Hyperbolic Surfaces with Arbitrarily Small Spectral Gap

Let $X=\mathrm{\Gamma }\setminus \mathbb{H}$ be a non-elementary geometrically finite hyperbolic surface and let $\delta$ denote the Hausdorff dimension of the limit set $\mathrm{\Lambda }(\mathrm{\Gamma })$. We prove that for every $\epsilon >0$ the surface $X$ admits a finite cover $X^{\mathrm{\prime }}$ such that the Selberg zeta function associated to $X^{\mathrm{\prime }}$ has a zero $s\mathrm{\ne }\delta$ with $ | \delta - s| < \varepsilon $\mathrm{.}For$ \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.