5.5 of [32] for more details), we know that #paΓ0XBmpeqq grows sub-exponentially in m, and in particular we have~~

κ1{2 #paΓ0 X Bmpeqq “ Opq q, with κ1 as in Claim (2). Claim (3) now follows from Claim (2). By Claims (1) and (3), we have now veriﬁed the conditions of the ﬂattening lemma (Lemma 4.3.9). We therefore apply it to obtain

~~´κ1 ||µ ˚ φ||2 ď BCq ||φ||2 for q large. 59~~

~~The following bound, which will be useful in a number of places, follows from direct calculation.~~

~~` Lemma 4.3.26. There is c˜ ą 0 such that, for any x, y P Σ with dθpx, yq ă 1, any admissible sequence pinq , . . . , i1, xq, and any |a| ă 1, |b| ă b0, we have~~

paq paq pfn ìbτnqpin,...,i1,yq´pfn ìbτnqpin,...,i1,xqq 1 ´ e ď c˜ ¨ dθpx, yq. ˇ ˇ ˇ ˇ Remark It is inˇ this type of bound that large values of |b| causeˇ problems. This character- izes the difference between Dolgopyat’s approach and that of Bourgain-Gamburd-Sarnak.

~~Before moving on we will establish another proposition; this one will furnish Lipschitz~~

~~` bounds on Mˆ ab,qH˜ . For real numbers a, b, for x, y P Σˆ with dθpx, yq ď θ, and a sequence~~

~~inq , . . . , imq`1, we deﬁne~~

~~a,b µ1 “ µ1p q (4.3.27) x,y,imq`1,...,inq paq paq epfnq `ibτnq qpinq ,...,i1,xq epfnq `ibτnq qpinq ,...,i1,yq δ . “ ´ cmq`1pimq`1,...,i1,xq i ,...,i 1 ÿmq ´ ¯~~

~~Proposition 4.3.28. As usual, we write n, m, r for nq, mq, rq. Let~~

~~a,b µ1 “ µ1p q (4.3.29) x,y,im`1,...,in~~

~~and~~

~~B1 B1 “ a,im`1,...,in~~

~~paq :“ cce˜ ηθ efr pin,...,im`1,ωpim`1qq~~

~~with c ą 1 the constant from Lemma 4.3.13. There exist constants κ ą 0, a0 ą 0, q1 ą~~

~~` 1,C ą 1 such that for any x, y P Σˆ with dθpx, yq ă 1 and any q ą q1, |a| ă a0, |b| ď b0~~

~~1 1 1 ´κ ||µ ˚ φ||2 ď B C q ||φ||2dθpx, yq~~

~~q for all φ P Eq . The constants C, κ may be chosen independent of q, x, a, b, and in, . . . , im`1.~~

~~Proof. Claim 1: A calculation similar to that for (4.3.23) yields~~

~~1 1 ||µ ||1 ď B dθpx, yq. 60~~

~~1 1 ´κ Claim 2: There exists κ ą 0 with ||µ ||8 ă B q . For any γ P SL2pqq, we estimate~~

paq paq |µ1|pγq “ epfn ìbτnqnpin,...,i1,xq ´ epfn ìbτnqnpin,...,i1,yq ˇ ˇ ˇi1,...,im:cm`1pim`1,...,i1,xq“γ ˇ ˇ ÿ ˇ ˇ paq paq paq ˇ ˇ fn pin,...,i1,xq pfn ìbτnqpin,...,i1,yq´pfn ìbτnqpin,...,i1,xq ˇ ď ˇe 1 ´ e ˇ i ,...,i :c i ,...,i ,x γ 1 m m`1ÿp m`1 1 q“ ˇ ˇ ˇ paq ˇ ˇ fn pin,...,i1,xq ˇ ď cd˜ θpx, yq e by Lemma 4.3.26 i ,...,i :c i ,...,i ,x γ 1 m m`1ÿp m`1 1 q“ The same argument as used in claim 2 of Proposition 4.3.22 now yields

~~1 1 ´κ ||µ ||8 ă B q~~

~~as expected. Claim 3: An argument similar to the one leading to claim 3 in the proof of Proposition 4.3.22 gives that~~

~~1 1 1 ´κ 1 1 1{10 |||πq1 pµ q|||8 ď B dθpx, yqpq q for q |q, q ą q .~~

~~The proposition now follows from the `2 ﬂattening lemma as in the proof of Proposition 4.3.22~~

4.3.5. Supremum bounds and Lipschitz bounds. The purpose of all the estimates in the last two subsections is to provide bounds on the congruence transfer operators. We’ll need to bound both the supremum norms and the Lipschitz constants. Start with the ˆ n ˜ supremum norm. We observe that pMab,qHqpx, gq has a good approximation by an appropriate sum of the convolutions, and use this fact, together with the convolution estimates in ˆ n ˜ propositions 4.3.22 and 4.3.28 to estimate the supremum and Lipschitz norms of pMab,qHq. ˜ ˜q For any q, for H P Eq , and a sequence inq , . . . , imq`1, deﬁne the function φ on SL2pqq by

~~φpgq “ φ ˜ pgq (4.3.30) H,pinq ,...,imq`1q : H˜ i , . . . , i , ω i , gc´1 i , . . . , i , ω i . (4.3.31) “ p nq mq`1 p mq`1qq rq´1p nq mq`1 p mq`1qqq~~

~~Note that ˜ q |φ| ď ||H||8 and φ P Eq . 61~~

~~Lemma 4.3.32. There exist C˜ ą 1 and a0 ą 0 such that the following holds for any~~

~~` q q, |a| ă a0, |b| ď b0, x P Σˆ , and any H˜ P E˜q :~~

~~ˆ nq ˜ a,b pMab,qHqpx, ¨q ´ µ ˚ φH,˜ i , ,i p¨q ˇ x,pinq ,¨¨¨ ,imq`1q p nq ¨¨¨ mq`1q ˇ ˇ imq`1,...,inq ˇ ˇ ÿ ˇ ˇ C˜ Lip H˜ θrˇ. ˇ ď dθ p q ˇ ˇ ˇ~~

~~Proof. Fix q, x, and H˜ and write n “ nq, r “ rq, m “ mq. Choose |a| ă a0, the constant from Lemma 4.3.13. For a sequence in, . . . , im`1, set~~

~~φpgq “ φ ˜ pgq. (4.3.33) H,pin,...,im`1q~~

~~Choose |a| ă a0, the constant from Lemma 4.3.13. We observe, as a consequence of the deﬁnitions and of Lemma 4.3.11, that~~

µa,b ˚ φ pgq x,pin,¨¨¨ ,im`1q pin,¨¨¨ ,im`1q i ,...,i m`ÿ1 n paq pfn `ibτnqpin,...,i1,xq ˜ ´1 “ e Hppin, . . . , im`1, ωpim`1qq, gcn pin, . . . , i1, xqq. i ,...,i 1 ÿ n We may therefore compute

ˆ n ˜ pM Hqpx, ¨q ´ µx, i , ,i ˚ φ ˜ p¨q ˇ ab,q p n ¨¨¨ m`1q H,pin,¨¨¨ ,im`1q ˇ ˇ im`1,...,in ˇ ˇ ÿ ˇ ˇ paq ˇ ď ˇ efn pin,...,i1,xq ˇ ˇ ˇ i ,...,i 1 ÿ n ˜ ˜ Hppin, . . . , i1, xq, ¨q ´ Hppin, . . . , im`1, ωpim`1qq, ¨q ˇ´ paq ¯ˇ ď ˇ efn pin,...,i1,xq Lip pH˜ qθr´1 ˇ ˇ dθ ˇ i ,...,i 1 ÿ n c Lip H˜ θr ď dθ p q

~~where c ą 1 is the constant from Lemma 4.3.13.~~

~~ˆ nq ˜ The next lemma provides bounds on the supremum norm for Mab,qH using the de- scription in terms of convolutions we just proved together with the convolution estimate, Proposition 4.3.22.~~

~~1 Lemma 4.3.34. There exist constants a0 ą 0, q1 ą 0, κ ą 0 such that the following holds q for any |a| ă a0, |b| ď b0, q ą q1 and H˜ P E˜q :~~

~~n 1 1 ||Mˆ q H˜ || ď q´κ ||H˜ || ` Lip pH˜ qθnq{2 . ab,q 8 2 8 dθ ´ ¯ 62~~

~~Proof. We choose a0 ą 0 small and q1 large as on Lemma 4.3.13 and Proposition 4.3.22. q Consider q ą q1, |a| ă a0, |b| ď b0, and a function H˜ P E˜q . We write n for nq and r for rq.~~

~~We recall the function φ on SL2pqq as in (4.3.31). Summing over all admissible sequences ˆ ` im`1, . . . , in and applying Lemma 4.3.32 and Proposition 4.3.22 we obtain, for x P Σ ,~~

~~ˆ n ˜ |Mab,qHpxq|~~

a,b ˜ ˜ r ď µ ˚ φ ˜ p¨q ` C Lipd pHqθ ˇ x,pin,¨¨¨ ,im`1q H,pin,¨¨¨ ,im`1q ˇ θ ˇim`1,...,in ˇ ˇ ÿ ˇ ˇ ´κ a ˇ r ď q C B |φ ˜ | ` C˜ Lip pH˜ qθ ˇ in,...,im`1 H,pin,¨¨¨ ,im`1q ˇ dθ ˇ i ,...,i ˇ m`ÿ1 n paq q´κcCeηθ H˜ efr pin,...,im`1,ωpim`1qq C˜ Lip H˜ θr ď || ||8 ` dθ p q i ,...,i m`ÿ1 n c2Cq´κeηθ H˜ C˜ Lip H˜ θr ď || ||8 ` dθ p q

~~1 ´κ1 2 η ´κ by Lemma 4.3.13. We may therefore choose κ ą 0 and q1 ą 1 so that q ą 2c Ce θ q~~

~~1 rq´nq{2 ´κ and 2θ C˜ ă q for all q ą q1 and hence obtain~~

~~1 1 ||Mˆ n H˜ || ď q´κ ||H˜ || ` Lip pH˜ qθn{2 ab,q 8 2 8 dθ ´ ¯ so long as q ą q1.~~

We’d like to iterate this argument, but before we can do that we need to estimate Lip Mˆ n H˜ . The proof of the next lemma is similar to the proof of the last one, though dθ p ab,q q slightly longer.

