A random cover of a compact hyperbolic surface has relative spectral gap frac{3}{16}-varepsilon

Abstract

Let X be a compact connected hyperbolic surface, that is, a closed connected orientable smooth surface with a Riemannian metric of constant curvature -1. For each nin {mathbf {N}}, let X_{n} be a random degree-n cover of X sampled uniformly from all degree-n Riemannian covering spaces of X. An eigenvalue of X or X_{n} is an eigenvalue of the associated Laplacian operator Delta _{X} or Delta _{X_{n}}. We say that an eigenvalue of X_{n} is new if it occurs with greater multiplicity than in X. We prove that for any varepsilon >0, with probability tending to 1 as nrightarrow infty , there are no new eigenvalues of X_{n} below frac{3}{16}-varepsilon . We conjecture that the same result holds with frac{3}{16} replaced by frac{1}{4}.