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Uniform exponential mixing for congruence covers of convex cocompact hyperbolic manifolds

Pratyush Sarkar

Abstract

Let $Γ$ be a Zariski dense convex cocompact subgroup contained in an arithmetic lattice of $\operatorname{SO}(n, 1)^{\circ}$. We prove uniform exponential mixing of the geodesic flow for congruence covers of the hyperbolic manifold $Γ\backslash \mathbb{H}^n$ avoiding finitely many prime ideals. This extends the work of Oh-Winter who proved the result for the $n = 2$ case. Following their approach, we use Dolgopyat's method for the proof of exponential mixing of the geodesic flow. We do this uniformly over congruence covers by establishing uniform spectral bounds for the congruence transfer operators associated to the geodesic flow. This requires another key ingredient which is the expander machinery due to Bourgain-Gamburd-Sarnak extended by Golsefidy-Varjú.

Uniform exponential mixing for congruence covers of convex cocompact hyperbolic manifolds

Abstract

Let be a Zariski dense convex cocompact subgroup contained in an arithmetic lattice of . We prove uniform exponential mixing of the geodesic flow for congruence covers of the hyperbolic manifold avoiding finitely many prime ideals. This extends the work of Oh-Winter who proved the result for the case. Following their approach, we use Dolgopyat's method for the proof of exponential mixing of the geodesic flow. We do this uniformly over congruence covers by establishing uniform spectral bounds for the congruence transfer operators associated to the geodesic flow. This requires another key ingredient which is the expander machinery due to Bourgain-Gamburd-Sarnak extended by Golsefidy-Varjú.

Paper Structure

This paper contains 26 sections, 53 theorems, 222 equations.

Table of Contents

  1. Introduction
  2. Outline of the proof of \ref{['thm:TheoremUniformExponentialMixingOfGeodesicFlow']}
  3. Organization of the paper
  4. Background and notations
  5. Markov sections
  6. Symbolic dynamics
  7. Thermodynamics
  8. Transfer operators
  9. Cocycles and congruence transfer operators
  10. Main technical theorem
  11. Spectral bounds for small $|b|$ using expander machinery
  12. Reductions
  13. Approximating the transfer operator
  14. Zariski density of Hp(y, z) in G
  15. L-2-flattening lemma
  16. ...and 11 more sections

Key Result

Theorem \oldthetheorem

There exist $\eta > 0$, $C > 0$ and a nontrivial proper ideal $\mathfrak{q}_0 \subset \mathcal{O}_{\mathbb K}$ such that for all square free ideals $\mathfrak{q} \subset \mathcal{O}_{\mathbb K}$ coprime to $\mathfrak{q}_0$, and for all $M$-invariant functions $\phi, \psi \in C^1(\Gamma_\mathfrak{q}

Theorems & Definitions (132)

  • Theorem \oldthetheorem
  • Remark
  • Corollary \oldthetheorem.1
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Remark
  • Definition \oldthetheorem
  • Remark
  • Definition \oldthetheorem
  • Theorem \oldthetheorem
  • ...and 122 more