~~1 Lemma 4.3.35. There exist C ą 0, q1 ą 0, κ ą 0, a0 ą 0 such that for all |a| ă a0, |b| ď q b0, q ą q1, and H˜ P E˜q , we have~~

~~n 1 1 Lip pMˆ q H˜ q ď q´κ ||H˜ || ` Lip pH˜ qθnq{2 . (4.3.36) dθ ab,q 2 8 dθ ´ ¯~~

~~Proof. Again, we choose a0 ą 0 small and q1 large as on Lemma 4.3.13 and propositions q 4.3.22, 4.3.28. Consider q ą q1, |a| ă a0, |b| ď b0, and a function H˜ P E˜q . We write n for~~

~~` l nq and r for rq. For x, y P Σˆ with xi “ yi for all i ď l (that is, with dθpx, yq ď θ ă 1q we 63 have~~

ˆ n ˜ ˆ n ˜ Mab,qHpx, gq ´ Mab,qHpy, gq ˇ paq ˇ ď ˇ efn pin,...,i1,xq ˇ ˇ ˇ i ,...,i 1 ÿ n ˜ ´1 ˜ ´1 |Hppin, . . . , i1, xq, gcn pin, . . . , i1, xqq ´ Hppin, . . . , i1, yq, gcn pin, . . . , i1, yqq|

~~paq paq ` epfn `ibτnqpin,...,i1,xq ´ epfn `ibτnqpin,...,i1,yq ˇ˜ ¸ ˇ i1,...,in ˇ ÿ H˜ˇ i , . . . , i , y , gc´1 i , . . . , i , y ˇpp n 1 q n p n 1 qq ˇ :“ W ` V. ˇ ˇ~~

~~The ﬁrst term W is bounded as~~

~~W c Lip H˜ θnd x, y (4.3.37) ď dθ p q θp q by Lemma 4.3.13. We estimate the other term as~~

paq paq V ď epfn ìbτnqpin,...,i1,xq ´ epfn ìbτnqpin,...,i1,yq ˇ ˇi1,...,in ´ ¯ ˇ ÿ ˇH˜ i , . . . , i , ω i , gc´1 i , . . . , i , x ˇ pp n m`1 p m`1qq n p n 1 qq paq ˇ paq `θr´1 Lip pH˜ q epfn ìbτnqpin,...,i1,xq ´ eˇpfn ìbτnqpin,...,i1,yq dθ ˇ i1,...,in ˇ ˇ ÿ ˇ ˇ :“ L ` K. ˇ ˇ

~~Next address K:~~

paq paq K θr´1 Lip H˜ epfn ìbτnqpin,...,i1,xq epfn ìbτnqpin,...,i1,yq “ dθ p q ´ i ,...,i 1 ÿ n ˇ ˇ ˇ paq paq paqˇ θr´1 Lip H˜ eˇ fn pin,...,i1,xq 1 epfn ìbτnqpin,...,i1,yq´pfn ˇìbτnqpin,...,i1,xqq ď dθ p q ´ i ,...,i 1 ÿ n ˇ ˇ paq ˇ ˇ cθ˜ r´1d x, y Lip H˜ efn pˇin,...,i1,xq ˇ ď θp q dθ p q i ,...,i 1 ÿ n by Lemma 4.3.26. A final application of Lemma 4.3.13 then gives

~~K ccθ˜ r´1 Lip H˜ d x, y . (4.3.38) ď dθ p q θp q~~

~~The bound on L uses the `2 ﬂattening lemma once again. We observe that~~

~~1a,b L “ µ ˚ φ ˜ ˇ x,y,i1,...,im Hpin,...,im`1qˇ ˇin,...,im`1 ˇ ˇ ÿ ˇ ˇ ˇ ˇ ˇ ˇ ˇ 64 for µ1, φ as in (4.3.29), (4.3.31) respectively. Proposition 4.3.28 then gives~~

~~1 µ1 φ C1q´κ B1 H˜ d x, y , | ˚ | ď a,im`1,...,in || ||8 θp q~~

~~and summation over in, . . . , im`1 yields~~

2 ´κ1 ˜ L ď C q dθpx, yq||H||8 (4.3.39) for an appropriately chosen constant C2 (more precisely, C2 “ c2cC˜ 1eηθ will do). Putting together the equations (4.3.37), (4.3.38), (4.3.39), we see that

~~1 Lip Mˆ n H˜ Cq˜ ´κ H˜ Lip H˜ θn{2 . (4.3.40) dθ p ab,q q ď || ||8 ` dθ p q ´ ¯ ˜ 2 1 2 ´κ2 1 for an appropriate constant C. Now choose κ “ κ {2 and q1 large enough that C q ă 2~~

~~for all q ą q1. Then~~

~~1 2 Lip pMˆ n H˜ q ď q´κ ||H˜ || ` Lip pH˜ qθn{2 . (4.3.41) dθ ab,q 2 8 dθ ´ ¯~~

~~Proof of Theorem 4.3.6 Combining Lemmas 4.3.34 and 4.3.35, we obtain that for some~~

~~κ1 ą 0,~~

~~1 Mˆ n H˜ θn{2 Lip Mˆ n H˜ q´κ H˜ θn{2 Lip H˜ || ab,q ||8 ` dθ p ab,q q ď p|| ||8 ` dθ p qq where n “ nq. Iterating, we obtain that for any l P N,~~

~~Mˆ ln H˜ θn{2 Lip Mˆ ln H˜ || ab,q ||8 ` dθ p ab.q q~~

~~1 q´lκ H˜ θn{2 Lip H˜ . ď p|| ||8 ` dθ p qq~~

~~It follows that ˆ ln ˜ ˆ ln ˜ ´lκ1 ˜ }Mab,qH}2 ď ||Mab,qH||8 ď q ||H||dθ as desired.~~

~~4.4 Uniform mixing of the BMS measure and the Haar measure~~

~~BMS We assume that Γ ă SL2pZq is convex cocompact. For each q P N, we denote by mq the BMS measure on ΓpqqzG induced bym ˜ and normalized so that its total mass is # SL2pqq. 65~~

4.4.1. Uniform exponential mixing Our aim in this subsection is to prove Theorem 3.1.2 using Theorem 4.1.25 on spectral bounds for the transfer operators. Although this argument is similar to that contained [17] and [1], we shall include it in order to understand the dependence of the implied constants on the level q. First we establish some more

~~notation. We ﬁx q such that ΓpqqzΓ “ SL2pqq. We recall the equivalence relation pu, tq „~~

~~pσu, t ´ τpuqq on Σ ˆ R and the suspension space~~

~~τ Σ :“ Σ ˆ R{„ .~~

~~Deﬁnition Similarly, we write~~

~~ˆ q,τ ˆ ` U :“ U ˆ SL2pqq ˆ R {pu, γ, t ` τpuqq „ pσˆpuq, γcpuq, tq. (4.4.1)~~

~~ˆ q,τ For a function φ : U Ñ C, we say φ P B0 if }φ}B0 ă 8 where~~

~~||φ||B0 :“ ||φ||8` 1 1 |φpu, γ, sq ´ φpu , γ, s q| 1 1 1 supt : u ‰ u , γ P SL2pqq, s P r0, τpuqq, s P r0, τpu qqu. dpu, u1q ` |s ´ s1|~~

~~We also say φ P B1 if }φ}B1 ă 8 where~~

~~ˆ ||φ||B1 :“ ||φ||8 ` suptVar0,τpuqpt ÞÑ φpu, γ, tqq : u P U, γ P SL2pqqu.~~

ˆ q,τ ˆ ˆ For a bounded measurable function φ : U Ñ C, we deﬁne the function φξ on UˆSL2pqq by τpuq ˆ ´ξt φξpu, γq :“ φpu, γ, tqe dt; ż0 we will sometimes regard this as a vector valued function on Uˆ. The following lemma can be easily checked.

~~ˆ τ ˆ ˆ SL2pqq Lemma 4.4.2. If ψ P B0 with γPΓ ψpu, γ, sq “ 0 for all pu, sq P U , then ψξ P WpU, C q when considered as a vector valuedř function.~~

~~For functions φ P B1 and ψ P B0, we deﬁne the correlation function:~~

~~τpuq ρ˜φ,ψptq :“ φpu, γ, s ` tqψpu, γ, sqdsdνpuq. (4.4.3) ˆ γ SL q U 0 P ÿ2p q ż ż 66~~

~~In order to establish an exponential decay forρ ˜φ,ψptq for a suitable class of functions φ, ψ, we consider its Laplace transform and relate it with the transfer operators. We decompose~~

~~ρ˜φ,ψptq as~~

τpuq ρ˜φ,ψptq “ φpu, γ, s ` tqψpu, γ, sqdsdνpuq ˆ γ SL q U maxp0,τpuq´tq P ÿ2p q ż ż maxp0,τpuq´tq ` φpu, γ, s ` tqψpu, γ, sqdsdνpuq ˆ γ SL q U 0 P ÿ2p q ż ż :“ ρφ,ψptq ` ρ¯φ,ψptq.

~~The reason for this decomposition is that the Laplace transform of ρφ,ψptq can be expressed neatly in terms of transfer operators (see Lemma 4.4.4 below). More importantly, the~~

~~Laplace transform of ρφ,ψ has better decay properties than the Laplace transform ofρ ˜φ,ψ; this is needed when we apply the inverse Laplace transform at the end of the argument.~~

~~Moreover, sinceρ ˜φ,ψptq “ ρφ,ψptq for all t ě sup τ, the exponential decay ofρ ˜φ,ψptq follows from that of ρφ,ψptq.~~

~~So, consider the Laplace transformρ ˆ of ρ: for ξ P C,~~

~~8 ´ξt ρˆφ,ψpξq “ e ρφ,ψptqdt. ż0~~

~~For the rest of the section, we shall use the notation~~

~~ξ “ a ´ ib.~~

~~The ﬁrst task is to writeρ ˆpξq in terms of the transfer operators:~~

~~Lemma 4.4.4. For φ P B1 and ψ P B0 and~~

8 k ˆ ˆ k ˆ ρˆφ,ψpξq “ λa φξpuq ¨ Mab,qψ´ξpuqdνpuq ˆ k“1 U ÿ ż where λa is the lead eigenvalue of L´pδ`aqτ as in Section 2. The right hand side should be SL q understood as an inner product between two vectors in C 2p q. 67

~~Proof. We calculate~~

τpuq 8 ´ξt ρˆφ,ψpξq “ e φpu, γ, s ` tqψpu, γ, aqdtdsdνpuq ˆ γ SL q U s“0 τpuq´s P ÿ2p q ż ż ż τpuq 8 “ e´ξpt´sqφpu, γ, tqψpu, γ, sqdtdsdνpuq ˆ γ SL q U 0 τpuq P ÿ2p q ż ż ż 8 τpuq τk`1puq “ e´ξpt´sqφpu, γ, tqψpu, γ, sqdtdsdνpuq “ ˆ γ SL q k“1 U 0 τkpuq P ÿ2p q ÿ ż ż ż 8 τpuq τpσˆkuq ´ξpt`τkpuq´sq k e φpσˆ puq, γckpuq, tqψpu, γ, sqdtdsdνpuq ˆ γ SL q k“1 U 0 0 P ÿ2p q ÿ ż ż ż 8 ´ξτkpuq ˆ k ˆ “ e φξpσˆ puq, γckpuqqψ´ξpu, γqdνpuq ˆ γ SL q k“1 U P ÿ2p q ÿ ż 8 k ˆ ˆ k ˆ “ λa φξpuq ¨ Mab,qψ´ξpuqdνpuq ˆ k“1 U ÿ ż ˆ˚ using the fact that L00pνq “ ν. ? 2 # SL pqqe|a| suppτq||φ|| ˆ ˆ 2 B1 Lemma 4.4.5. If φ P B1, then ||φξ||2 ď ||φξ||8 ď maxp1,|b|q .

