Uniform exponential mixing and resonance free regions for convex cocompact congruence subgroups of $\operatorname{SL}_2(\mathbb{Z})$

TL;DR

This paper proves uniform exponential mixing and establishes resonance-free regions for congruence subgroups of SL(2,Z), using dynamical systems and expander graph techniques to analyze spectral properties uniformly over levels.

Contribution

It extends Dolgopyat's method to uniform settings over congruence covers, providing new spectral bounds and resonance-free regions for these groups.

Findings

Uniform exponential decay of matrix coefficients for congruence subgroups.

Establishment of a uniform resonance-free half-plane for the Laplacian.

Application of expander graph theory to spectral analysis.

Abstract

Let $\Gamma<\mathrm{SL}_2(\mathbb{Z})$ be a non-elementary finitely generated subgroup and let $\Gamma(q)$ be its congruence subgroup of level $q$ for each $q\in \mathbb{N}$. We obtain an asymptotic formula for the matrix coefficients of $L^2(\Gamma (q) \backslash \mathrm{SL}_2(\mathbb{R}))$ with a {\it uniform} exponential error term for all square-free $q$ with no small prime divisors. As an application we establish a uniform resonance-free half plane for the resolvent of the Laplacian on $\Gamma (q) \backslash \mathbb{H}^2$ over $q$ as above. Our approach is to extend Dolgopyat's dynamical proof of exponential mixing of the geodesic flow uniformly over congruence covers, by establishing uniform spectral bounds for congruence transfer operators associated to the geodesic flow. One of the key ingredients is the expander theory due to Bourgain-Gamburd-Sarnak.