Hyperbolic Surfaces with Arbitrarily Small Spectral Gap

TL;DR

This paper demonstrates that hyperbolic surfaces can be covered to have arbitrarily small spectral gaps, using combinatorial and thermodynamic methods depending on the Hausdorff dimension of the limit set.

Contribution

It introduces a method to construct finite covers of hyperbolic surfaces with spectral gaps arbitrarily close to zero, extending previous understanding of spectral properties.

Findings

Existence of finite covers with zeros of the Selberg zeta function close to the Hausdorff dimension

Use of expander graphs for elta > 1/2 cases

Application of thermodynamic formalism for elta q; 1/2 cases

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.