~~Proof. This follows from integration by parts in the ﬂow direction. ? # SL pqqe|a| suppτqp3 suppτq`Lip pτqq||ψ|| ˆ 2 d B0 Lemma 4.4.6. If ψ P B0, then ||ψξ||1,b ď maxp1,|b|q .~~

~~Proof. The trivial bound varr0,τpuqqψpu, γ, ¨q ď ||ψ||B0 suppτq provides~~

|a| suppτq 2 # SL2pqqe suppτq||ψ||B ||ψˆ || ď 0 . (4.4.7) ξ 8 max 1, b a p | |q 1 On the other hand consider any u, u P Uˆ, γ P SL2pqq, and suppose, without loss of generality, that τpu1q ě τpuq. Then

~~ˆ ˆ 1 |ψξpu, γq ´ ψξpu , γq| τpuq τpu1q ď |ψpu, γ, tq ´ ψpu1, γ, tq|e|a|tdt ` |ψpu1, γ, tq|e|a|tdt ż0 żτpuq 1 |a| suppτq ď dpu, u qe psuppτq||ψ||B0 ` Lipdpτq||ψ||8q .~~

~~Together with (4.4.7), this proves the claim.~~

~~We will now use the spectral bounds (Theorem 4.1.25) to prove a rate of decay for the correlation functions. 68~~

~~1 Proposition 4.4.8. Let a0, q0, q0 be as in Theorem 4.1.25. There exist C ą 0, η ą 0 such 1 that for all square free q with pq, q0q0q “ 1, we have~~

C ´ηt |ρ˜φ,ψptq| ď Cq ||φ||B1 ||ψ||B0 e for all φ B and ψ B satisfying ψ u, γ, s 0. P 1 P 0 γPSL2pqq p q “ ř Proof. We will establish that the Laplace transformρ ˆφ,ψ extends to an appropriate half plane and then apply the inversion formula. Lemma 4.4.4 gives 8 k ˆ ˆ k ˆ ρˆφ,ψpξq “ λa φξpuq ¨ Mab,qψ´ξpuqdνpuq, ˆ k“0 U ÿ ż for

k ˆ ˆ k ˆ k ˆ k ˆ λa φξpuq ¨ Mab,qψ´ξpuqdν ď λa||Mab,qψ´ξ||2||φξ||2 ˆ żU k C ´k ˆ ď λaCq e ||ψ´ξ||1,b||φξ||2 1 C1 ´k k C q e ď λ ||ψ||B ||φ||B , a maxp1, |b|q2 0 1 where C1 is given by Lemmas 4.4.5 and 4.4.6; this is clearly summable so long as we choose

~~a0 small enough that~~

~~{2 max λa ď e . |a|ďa0~~

~~This computation also gives that for some absolute constant C1 ą 0, C1 C1q |ρˆ pξq| ď ||ψ||B ||φ||B φ,ψ 1 ` |b|2 0 1~~

~~for all ξ with |~~

~~C t ą suppτq, and sinceρ ¯φ,ψ is bounded as q ||ψ||B0 ||φ||B1 , the result follows. 69~~

~~q,τ We can convert a function φ on ΓpqqzG to give a function φt on Uˆ as follows: for t ą 0, u P Uˆi, 0 ď s ď τpuq and γ P ΓpqqzΓ, we setu ˜ to be the lift of u to U˜, and~~

~~φtpu, γ, sq :“ φpγru,˜ y˜sat`sqdνupy˜q (4.4.10) ˜ żSi where νu is the probability measure on S˜i conditioned from the measure ν at u. For a general s ą 0, we deﬁne~~

~~k φtpu, γ, sq :“ φtpσˆ puq, γckpuq, s ´ τkpuqq (4.4.11)~~

~~k where k P N is such that 0 ď s ´ τkpuq ď τpσˆ puqq. By the equivalence relation (4.4.1), this ˆ deﬁnes φt on all of U ˆ SL2pqq ˆ Rě0.~~

~~Lemma 4.4.12. There exists C ą 0 such that, for any y˜ P S˜i, we have~~

~~´t |φpγru,˜ y˜sa2t`sq ´ φtpu, γ, s ` tq| ď Ce ||φ||C1 .~~

˜ k Proof. Let u, u,˜ y˜ P Si be as above. Choose k P N such that 0 ď t ` s ´ τkpuq ď τpσ uq, and 1 k 1 1 1 1 write u “ σˆ u P Uj. Setu ˜ to be the lift of u to U˜j. If y P Sj with lifty ˜ P S˜j, then the

~~1 1 deﬁnition of the cocycle c tells us that both γru,˜ y˜saτkpuq and γckpuqru˜ , y˜ s lie in the stable leaf of γckpuqR˜j Ă G. It follows that for some c1 ą 0,~~

~~1 1 ´p2t`s´τkpuqq ´t dpγru,˜ y˜sa2t`s, γckpuqru˜ , y˜ sa2t`s´τkpuqq ď c1e ď c1e , and so that~~

~~1 1 ´t 1 |φpγru,˜ y˜sa2t`sq ´ φpγckpuqru˜ , y˜ sa2t`s´τkpuqq| ď c1||φ||C e .~~

~~1 Integrating this inequality overy ˜ P Sj and using (4.4.11) which gives~~

~~1 1 φpγckpuqru˜ , y˜ sa2t`s´τ puqqdνu “ φtpu, γ, s ` tq ˜ k żSj this gives the required result.~~

~~We therefore have the following lemma (cf. [1, Lemma 8.2])~~

~~Lemma 4.4.13. There are constants η ą 0,C ą 0 independent of ψ, φ, q such that~~

~~BMS ρ˜φt,ψ0 ptq ´ηt pφ ˝ a2tq ¨ ψ dmq ´ ă C ¨ # SL2pqq ¨ ||φ||C1 ||ψ||8e ˇ ΓpqqzG νpτq ˇ ˇż ˇ ˇ ˇ for all φ, ψˇ C1 Γ q G . ˇ ˇ P p p qz q ˇ 70~~

~~Proof of Theorem 3.1.2 . We assume that mBMSpΓzGq “ 1 without loss of generality, so~~

~~BMS 1 that the total mass of mq is equal to # SL2pqq. Fix q with pq, q0q0q “ 1 and compactly supported functions ψ, φ P C1pΓpqqzGq. We write~~

~~ψ “ ψ1 ` ψ2~~

1 2 2 where ψ is (left) Γ-invariant, and ψ satisﬁes γPΓpqqzΓ ψ pγxq “ 0 for all x P ΓpqqzG. Exponential mixing of ψ1 (with constant independentř of q) follows from the bounds established in section 4.2 together with the complex RPF theorem, as was carried out in the work of Dolgopyat and Stoyanov [53]. So we can and shall assume that ψ “ ψ2, so that

~~γPΓpqqzΓ ψpγxq “ 0 for all x P ΓpqqzG.~~

~~ř We consider the functions φt, ψ0 as deﬁned in (4.4.10); note that ψ0 satisﬁes γPΓ ψ0pu, γ, sq “~~

~~0 and that ||ψ0||B0 ! ||ψ||C1 . We also need to bound ||φt||B1 . It’s clear that sup |řφt| ď ||φ||C1 . On the other hand, we know that, for ﬁxed u, and s such that pu, sq is not of the form~~

1 pu , 0q, φtpu, sq is diﬀerentiable in the ﬂow direction with derivative bounded by ||φ||C1 . On sup τ 1 the other hand there are at most inf τ ` 1 values of s such that pu, sq „ pu , 0q. Each of

~~these may be a discontinuity, but each jump is at most 2||φ||C1 . We can therefore bound the variation as~~

~~sup τ var ps Ñ φ pu, γ, sqq ď τpuq ` 2 ` 1 ||φ|| 1 . r0,τpuqq t inf τ C ´ ´ ¯¯ In other words~~

~~||φt||B1 ! ||φ||C1 .~~

~~Now calculate; for any t ą 0~~

BMS φpga2tqψpgqdm ˇ ΓpqqzG ˇ ˇż ˇ ˇ ˇ ρ˜φt,ψ0 ptq ´ηt ˇď ` Cp# SL2pqqq||ˇ φ|| 1 ||ψ|| 0 e ˇ νpτq ˇ C C ˇ ˇ ˇ 1 C1 ˇ ´η1t Cˇ q ψˇ φ φ 1 ψ 0 e ď ˇ p|| ˇ0||B0 || t||B1 ` || ||C || ||C q 2 C2 ´η1t ď C q ||φ||C1 ||ψ||C0 e

~~for some C1,C2, η1 ą 0, by Lemma 4.4.13 and Proposition 4.4.8. 71~~

~~4.4.2. Exponential decay of the matrix coeﬃcients Let Γ be a geometrically ﬁnite~~

~~BR BR subgroup of PSL2pRq. We begin by recalling the deﬁnitions of measures m , m ˚ and mHaar. Similar to the deﬁnition of the BMS measure~~

~~BMS δβ ` po,uq δβ ´ po,uq PS ` PS ´ dm˜ puq “ e u e u dµo pu qdµo pu qds~~

~~BR BR BR˚ BR˚ Haar given in section 2, the measuresm ˜ “ m˜ Γ ,m ˜ “ m˜ Γ andm ˜ on PSL2pRq are deﬁned as follows:~~

