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 $ \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.