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Mixing Properties of Flows on Geometrically Finite Hyperbolic Manifolds

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Mixing Properties of Flows on Geometrically Finite Hyperbolic Manifolds 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 ¤ λ1pqq of the hyperbolic Laplacian on Xpqq satisfy λ0pqq “ 0 and λ1pqq ¥ 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~ ¡ 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 ques- tions, 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.
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