~~BR β ` po,uq δβ ´ po,uq ` PS ´ dm˜ puq “ e u e u dmopu qdµo pu qds;~~

~~BR˚ δβ ` po,uq β ´ po,uq ´ PS ` dm˜ puq “ e u e u dmopu qdµo pu qds;~~

~~Haar β ` po,uq β ´ po,uq ` ´ dm˜ puq “ e u e u dmopu qdmopu qds~~

~~2 where mo is the invariant measure on BpH q under the stabilizer of o. These measures are all left Γ-invariant and induce measures on ΓzG, which we will denote by mBR, mBR˚ , mHaar respectively.~~

~~Let 1 0 1 s N “ tns :“ : s P Ru and H “ ths :“ : s P Ru. ¨s 1˛ ¨0 1˛ ˝ ‚ ˝ ‚ For g P G, denote by g˘ the forward and backward end points of the geodesic determined by g and set~~

~~` ´ αpg, ΛpΓqq :“ inft|s| : pgnsq P ΛpΓqu ` inft|s| : pghsq P ΛpΓqu ` 1.~~

~~It follows from the continuity of the visual map that for any compact subset Q Ă G,~~

~~αpQ, ΛpΓqq :“ sup αpg, ΛpΓqq ă 8.~~

If Γ1 is a normal subgroup of Γ of ﬁnite index, then ΛpΓq “ ΛpΓ1q, and hence αpQ, ΛpΓqq “ αpQ, ΛpΓ1qq. Therefore the following theorem implies that Theorem 3.1.4 can be deduced from Theorem 3.1.2; note that even though we need the following theorem only for Γ convex cocompact in this chapter, we record it for a general geometrically ﬁnite group Γ of G for future reference. Let π : G Ñ ΓzG be the canonical projection. 72

~~Theorem 4.4.14. Let Q Ă G be a compact subset. Suppose that there exist constants~~

~~1 cΓ ą 0 and ηΓ ą 0 such that for any Ψ, Φ P C pΓzGq supported on πpQq,~~

BMS BMS BMS m pΨq¨m pΦq ´ηΓt Ψ ga Φ g dm O c Ψ 1 Φ 1 e (4.4.15) p tq p q “ mBMSpΓzGq ` p Γ ¨ || ||C || ||C ¨ q żΓzG where the implied constant depends only on Q. Then for any Ψ, Φ P C1pΓzGq supported on πpQq, as t Ñ `8,

p1´δqt Haar e ΨpgatqΦpgqdm żΓzG mBRpΨq¨mBR˚ pΦq ´η1 t O c Ψ 1 Φ 1 e Γ (4.4.16) “ mBMSpΓzGq ` p Γ ¨ || ||C || ||C ¨ q where η1 “ ηΓ and the implied constant depends only on Q and αpQ, ΛpΓqq. Γ 8`2ηΓ

The rest of this section is devoted to the proof of this theorem. The proof involves eﬀectivizing the original argument of Roblin [50], extended in [52], [40], [36], while making the dependence of the implied constant on the relevant functions precise.

~~For ą 0 and a subset S of G, S denotes the set ts P S : dps, eq ď u. Let~~

~~P :“ HA.~~

~~Then the sets B :“ PN, ą 0 form a basis of neighborhoods of e in G.~~

~~For g P PSL2pRq, we deﬁne measures on gN:~~

~~Leb βpgnq` po,gnq ` dµ˜gN pgnq “ e dmopgn q;~~

~~PS δβpgnq` po,gnq PS ` dµ˜gN pgnq “ e dµo opgn q.~~

~~If x “ rgs P ΓzG, for a compact subset N0 of N such that gN0 injects to ΓzG, and for~~

Leb PS a function ψ on xN0, we write dµxN pψq and dµxN pψq for the push-forward of the above Leb measures to xN0 via the isomorphism gN0 with xN0. The measure dµ˜gN pgnq is simply the Leb Haar measure on N, and hence we write dn “ dµ˜gN pgnq. The quasi-product structure ofm ˜ BMS is a key ingredient in the arguments below: for

~~Ψ P C0pGq supported on gB for all ą 0 small,~~

~~BMS PS m˜ pΨq “ Ψpgpnqdµ˜gpN pgpnqdνgP pgpq żgP żgpN 73~~

~~δβpgpq´ po,gpq PS ´ where dνgP pgpq “ e dµo opgp qds for s “ βgp´ po, gpq. In the rest of this section, we ﬁx a compact subset Q of G, and assume that the hy-~~

~~potheses of Theorem 4.4.14 are satisﬁed for functions supported in πpQq. Let 20 ą 0 be the injectivity radius of πpQq. Fix x “ rgs P πpQq and functions Ψ, Φ P C1pΓzGq which are~~

~~supported in xB0{2.~~

Proposition 4.4.17. Fix y xP and put φ : Φ C1 yN . Then for t 1, P 0 “ |yN0 P p 0 q ą µPS pφq PS yN BMS ´η1t ΨpynatqφpynqdµyN pynq “ BMS m pΨq ` OpcΓ}Ψ}C1 }φ}C1 e q yN |m | ż 0 where η1 “ ηΓ{p4 ` ηΓq and the implied constant depends only on Q and αpQ, ΛpΓqq.

´1 Proof. Set R0 :“ αpy, ΛpΓqq`2. For a suﬃciently small P p0, 1q, if we set t0 :“ logpR0 q, y0 “ yat0 , then νpy0Pq ą 0. Hence we may choose a smooth positive function ρ supported ´3 1 : on y0P such that νpρq “ 1 and that }ρ}C1 ! . Deﬁne a C -function Φ supported on y0PN0 as follows:

~~´δt0´δβ ` pnp,pnq : np Φ py0pnq :“ e φpy0npa´t0 qρpy0pq~~

~~´1 BMS : PS where np P N is the unique element such that p np P nP . We have m pΦ q “ µyN pφq. Now by the hypothesis of Theorem 4.4.14, we have~~

PS : ΨpynatqφpynqdµyN pynq “ p1 ` Opqqxat´t0 Ψ, Φ ymBMS yN ż 0 µPS pφq yN BMS ´3 ´ηpt´t0q “ p1 ` Opqq BMS m pΨq ` OpcΓ e q ˜|m | ¸ µPS φ yN p q BMS η ´η´3 ´ηt “ m pΨq ` Op ` cΓR e q |mBMS| 0 where the implied constant depends only on the C1-norms of Ψ and φ and Q. By taking

´ηt{p4`ηq “ e and by setting η1 :“ η{p4 ` ηq, we obtain µPS pφq PS yN BMS η ´η1t ΨpynatqφpynqdµyN pynq “ BMS m pΨq ` OpcΓR0e q. yN |m | ż 0 Since R0 is bounded above by αpQ, ΛpΓqq, this proves the claim.

Proposition 4.4.18. Keeping the same notation as in Proposition 4.4.17, we have µPS pφq p1´δqt yN BR ´η1t{2 e Ψpynatqφpynqdn “ BMS m pΨq ` OpcΓ}Ψ}C1 }φ}C1 e q yN |m | ż 0 where the implied constant depends only on Q and αpQ, ΛpΓqq. 74

~~Proof. We deduce this proposition from Proposition 4.4.17 by comparing the two integrals on the left hand sides via transversal intersections.~~

~~˘ 1 Deﬁne φ P C pyNq by~~

` 1 ´ 1 φ pynq “ sup φpynn q and φ pynq “ inf φpynn q. (4.4.19) 1 1 n PN n PN ` ` Fix R1 :“ αpQ, ΛpΓqq`1. For each p P P0 , let Np :“ tn P N : ppnq “ ns for some |s| ă R1u; PS PS then µxpN pxpNpq ą 0, and the map xp ÞÑ µxpN pxpNpq is a positive smooth function on xP . Set B1 : pN ; we may assume that the map g xg is injective on B1 by 0 0 “YpPP0 p Ñ 0 replacing 0 by a smaller number if necessary. Deﬁne the ﬁnite set

~~Pxptq :“ tp P P0 : xpn P supppφqat for some n P Npu.~~

~~1 1 Deﬁne functions ψ and Ψ supported on xP0 and xB0 respectively:~~

~~ψpxpq :“ Ψpxpnqdn and Ψ1pxpnq :“ ψpxpq for pn P B1 . µPS pxpN q 0 xpN xpN p ż 0 BMS 1 BR We then have m pΨ q “ νxP pψq “ m pΨq, and we can ﬁnd~~

~~1 1 1 1 BMS 1 BMS 1 C -approximations Ψ,´ ď Ψ ď Ψ,` such that m pΨ,˘q “ m pΨ q ` Opq, and 1 ´1 }Ψ,˘}C1 “ Op }Ψ}C1 q. The following computation holds for all small 0 ă ! 0:~~

p1´δqt e Ψpynatqφpynqdn żyN ´δt ˘ “ p1 ` Opqqe ψpxpqφ ´t pxpa tq ce 0 ´ p P t Pÿxp q 1 ˘ PS “ p1 ` Opqq Ψ pynatqφ 1 ´t pynqdµ pynq c p0`R1qe yN żyN 1 ˘ PS “ p1 ` Opqq Ψ pynatqφ 1 ´t pynqdµ pynq ,˘ c p0`R1qe yN żyN mBRpΨqµPS pφq ´t yN ´1 ´η1t 1 O O R e O c Ψ 1 φ 1 e “ p ` p q ` pp 0 ` 1q qq |mBMS| ` p Γ } }C } }C q ˆ ˙ by Proposition 4.4.17 (we refer [40] and [36] for details in this step).

~~Therefore taking “ e´η1t{2,~~

mBRpΨqµPS pφq p1´δqt yN ´η1t{2 e Ψ yna φ yn dn O c Ψ 1 φ 1 e p tq p q “ |mBMS| ` p Γ} }C } }C q żyN where the implied constant depends only on 0 and R1, and hence only on Q and αpQ, ΛpΓqq. 75

~~In order to ﬁnish the proof of Theorem 4.4.14, we ﬁrst observe that by the partition of~~

~~unity argument, it suﬃces to prove the claim for Φ and Ψ supported on xB0{2 for x P Q. We note that dmHaarppnq “ dpdn where dp is a left Haar measure on P , and hence~~

~~Haar ΨpxatqΦpxqdm pxq “ ΨpxpnatqΦpxpnqdndp. Γ G xp zP xpN ż z ż P 0 ż 0~~

~~Hence applying Propositions 4.4.17 and 4.4.18 for each y “ xp P xP0 , we deduce that~~

p1´δqt Haar e ΨpxatqΦpxqdm pxq żΓzG mBRpΨqµPS pΦ| q xpN xpN0 ´η1t{2 “ ` Opc }Ψ} 1 }Φ| } 1 e q dp |mBMS| Γ C xpN0 C xp xP ż P 0 ˆ ˙ BR BR˚ m pΨqm pΦq ´η1t{2 O c Ψ 1 Φ 1 e “ |mBMS| ` p Γ} }C } }C q where the implied constant depends only on Q and αpQ, ΛpΓqq . This ﬁnishes the proof.

