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The relation between expander graphs and spectral gap for eigenvalues of theLaplacian on Riemannian manifolds is well-known from the work of Brooks,Burger, Sunada, and Bourgain--Gamburd--Sarnak. In this paper we prove ananalogous result for resonances of convex compact hyperbolic surfaces. Givensuch a surface $X$, let $\delta$ be the dimension of its limit set, let$(X_n)_{n\in \mathbb{N}}$ be a family of finite coverings of $X$, and let$(\mathcal{G}_n)_{n\in \mathbb{N}}$ be the associated family of Schreier cosetgraphs with respect to the Schottky generating set for the fundamental group of$X$. We prove that if the surfaces $X_n$ have a uniform resonance-free regionin the vicinity of $\delta$, then the graphs $\mathcal{G}_n$ form a family ofexpanders. In the other direction, we prove that if $(\mathcal{G}_n)_{n\in\mathbb{N}}$ is a family of two-sided expanders, then $(X_n)_{n\in\mathbb{N}}$has a uniform spectral gap in the following sense: there exists some $\eta > 0$such that for all $n\in \mathbb{N}$ the cover $X_n$ has no resonances in thehalf-plane $\mathrm{Re}(s)\geqslant \delta-\eta$, except for the "trivial"resonance at $s=\delta.$ This result implies uniform spectral gap for thefamily of congruence covers $X(q)$ of $X$ where $q$ runs over all positiveintegers. This was established by Oh--Winter for all square-free $q\in\mathbb{N}$ with no small prime divisors, and extended byBourgain--Kontorovich--Magee to all $q\in \mathbb{N}$ with no small primedivisors. We construct further explicit families $(X_n)_{n\in \mathbb{N}}$ withuniform spectral gap. We also prove the existence of a new "universal"resonance-free region for $X$, by which we mean a region in the complex planethat contains no resonances of any finite cover of $X$, refining a result ofMagee--Naud. Our methods rely on the thermodynamic formalism for twistedSelberg zeta functions.

Uniform resonance free regions for convex cocompact hyperbolic surfaces and expanders