Uniform resonance free regions for convex cocompact hyperbolic surfaces and expanders

TL;DR

This paper establishes uniform spectral gaps for families of coverings of convex cocompact hyperbolic surfaces with expanders, introduces a new resonance-free region, and extends previous results to settings without $L^2$-eigenvalues.

Contribution

It extends spectral gap results to infinite-area surfaces using resonances, and introduces a universal resonance-free region for any convex cocompact surface.

Findings

Uniform spectral gap for coverings with expander Schreier graphs

A new universal resonance-free region for convex cocompact surfaces

Extension of spectral gap results to settings without $L^2$-eigenvalues

Abstract

We prove that every family of coverings of any infinite-area, convex cocompact hyperbolic surface has uniform spectral gap, provided that the associated Schreier graphs form a family of two-sided expanders. This extends the results of Brooks, Burger, and Bourgain-Gamburd-Sarnak to a setting where the Laplacian has no $L^2$-eigenvalues. In particular, the notion of spectral gap needs to be redefined in terms of the resonances of the Laplacian. As an immediate corollary, we obtain uniform spectral gap for congruence covers of convex cocompact surfaces, a result previously established by Oh-Winter and Bourgain-Kontorovich-Magee. Moreover, given any convex cocompact hyperbolic surface $X$, we provide a new "universal" resonance-free region for $X$, by which we mean a region in the complex plane that contains no resonances for any finite cover of $X$. This enlarges the universal resonance-free region given by Magee-Naud. Our methods rely on the thermodynamic formalism for twisted Selberg zeta functions.