~~4.5 Zero-free regions for the Selberg zeta functions~~

Let Γ ă SL2pZq be as in Theorem 3.1.4 and suppose that δ ď 1{2. Let PqpT q denote the set of all primitive closed geodesics in Γpqqz PSL2pRq of length at most T and let `pCq denote the length of C. In [34], it was shown that Theorem 3.1.4 implies the following:

~~1 Theorem 4.5.1. There exist C ą 0 and 0 ą 0 such that for all square free q ě 1 with pq, q0q “ 1,~~

~~1 δT C pδ´0qT #PqpT q “ li pe q ` Opq e q (4.5.2)~~

~~x dx where lipxq “ 2 log x . ş For each q P N, set q ě 0 be the supremum of 0 ď ď δ such that t~~

~~[37], the precise relation of an exponential error term in the prime geodesic theorem and q was studied, and in particular it follows from [37, Theorem 1.4] that for 0 as in (4.5.2),~~

~~q ě 20 for each q satisfying (4.5.2). Hence Theorem 3.1.6 follows. Chapter 5~~

~~Mixing properties of frame ﬂow~~

In this chapter we discuss the question of lifting the mixing statement from geodesic to frame ﬂow. Throughout this chapter we choose G to be a simple Lie group of real rank one, as in Section 2.3, and aim to prove Theorem 3.2.1. We retain the notation for subgroups

~~N `,N ´, A, M, K in G, for the base point o P X˜ :“ G{K, for the base tangent vector~~

~~1 ` v0 P T X˜, and for the visual maps v ÞÑ v PBX˜.~~

The strategy was worked through by Brin, Gromov, and Pesin for compact manifolds. The key points in our setting are to understand the Bruhat decomposition for a rank one Lie group, and to understand the relationship between so-called Brin-Pesin moves and the limit set ΛpΓq.

~~5.1 Further properties of rank one groups~~

~~5.1.1. Properties of N ´,N `, and M It is well known that~~

~~´ ` P :“ PpRq “ MAN is the stabilizer subgroup of v0 . On the other hand the map~~

~~N ´ ÑBX,˜ n ÞÑ n´~~

~~˜ ` ´ ´ is a diﬀeomorphism onto BX ´ tv0 u, so N doesn’t stabilize v0 . We denote the inverse map by~~

~~˜ ` ´ BX ´ tv0 u Ñ N , ξ ÞÑ npξq.~~

~~76 77~~

~~` ´ Similarly N MA is the stabilizer of v0 , while~~

~~N ` ÑBX,˜ h ÞÑ h`~~

~~˜ ´ is a diﬀeomorphism onto BX ´ tv0 u, whose inverse we denote by ξ ÞÑ hpξq. Since G is a rank one group there is a root α ą 0 such that~~

~~` TeN “ n “ nα ` n2α where nα, n2α are the α and 2α root spaces for at respectively, satisfying~~

~~αt e X if X P nα, and Adat X “ (5.1.1) $ 2αt & e X if X P n2α. The following facts are straightforward.%~~

~~Fact 5.1.2. The dimension of the center of M is at most one.~~

~~Proposition 5.1.1. Suppose that Γ ă G is Zariski dense and that ξ PBX˜. Then~~

~~´ ` 1. the set thpγξq : γ P Γ, γξ ‰ v0 u is Zariski dense in N , and~~

~~` ´ 2. the set tnpγξq : γ P Γ, γξ ‰ v0 u is Zariski dense in N .~~

~~5.2 Algebraic properties of the Bruhat decomposition.~~

~~For the rest of this section, a blackboard bold character (such as G) will always refer to an~~

R-algebraic subgroup of GLnpCq. We will denote the real points of such a group by GpRq. It will be necessary to use both the Zariski topology and the Euclidean topology and to switch regularly between the two. A subscript “0” will denote the identity component of a group in the Zariski topology, so for instance G0 will be the Zariski identity component of G.A subscript “+” will mean the Euclidean identity component, e.g. GpRq` is the Euclidean identity component in the real points of G. The words open, closed, dense, compact, and so forth used without qualiﬁcation will refer to the Euclidean topology. When referring to the Zariski topology we will be explicit. All of the groups we have discussed so far are actually algebraic. More precisely, there is a Zariski connected R-algebraic subgroup G ă GLnpCq with R-rank one and G “ GpRq` 78 is the identity component in the group of real points. We denote by K the Zariski closure of K in G. Algebraicity of compact real matrix groups implies that KpRq “ K. We choose

a maximal R-split torus S ă G such that SpRq` “ A. Let M “ ZKpSq and again note that M “ MpRq ă K. Finally let N´ be the negative root group of S and N` the opposite horospherical group. Once again we have N ´ “ N´pRq and N ` “ N`pRq. Let P be the minimal R-parabolic subgroup of G containing S and N´, and let W “ N`.P Ă G. The Bruhat decomposition tells us that W is Zariski open in G and that the product map

~~` N ˆ P Ñ W is an R-isomorphism of varieties. Now let L “ ZGpSq be the centralizer of S in G. This is a Levi factor for P, so the product map~~

~~` ´ W “ N ˆ L ˆ N (5.2.1) is also an R-isomorphism of varieties. We now collect some standard facts about these decompositions.1~~

~~Lemma 5.2.1. We have~~

~~1. The group M is connected.~~

~~2. LpRq` “ M`A and L “ M0.S.~~

~~3. The product map~~

~~` ´ N ˆ M ˆ A ˆ N Ñ WpRq X G~~

~~is a diﬀeomorphism.~~

~~Remark Item 1 above is the stage at which ﬁnite covers of PSL2pRq must be handled separately.~~

~~We are now in a position to state one of the two technical observations we need from this section. Let W “ WpRq`. We write~~

~~` ´ ζ “ ζN ` ˆ ζM ˆ ζA ˆ ζN ´ : W Ñ N ˆ M ˆ A ˆ N . for the inverse to the multiplication map.~~

~~1Some additional details on the background are given in the initial arXiv version of this work. 79~~

~~Proposition 5.2.2. Suppose that Γ ă G is Zariski dense. Then ζM pΓ X W q is Zariski dense in M, and generates a dense subgroup xζM pΓ X W qy ă M.~~

~~Before proving this we need two observations.~~

~~Remark We note that:~~

~~1. the intersection M0 X S is ﬁnite;~~

~~2. let~~

~~p : M0 ˆ S Ñ L~~

~~be the product map and suppose that Γ Ă L is Zariski dense. Then the pre-image ´1 p pΓq Ă M0 ˆ S is Zariski dense.~~

Proof of Proposition 5.2.2. Suppose that Γ Ă G is Zariski dense. Then Γ X W “ Γ X W is Zariski dense in W. If we write ζL for the L component of the isomorphism W Ñ ` ´ N ˆ L ˆ N above, then we conclude that ζLpΓ X W q Ă L is Zariski dense. By Lemma 5.2, it follows that

~~˜ ´1 Γ:“ p pζLpΓ X W qq Ă M0 ˆ S~~

~~˜ is Zariski dense, where p : M0 ˆ S Ñ L is the product map. But Γ is very nearly the same as ζM ˆ ζApΓ X W q; in fact~~

~~´1 Γ˜ “ tpg ζM pγq, gζApγqq : γ P Γ X W u. gP X Mď0 S~~

~~This is a union of ﬁnitely many sets and it follows (since M0 ˆ S is irreducible) that each set~~

´1 tpg ζM pγq, gζApγqq : γ P Γ X W u is Zariski dense. In particular, the case g “ e implies that ζM pΓXW q Ă M0 is Zariski dense as required. The second statement is quicker to prove. The subgroup H ă M generated by

~~ζM pΓ X W q is Zariski dense in M. The Euclidean closure H¯ is therefore both Zariski closed (by algebraicity of compact matrix groups) and Zariski dense, so H¯ “ M. 80~~

5.2.1. Brin-Pesin moves, the transitivity group, and mixing of the frame ﬂow This section is the heart of the proof of Theorem 3.2.1. We begin by deﬁning the transitivity groups and investigating their basic properties in subsection 5.2.2. The rest of the section is then concerned with showing that the transitivity groups are dense in M ˆ A whenever Γ ă G is Zariski dense. The main result is Theorem 5.2.2.

~~5.2.2. Deﬁntion and basic properties of the transitivity group.~~

~~Deﬁnition Fix g P supppmBMSq. We will say that g1 P supppmBMSq is accessible from g if~~

~~´ ` there is a sequence hi P N Y N , i “ 1 . . . k and a γ P Γ such that~~

~~BMS gh1h2, . . . hr P supppm q for all 0 ď r ď k, and~~

~~1 γgh1h2 . . . hk “ g .~~

~~Remark For lattices the BMS measure has full support, so the ﬁrst condition can be ignored.~~

~~Remark The “move”g ˜ Ñ gh˜ for h P N ´ Y N ` is referred to as a Brin-Pesin move. We’re interested in points connected by sequences of Brin-Pesin moves.~~

~~For g P supppmBMSq we denote by~~

~~HΓpgq :“ tpm, aq P M ˆ A : gma is accessible from gu.~~

~~BMS Lemma 5.2.3. For every g P supppm q, the subset HΓpgq Ă MA is a subgroup; we will call it the transitivity group of Γ at g.~~

Remark Brin gave a related deﬁnition [12] for compact manifolds, and Flaminio-Spatzier generalized that idea to geometrically ﬁnite hyperbolic manifolds [20]; their notion corresponds to our “strong transitivity group”, which we will discuss shortly.

