Hyperbolic Surfaces with Arbitrarily Small Spectral Gap

Abstract

Let $X=\Gamma \setminus \mathbb{H}$ be a non-elementary geometrically finite hyperbolic surface and let $\delta$ denote the Hausdorff dimension of the limit set $\Lambda (\Gamma )$. We prove that for every $\epsilon >0$ the surface $X$ admits a finite cover $X^{\prime }$ such that the Selberg zeta function associated to $X^{\prime }$ has a zero $s\ne \delta$ with $|\delta -s|<\epsilon$. For $\delta >\frac{1}{2}$ we exploit the combinatorial interpretation of spectral gap in terms of expander graphs. For $\delta \le \frac{1}{2}$ the proof is based on the thermodynamic formalism approach for L-functions associated to hyperbolic surfaces and an analogue of the Artin-Takagi formula for these L-functions.