Hyperbolic Surfaces with Arbitrarily Small Spectral Gap

Abstract

Let $ X = Γ\setminus \mathbb{H} $ be a non-elementary geometrically finite hyperbolic surface and let $ δ$ denote the Hausdorff dimension of the limit set $ Λ(Γ) $. 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 δ$ with $ | δ- s| < \varepsilon $. For $ δ> \frac{1}{2} $ we exploit the combinatorial interpretation of spectral gap in terms of expander graphs. For $ δ\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.