~~Proof. We ﬁrst check that the transitivity group contains inverses. Suppose that pm, aq P~~

~~´ ` HΓpgq. Take a sequence h1 . . . hk P N Y N and γ P Γ for ma, i.e.~~

~~BMS gh1h2, . . . hr P supppm q for all 0 ď r ď k, and 81~~

~~γgh1h2 . . . hk “ gma.~~

~~´1 ´1 ´1 ´1 The conjugate inverse sequence pmaqhk pmaq ,... pmaqh1 pmaq satisﬁes~~

~~´1 ´1 ´1 ´1 ´1 ´1 gmahk . . . hr pmaq “ γgh1 . . . hkhk . . . hr pmaq~~

~~´1 “ γgh1 . . . hr´1pmaq~~

~~BMS which is in supppm q because gh1 . . . hr´1 is. Similarly~~

~~´1 ´1 ´1 ´1 ´1 ´1 ´1 γ gmahk . . . h1 pmaq “ gh1 . . . hkhk . . . h1 pmaq~~

~~“ gpmaq´1.~~

~~Next we check that products are contained in the transitivity group. Suppose that ma~~

~~˜ ´ ` andm ˜ a˜ are both in HΓpgq, and take admissible sequences hi, hi P N Y N and γ, γ˜ P Γ such that ˜ ˜ gma “ γgh1 . . . hk and gm˜ a˜ “ γg˜ h1 ... hl.~~

~~˜ ˜ ´1 ´1 We claim that h1 ... hl, pm˜ a˜q h1m˜ a,˜ . . . pm˜ a˜q hkm˜ a˜ is a sequence for mam˜ a˜ together with γγ˜. The argument is similar to the one above.~~

~~´ ` BMS Lemma 5.2.4. If γ0 P Γ, u P N Y N Y A, and if g, γ0gu P supppm q, then HΓpgq “~~

~~HΓpγ0guq as subgroups of M ˆ A.~~

~~Proof. Suppose that h1 . . . hk and γ is a sequence for ma P HΓpgq. Then~~

~~´1 • h1, . . . hk and γ0γγ0 is a sequence for ma in HΓpγ0gq,~~

~~´1 ´1 • u , h1, . . . hk, pamq upamq and γ is a sequence for ma in HΓpguq whenever u P N ´ Y N `, and~~

~~´1 ´1 • u h1u, . . . , u hku and γ is a sequence for ma in HΓpguq whenever u P A.~~

~~Lemma 5.2.5. If m P M and g P supppmBMSq, then~~

~~´1 HΓpgmq “ m HΓpgqm. 82~~

~~´1 ´1 Proof. Suppose that h1 . . . hk and γ is a sequence form ˜ a˜ P HΓpgq. Then m h1m, . . . m hkm~~

~~´1 and γ is a sequence for m m˜ am˜ in HΓpgmq.~~

Remark We have just seen that the transitivity group HΓpgq depends very little on which g we choose. In particular, the question we are most interested in, whether HΓpgq ă MA is dense, is entirely independent of the choice of g P supppmBMSq. For this reason we will often refer to the transitivity group without reference to its base point. Note that the transitivity group does depend very much on the discrete subgroup Γ.

~~With these basics in hand we deﬁne the strong transitivity group~~

~~s HΓpgq “ pM ˆ teuq X HΓpgq~~

~~w and the weak transitivity group HΓ pgq, which is the projection πM pHΓpgqq of the transitivity group to the M coordinate. Observe that~~

~~s w HΓpgq ă HΓ pgq ă M.~~

~~To ﬁnish this subsection we will use the algebraic properties of the Bruhat decomposition to show that the transitivity group is reasonably large. Let~~

~~` ´ W “ tg P G such that g ‰ v0 u~~

~~and let~~

~~` ´ ζ “ ζN ` ˆ ζM ˆ ζA ˆ ζN ´ : W Ñ N ˆ M ˆ A ˆ N~~

~~be the inverse to the product map as before.~~

~~BMS ´1 w Lemma 5.2.6. Let g P supppm q. Then ζM pg Γg X W q Ă HΓ pgq.~~

~~Proof. Suppose that g´1γg “ hman (with h P N `, m P M, a P A, n P N ´). Then γg “~~

~~´1 ´1 ´1 ´1 gpg γgq “ ghman. Thus n , pmaq h ma, and γ give a sequence for ma in HΓpgq.~~

~~Corollary 5.2.7. Let g P supppmBMSq. If Γ ă G is Zariski dense, then the weak transitivity~~

~~w group HΓ pgq is dense in M.~~

~~Proof. This follows immediately from Lemma 5.2.6 and Proposition 5.2.2. 83~~

Remark In the real hyperbolic case, Corollary 5.2.7 follows from the arguments of Flaminio- Spatzier [20]. This is enough to prove ergodicity of the frame ﬂow; mixing of frame ﬂow, however, requires density of the transitivity group HΓpgq in MA, which does not follow immediately from Corollary 5.2.7 unless M is semisimple.

~~Corollary 5.2.8. Let g P supppmBMSq. If Γ ă G is Zariski dense, then the closure of the strong transitivity group contains the commutator rM,Ms.~~

~~w s Proof. For mi P HΓ pgq and pmi, aiq P HΓpgq the commutator lands in HΓpgq:~~

~~s rpm1, a1q, pm2, a2qs “ prm1, m2s, eq P HΓpgq.~~

~~s w w It follows that HΓpgq ą rHΓ pgq, HΓ pgqs “ rM,Ms.~~

Remark If M is semisimple, then this fact, together with non-arithmeticity of the length spectrum of Γ, is enough to prove density of HΓpgq ă MA. In the case that M has non- trivial center, however, a more careful approach is needed.

~~5.2.3. Smooth constraints on the limit set. Our next target is Corollary 5.2.12. In order to motivate our approach, however, we will start with an example,~~

Example Let G PSL Isom 2 . We pick the natural base point i 2 , and “ 2pRq “ `pHRq P HR the natural (upward pointing) unit tangent vector. Then A ă G is the diagonal subgroup and M is trivial. A short calculation gives

1 0 1 x ` ´ N “ : x P R and N “ : x P R . $¨ ˛ , $¨ ˛ , & x 1 . & 0 1 . ˝ ‚ ˝ ‚ We also identify 2%with in the- usual way.% Assume that Γ -PSL is non- BHR R Y t8u ă 2pRq elementary and that t0, 1, 8u Ă ΛpΓq. Then e P supppmBMSq, and the transitivity group

~~HΓpeq is dense in MA “ A.~~

~~2 Proof. The ﬁrst claim is clear. For the second, we deﬁne a smooth map f : BpH q´t0, 8u Ñ~~

~~A by sending ξ to g6 as in the following picture. 84~~

~~e g1 g1 g3~~

~~g2 g2~~

~~0 1 0 1 0 1 g3 g4 g4~~

~~g5 g5 0 ξ 0 ξ 0 ξ~~

~~Algebraically this is just the map~~

1 1 1 0 1 1 1 0 1 ξ´1 1 0 F : ξ ÞÑ ¨ 0 1 ˛ ¨ ´1 1 ˛ ¨ 0 1 ˛ ¨ ´ξ 1 ˛ ¨ 0 1 ˛ ¨ ´ξ 1 ˛ ˝ ‚˝ ‚˝ ‚˝ ‚˝ ‚˝ ‚ ´ξ 0 “ ¨ 0 ´ξ´1 ˛ ˝ ‚ ξ 0 “ . ¨ 0 ξ´1 ˛ ˝ ‚ 2 The points are that this map is a local diﬀeomorphism, that it carries 1 P BpH q to the identity in A » R, and that it carries the limit set into the transitivity group. Since Γ is not elementary we know that 1 is not isolated in the limit set, and so that e is not isolated

~~in the transitivity group HΓpeq. But a subgroup of R where the identity is not isolated is necessarily dense.~~

Remark For G “ SL2pRq, this argument is sufficient to prove that the transitivity groups contain M “ t˘ Idu. Together with the results of the next two sections that suffices to prove mixing of “frame flow” for SL2pRq in the sense of Theorem 3.2.1.

We now aim to prove density of the transitivity groups for general G and Γ ă G Zariski dense. The strategy is roughly the same as that in the example above; construct a map from the limit set to the transitivity group and then use smoothness properties of that map and density properties of the limit set to conclude that the transitivity group is dense. Unfortunately, we cannot write our map down explicitly in the general case. Let 85

~~M˜ “ rM,Ms be the commutator subgroup of M and M ab “ M{M˜ be the abelianization. We will writeπ ˜ : M ˆ A Ñ M ab ˆ A for the projection map.~~

~~Proposition 5.2.9. Suppose that Γ ă G is Zariski dense. Fix g P supppmBMSq. There is a neighborhood U of g´ in BX˜ and a smooth map~~

~~Φ: U Ñ M ˆ A~~

~~such that~~

~~• Φ maps ΛpΓq X U into HΓpgq,~~

~~• Φpg´q “ pe, eq is the identity, and~~

~~˜ ab • the diﬀerential map dpπ˜ ˝ Φq :Tg´ pBXq Ñ TepM ˆ Aq is surjective.~~

There are two cases here depending on whether the abelianization of MA has dimension one or two (cf. Fact 5.1.2). We will consider the latter case; the former essentially amounts to non-arithmeticity of the length spectrum (see [27]) and can be proved using simpliﬁed versions of Lemma 5.2.10 and Corollary 5.2.11.

~~Lemma 5.2.10. Suppose that the abelianization of MA is two dimensional. Let gˆ Ă g be the subspace spanned by the Lie algebras of N `,N ´, and M˜ . There exist tangent vectors~~

~~` ´ V P TeN and Yi P TeN such that~~

~~trV,Y1s, rV,Y2su is a basis of g{gˆ.~~

~~Proof. Fix a basis Z3,...Zr for gˆ. We consider the map~~

~~r ` 2 ´ 2 Ξ: pTeN q ˆ pTeN q Ñ g “ R ľ deﬁned by~~

~~pV1,V2,Y1,Y2q Ñ rV1,Y1s ^ rV2,Y2s ^ Z3 ^ ... ^ Zr.~~

~~Observe that this map is linear in all four variables, and that it has a symmetry~~

~~ΞpV1,V2,Y1,Y2q “ ´ΞpV2,V1,Y2,Y1q. 86~~

~~Note that Ξ cannot be identically zero; semisimplicity of g implies that~~

~~´ ` g “ rg, gs Ă gˆ ` rTeN , TeN s,~~

~~´ ` so vectors in rTeN .TeN s must span g{gˆ.~~

~~` ´ Our task is to ﬁnd some trio V P TeN and Y1,Y2 P TeN such that ΞpV,V,Y1,Y2q is non-zero. We will argue by contradiction. If no such trio exists, then we get a second symmetry~~

~~ΞpV1,V2,Y1,Y2q “ ´ΞpV2,V1,Y1,Y2q,~~

~~by linearity.~~

~~We want one more symmetry of Ξ. Let k0 P K be a representative of the non-trivial element of the Weyl group. Then~~

~~• Adk0 TeA “ TeA,~~

~~• Adk0 TeM “ TeM,~~

~~˘ ¯ • Adk0 TeN “ TeN , and~~

~~• The vectors Adk0 pZ3q ... Adk0 pZrq still give a basis for gˆ.~~

~~´ ` Now let Y P TeN and Vi P TeN . Then our contradictory assumption implies that~~

