Hyperbolic Surfaces with Arbitrarily Small Spectral Gap

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 $\varepsilon > 0$ the surface $X$ admits a finite cover $X'$ such that the Selberg zeta function associated to $X'$ has a zero $s\neq \delta$ with $| \delta - s| < \varepsilon$. For $\delta > \frac{1}{2}$ we exploit the combinatorial interpretation of spectral gap in terms of expander graphs. For $\delta \leq \frac{1}{2}$ 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.