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Generalization of Selberg's $3/16$ theorem for convex cocompact thin subgroups of $\operatorname{SO}(n, 1)$

Pratyush Sarkar

Abstract

Let $Γ$ be a convex cocompact thin subgroup of an arithmetic lattice in $\operatorname{SO}(n, 1)$. We generalize Selberg's $\frac{3}{16}$ theorem in this setting, i.e., we prove uniform exponential mixing of the frame flow and obtain a uniform resonance-free half plane for the congruence covers of the hyperbolic manifold $Γ\backslash \mathbb H^n$. This extends the work of Oh-Winter who established the $n = 2$ case. The theorem follows from uniform spectral bounds for the congruence transfer operators with holonomy. We employ Sarkar-Winter's frame flow version of Dolgopyat's method uniformly over the congruence covers as well as Golsefidy-Varjú's generalization of Bourgain-Gamburd-Sarnak's expansion machinery by using the properties that the return trajectory subgroups are Zariski dense and have trace fields which coincide with that of $Γ$. These properties follow by proving that the return trajectory subgroups have finite index in $Γ$.

Generalization of Selberg's $3/16$ theorem for convex cocompact thin subgroups of $\operatorname{SO}(n, 1)$

Abstract

Let be a convex cocompact thin subgroup of an arithmetic lattice in . We generalize Selberg's theorem in this setting, i.e., we prove uniform exponential mixing of the frame flow and obtain a uniform resonance-free half plane for the congruence covers of the hyperbolic manifold . This extends the work of Oh-Winter who established the case. The theorem follows from uniform spectral bounds for the congruence transfer operators with holonomy. We employ Sarkar-Winter's frame flow version of Dolgopyat's method uniformly over the congruence covers as well as Golsefidy-Varjú's generalization of Bourgain-Gamburd-Sarnak's expansion machinery by using the properties that the return trajectory subgroups are Zariski dense and have trace fields which coincide with that of . These properties follow by proving that the return trajectory subgroups have finite index in .

Paper Structure

This paper contains 30 sections, 39 theorems, 144 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Historical background
  3. Main result
  4. Applications
  5. Outline of the proof of \ref{['thm:TheoremUniformExponentialMixingOfFrameFlow']}
  6. Organization of the paper
  7. Preliminaries
  8. Convex cocompact groups
  9. Patterson--Sullivan density
  10. Bowen--Margulis--Sullivan measure
  11. Coding of the geodesic flow and its associated cocycles
  12. Markov sections
  13. Symbolic dynamics
  14. Thermodynamics
  15. Cocycles and compatible Markov sections
  16. ...and 15 more sections

Key Result

Theorem \oldthetheorem

There exist $\eta > 0$, $C > 0$, $r \in \mathbb N$, 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 $\mathfrak{q}_0$, $\phi \in C_{\mathrm{c}}^r(\Gamma_\mathfrak{q} \backslas

Figures (2)

  • Figure 1: The Markov property.
  • Figure 2: This illustrates the idea of the proof of \ref{['pro:LimitSetH(yy)RadialEqualsLimitSetH(yy)EqualsLimitSetGamma']}. Note that the actual positions of $v_q$ and $\mathcal{P}^{-(q + r)}(v_q)$ are perturbations of what is shown.

Theorems & Definitions (95)

  • Conjecture \oldthetheorem
  • Conjecture \oldthetheorem
  • Theorem \oldthetheorem
  • Remark
  • Remark
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Definition \oldthetheorem: Limit set
  • ...and 85 more