~~0 “ ΞpAdk0 Y, Adk0 Y, Adk0 V1, Adk0 V2q~~

~~“ Adk0 prY,V1sq ^ Adk0 prY,V2sq ^ Z3 ^ ... ^ Zr~~

~~“ Adk0 prY,V1sq ^ Adk0 prY,V2sq ^ Adk0 Z3 ^ ... ^ Adk0 Zr~~

~~“ rY,V1s ^ rY,V2s ^ Z3 ^ ... ^ Zr~~

~~“ ΞpV1,V2,Y,Y q.~~

~~This gives our ﬁnal symmetry ΞpV1,V2,Y1,Y2q “ ´ΞpV1,V2,Y2,Y1q. But any tensor satisfying these three symmetries is identically zero, which is a contradiction.~~

~~Corollary 5.2.11. Fix Yi as in Lemma 5.2.10. There is a non-empty Zariski open set~~

~~` L Ă N such that tAdhY1, AdhY2u is a basis of g{gˆ whenever h P L. 87~~

~~Proof. Fix a basis Z3,...,Zr for gˆ and consider the function~~

~~r Ξ:ˆ N ` Ñ g ľ h ÞÑ AdhY1 ^ AdhY2 ^ Z3 ^ ... ^ Zr.~~

~~The ﬁrst observation is that Ξˆ is non-constant; for any curve αptq passing through the identity of N ` we have~~

d2 Ξˆpαptqq “ Ξpα1p0q, α1p0q,Y ,Y q dt2 1 2 at t “ 0, which is non-zero for α1p0q “ V as in Lemma 5.2.10. It follows that Ξˆ is non- constant algebraic. We choose L to be the complement of the pre-image of zero.

~~` Proof of Proposition 5.2.9. Let L Ă N be the set from Corollary 5.2.11. Choose h0 P L~~

~~` such that pgh0q is in the (Zariski dense) limit set ΛpΓq. We deﬁne the function~~

~~Φ:˜ N ´ ˆ N ` ˆ N ´ Ñ G by~~

~~˜ ´1 Φpn1, h2, n3q “ h0n1h0 h2n3.~~

~~By the implicit function theorem we may choose a neighborhood U˜ of e in N ´ and smooth~~

~~` ´ functions f2 : U˜ Ñ N and f3 : U˜ Ñ N such that~~

~~˜ ´ ´ ˜ ` ` pΦpn1, f2pn1q, f3pn1qqq “ v0 and pΦpn1, f2pn1q, f3pn1qqq “ v0~~

~~for all n1 P U˜. It follows that Φˆpn1q :“ Φ˜pn1, f2pn1q, f3pn1qq is a smooth map U˜ Ñ MA. Let~~

~~˜ ` ´1 ˜ U “ tξ PBX ´ tgh0v0 u : nppgh0q ξq P Uu, and~~

~~´1 Φpξq “ Φˆpnppgh0q ξqq.~~

~~´1 If we ﬁx g and h0 and write nξ :“ nppgh0q ξq, then we can express this explicitly as~~

~~´1 Φpξq “ h0.nξ.h0 f2pnξq.f3pnξq. 88~~

~~g` g` ` ` gh0 gh0~~

~~g g1 :“ gh0 g1 g2 :“g1nξ~~

~~ξ ξ g´ g´ with g1 :“ gh0 with g2 :“ g1nξ g` g` ` ` gh0 gh0~~

~~g2 g3 g3 gΦpξq~~

~~ξ ξ g´ g´ ´1 with gΦ ξ g f n with g3 :“ g2h0 f2pnξq p q “ 3 3p ξq~~

~~We can now check the required properties directly. The subset U is indeed a neighborhood of g´ in BX˜. The function Φ is a composition of smooth maps, so is smooth. If ξ P ΛpΓqXU, then~~

~~´1 h0, nξ, h0 f2pnξq, f3pnξq~~

~~´ and e P Γ form a sequence for Φpξq in HΓpgq (see the picture). If Yi P TeN are the tangent vectors from Lemma 5.2.10, then tdΦ˜pY1q, dΦ˜pY2qu spans g{gˆ; in fact~~

~~˜ dΦpYiq “ Adh0 pYiq in g{gˆ.~~

~~˜ ab It follows that dpπ˜ ˝ Φq :Tg´ pBXq Ñ TepM Aq is surjective as required.~~

~~Proposition 5.2.9 is useful because of the following corollary.~~

~~Corollary 5.2.12. Suppose that Γ ă G is Zariski dense and there exists g P supppmBMSq~~

~~ab 1 ´ such that π˜pHΓpgqq ă M ˆ A is not dense. Then there is a neighborhood U of g in BX˜ and a smooth embedded submanifold S Ă U 1 of codimension one such that ΛpΓq X U 1 Ă S. 89~~

Proof. We consider the projection H “ π˜pHΓpgqq of the transitivity group to the abelianization M abA. If H is not dense, then H ă M abA is a proper Lie subgroup, so is a smooth embedded submanifold of M abA with positive codimension. Now apply the pre- image theorem to the compositionπ ˜ ˝ Φ (see Proposition 5.2.9); we conclude that there is a neighborhood U 1 of g´ in BX˜ and an embedded submanifold S “ pπ˜ ˝ Φq´1pHq X U 1 with ΛpΓq X U 1 Ă S as required.

~~Remark There’s a reasonable question of whether we can extend this corollary to show that~~

~~ΛpΓq is globally contained in a smooth submanifold of the boundary. The next argument will show that we can.~~

~~5.2.4. Algebraic constraints on ΛpΓq: an example. Our next task is to show that~~

~~ab the the projectionπ ˜pHΓpgqq ă M ˆ A is dense whenever Γ ă G is Zariski dense and g P supppmBMSq. We seek a contradiction with the outcome of Corollary 5.2.12. Begin again with an example.~~

Example Consider G “ PSL2pCq acting as the isometry group of real hyperbolic three space 3 . Identify the boundary 3 with 2 in the usual way. Suppose that HR BpHRq R Y t8u Γ ă G is discrete. If 2 0 alog 2 “ P Γ, ¨ 1 ˛ 0 2 ˝ ‚ 2 then p0, 0q P ΛpΓq. If in addition there is a smooth curve S Ă R passing through p0, 0q, and a positive such that

~~2 ΛpΓq X tpx, yq P R : |px, yq| ă u Ă S, then Γ is not Zariski dense in PSL2pCq, considered as a real algebraic group.~~

It is clear that p0, 0q P ΛpΓq. Shrinking if necessary, we may assume that S is given either as the graph of a smooth real function y “ fpxq or as the graph of a smooth real function x “ gpyq. For simplicity assume the former. There is, therefore, some constant C ą 0 with

~~|y ´ f 1p0qx| ď Cx2 for all pairs px, yq P S with |px, yq| ă , and so for all suﬃciently small pairs px, yq P ΛpΓq. 90~~

~~1 Lemma 5.2.13. If px0, y0q P ΛpΓq ´ t8u, then y0 ´ f p0qx0 “ 0.~~

~~Proof. Since alog 2 P Γ, and since the limit set is Γ-invariant, we have~~

~~´k k k alog 2px0, y0q “ px0{2 , y0{2 q P ΛpΓq~~

~~k k for all k. For k large enough we have that |px0{2 , y0{2 q| ă , and so that~~

~~1 |y0 ´ f p0qx0| x0 2 ď C . 2k 2k ´ ¯ 1 2 k But this implies that |y0 ´ f p0qx0| ă Cx0{2 for all k, which in turn implies that y0 ´ 1 f p0qx0 “ 0 as expected.~~

~~2 From the lemma it follows that ΛpΓq X R is contained in some straight line, so is not 3 Zariski dense. It also follows that the orbit Γp0, 0q Ă BpH q is not Zariski dense, and so that~~

~~Γ ă PSL2pCq is not Zariski dense. This completes the example.~~

5.2.5. Algebraic constraints on ΛpΓq: the general case. The example above actually contains all the necessary ideas to prove our next proposition. We now repeat the argument in a more general setting and with fewer simplifying assumptions.

~~Proposition 5.2.14. Suppose that there is a loxodromic element γ of Γ whose expanding~~

´ ` ﬁxed point is v0 and whose contracting ﬁxed point is v0 (so γ “ ma P MA). Suppose ´ ˜ further that there exists a neighborhood U of v0 in BX, and a smooth submanifold S Ă U of codimension one, such that ΛpΓq X U Ă S. Then Γ ă G is not Zariski dense.

´ ˜ ` Proof. We identify N with BX ´ tv0 u as usual by taking n to its negative end point ´ ´ ´ n . We further identify TeN and N via the exponential map. The action of γ on BX˜ corresponds to the action of γ on N ´ by conjugation, and so to the adjoint action of γ on

´ ´ TeN . The action of a on TeN is diagonalizable over R. In light of 5.1.1 we may choose 2 α˜ such that all eigenvalues of a are in tα,˜ α˜ u. We choose a basis v1, . . . vk, w1 . . . ql with

~~2 eigenvaluesα ˜ for the vi andα ˜ for the wj. We choose the corresponding coordinates xi, yj~~

~~´ ´ on TeN . Since v0 is an expanding ﬁxed point we know thatα ˜ ą 1. Since m commutes ´ k with a, the m action on TeN “ R respects the eigenspaces of a. 91~~

~~´ ´ Now consider the hypersurface S Ă TeN . We know that 0 P S. Shrinking U Ă TeN if necessary, we may assume that there is a coordinate xj (or yj) such that S is the graph~~

~~k S “ tpx1, . . . ykq P R such that xj “ fpx1,... xˆj . . . ykqu or~~

k S “ tpx1, . . . ykq P R such that yj “ fpx1,... yˆj . . . ykqu of a smooth function f. For simplicity we shall concentrate on the former case; the latter requires only minor modiﬁcations. Shrinking U again we may assume that U is pre-compact, and that f extends smoothly to the closure U. Let ppx1, x2,... xˆj, . . . ykq be the degree 2 Taylor polynomial for f. There is a constant C ą 0 such that

~~3 |fpx1,... xˆj . . . ykq ´ ppx1,... xˆj . . . ykq| ă C|px1,... xˆj . . . ykq|~~

~~for all x P U. Now a acts on the set of polynomials in the xi, yj. The action is diagonalizable,~~

~~2 so let p1, . . . p4 be the eigenfunction decomposition of xj ´ p with eigenvaluesα ˜ ă α˜ ă~~

~~3 4 α˜ ă α˜ . Note that p1 is non-trivial as it has the xj term. As in the example we will show~~

~~´ ´r that p1 vanishes on ΛpΓq. To this end ﬁxx ˆ “ pxˆ1 ... yˆkq P TeN X ΛpΓq. Then γ xˆ P ΛpΓq for all r, and γ´rxˆ P ΛpΓq X U Ă S once r is suﬃciently large. It follows that~~

~~4 ´r ´r 3 | pipγ xˆq| ď C|γ xˆ| 1 ÿ for all large r. Thus~~

~~4 ´r ´r ´kr ´r ´3r ´r 3 |α˜ p1pm xˆq| ď | α˜ pkpm xˆq| ` Cα˜ |m xˆ| k“2 ÿ for all large r. Rearranging we obtain~~

~~4 ´r p1´kqr ´r ´2r ´r 3 |p1pm xˆq| ď | α˜ pkpm yq| ` Cα˜ |m y| . k“2 ÿ ´r Now choose a subsequence rj Ñ 8 such that m j Ñ e. The left hand side converges to~~

~~|p0pyq|, while the right is bounded by~~

~~Corollary 5.2.15. If there exist ξ P ΛpΓq, a neighborhood U of ξ in the boundary BX˜, and a smooth codimension one submanifold S of U such that ΛpΓq X U Ă S, then Γ ă G is not~~

~~ab Zariski dense. In particular, if Γ ă G is Zariski dense, then π˜pHΓpgqq is dense in M ˆ A.~~

~~Proof. Let ξ, U, S be as in the statement. Choose a loxodromic element γ0 P Γ whose~~

~~expanding end point falls in U, therefore in S. Let ξ0 and ξ8 be the expanding and~~

~~contracting end points of γ0. Choose an element g P G such that gp0q “ ξ0 and gp8q “ ξ8.~~

´1 ´1 ´1 ´1 Then g Γg contains the elementγ ˜0 “ g γg P MA and has limit set Λpg Γgq “ g ΛpΓq. The sets g´1U and g´1S satisfy the conditions of the Proposition 5.2.14, so we see that g´1Γg is not Zariski dense. But then neither is Γ. The second statement follows from the ﬁrst statement via Corollary 5.2.12.

~~Theorem 5.2.2. If Γ ă G is Zariski dense and g P supppmBMSq, then the transitivity group~~

~~HΓpgq is dense in M ˆ A.~~

~~Proof. Consider the principal bundle~~

~~π˜ 0 ÝÝÝÝÑrM,Ms ÝÝÝÝÑ M ˆ A ÝÝÝÝÑ M ab ˆ A ÝÝÝÝÑ 0.~~

~~s We already know that HΓpgq ą rM,Ms. It follows that HΓpgq is a union of ﬁbers ofπ ˜. But ab π˜pHΓpgqq Ă M ˆ A is dense, so (using the compactness of the ﬁbers) HΓpgq meets every~~

~~ﬁber ofπ ˜. Thus HΓpgq Ă M ˆ A is dense as claimed.~~

~~5.3 Invariant functions for the horospherical groups~~

~~This section contains the measure theory we need to prove mixing of frame ﬂow. The aim is to prove Corollary 5.3.1. We assume ﬁniteness of the BMS measure throughout this section.~~

~~The arguments of the next two sections also apply to the M-invariant lifts of ﬁnite,~~

A-invariant and A ergodic quasi-product measures on ΓzG{M, requiring only cosmetic modiﬁcation. We concentrate on the BMS measure for concreteness and because it is the example of principal interest to us.

~~5.3.1. Periods of Σ˜-measurable functions. 93~~

~~Deﬁnition We write BpGq for the Borel sigma algebra on G and deﬁne subalgebras~~

~~˘ Σ˘ “ tB P BpGq : B “ ΓBN u consisting of all Borel measurable sets that are invariant under Γ and the stable (unstable) horospherical group. We further deﬁne~~

~~˜ Σ “ Σ´ ^ Σ`;~~

~~˜ in other words, a Borel measurable set B Ă G belongs to Σ if there exist B˘ P Σ˘ such BMS ˜ thatm ˜ pB4B˘q “ 0. Note that the deﬁnition of Σ depends on the group Γ.~~

~~Our task now is to prove that Σ˜ is trivial whenever the transitivity groups are dense in MˆA. Let ψ be a bounded Σ˜ measurable function on G. We assume without loss of generality that~~

~~ψ is strictly invariant for the left Γ action. Let ψ˘ be bounded Σ˘ measurable functions BMS satisfying ψ˘ “ ψ almost everywhere (with respect tom ˜ ). Let~~

~~ψ|gMA is measurable, and $ , E “ ’gMA : ψpgmaq “ ψ`pgmaq “ ψ´pgmaq / ’ / &’ ./ for Haar almost every ma P MA ’ / Ă ’G{MA. / %’ -/~~

~~It will also be useful to identify G{MA with B2X˜ via the map gMA ÞÑ pg´, g`q. Recall the~~

~~2 Patterson-Sullivan measure σo from Section 2. By Fubini, E has full measure in pB X,˜ σo ˆ~~

~~σoq. For ą 0 let M be the ball around the identity in M, and deﬁne~~

~~ ψ pgq “ ψpgmatqdtdm, and żM ż0 ψ˘pgq “ ψ˘pgmatqdtdm. żM ż0 Note that if gMA P E, then the functions ψ , ψ˘ are continuous, well deﬁned, and identical on gMA. Now let~~

~~´ 1 1 E “ tξ P ΛpΓq : pξ, η q P E for σo almost every η PBX˜ ´ tξuu~~

~~and~~

~~` 1 1 E “ tη P ΛpΓq : pξ , ηq P E for σo almost every ξ PBX˜ ´ tηuu.~~

~~Another application of Fubini tells us that E` and E´ both have full measure in BX˜. 94~~

~~Lemma 5.3.1. Let ą 0. Suppose that gMA P E, that g` P E`, that g´ P E´, and that~~

~~ pm0, a0q P HΓpgq Ă M ˆ A. Then ψ pgm0a0q “ ψ pgq.~~

~~Proof. Consider a sequence gm0a0 “ γgh1h2 . . . hk for m0a0 in the transitivity group. We~~

~~will say that this sequence is permissible if gh1 . . . hr P E for all 1 ď r ď k. In this case~~

~~ ψ pgq “ ψ pgh1q “ ... “ ψ pgh1 . . . hkq “ ψ pγgh1 . . . hkq “ ψ pgm0a0q,~~

` ´ using invariance of ψ` under N , invariance of ψ´ under N , invariance of ψ under Γ, and the fact that all three agree on E. More generally, we can approximate any element of the transitivity group by a per-

~~missible sequence. Again suppose that m0a0 P HΓpgq and pick a sequence gm0a0 “~~

~~´ ` γgn1h1 . . . nkhk for m0a0 in the transitivity group with ni P N and hi P N . By in-~~

~~plq ´ plq ` duction on i we can choose sequences ni Ñ ni in N and hi Ñ hi in N such that~~

~~plq plq plq plq plq plq 1. γgn1 h1 . . . nk hk “ gm a P gMA,~~

~~plq plq plq plq ` ` plq plq plq plq ´ ´ 2. pgn1 h1 . . . nr hr q P E and pgn1 h1 . . . hr´1nr q P E for each l and r, and~~

~~plq plq plq plq plq plq plq plq 3. gn1 h1 . . . nr hr P E and gn1 h1 . . . hr´1nr P E for each l and r.~~

~~plq plq plq plq plq plq In other words gm a “ γgn1 h1 . . . nk hk is a permissible sequence for each l. We then have~~

~~ plq plq ψ pgm0a0q “ lim ψ pgm a q “ ψ pgq lÑ8 as required.~~

~~BMS Theorem 5.3.1. If HΓpgq ă MA is dense for any g P supppm q, then Σ˜ is trivial.~~

Proof. Let ψ be some bounded Σ˜ measurable function. Then ψ is right MA-invariant for any ą 0. Thus ψ is right MA-invariant. But the right MA action on pΓzG, mBMSq is ergodic [15], so ψ is constant almost everywhere.

~~5.4 Triviality of Σ˜ and mixing of the frame ﬂow~~

We now have almost all the technical pieces in place to prove Theorem 3.2.1. The last piece of the argument is to show that mixing follows from triviality of Σ.˜ We assume ﬁniteness of the BMS measure throughout this section. 95

~~Proposition 5.4.1. If Σ˜ is trivial (i.e. if every Σ˜-measurable set is either null or co-null for the BMS measure), then the A action on pΓzG, mBMSq is mixing.~~

~~The key point in the proof is the following lemma (Lemma 1 from [2]).~~

Lemma 5.4.2. Let pX, B, m, patqtPRq be a measure preserving dynamical system where pX, Bq is a standard Borel space, m a ﬁnite Borel measure and at an action of R on X by measure preserving transformations. Let

2 ψ P L pX, mq be a real valued function on X such that X ψdm “ 0. 2 If there exist tn Ñ 8 in R such that ψ ˝ atn does notş converge weakly to zero in L , then 2 there is a sequence sm P R tending to `8 and a non-constant function φ P L pX, mq such that

~~2 ψ ˝ asn Ñ φ and ψ ˝ a´sn Ñ φ in the weak L topology.~~

~~Proof of Proposition 5.4.1. Suppose that the right A action is not mixing. Then we may choose a uniformly continuous function~~

~~ψ :ΓzG Ñ R~~

~~BMS 2 BMS such that ΓzG φdm “ 0 and ψ ˝ at does not converge weakly to zero in L pΓzG, m q.~~

~~By Lemmaş 5.4.2 we get a non-constant function φ and sn Ñ `8 such that~~

~~2 ψ ˝ asn Ñ φ and ψ ˝ a´sn Ñ φ in the weak L topology.~~

~~Passing to a subsequence snk we may assume (by Banach-Saks) that the Cesaro averages~~

1 R 1 R S pψq :“ ψ ˝ a and S˜ pψq :“ ψ ˝ a R R snr R R ´snr r“1 r“1 ÿ ÿ 2 converge strongly to φ in L . Passing to further subsequences we may assume that SRl pψq ˜ and SRl pψq converge almost everywhere to φ. ˜ It is standard to show that any almost everywhere limit of SRl pψq (respectively SRl pψq) ´ ` is N -invariant (respectively N -invariant). So on the one hand φ “ liml SRl pψq is Σ´ ˜ measurable, while on the other φ “ liml SRl pψq is Σ` measurable. Thus φ is a non-constant Σ˜ measurable function, and Σ˜ cannot be trivial. 96

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