A random cover of a compact hyperbolic surface has relative spectral gap 3/16 - ϵ

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 n ∈ N, let Xn be a random degree-n cover of X sampled uniformly from all degree-n Riemannian covering spaces of X. An eigenvalue of X or Xn is an eigenvalue of the associated Laplacian operator ΔX or ΔXn. We say that an eigenvalue of Xn is new if it occurs with greater multiplicity than in X. We prove that for any ε > 0, with probability tending to 1 as n → ∞, there are no new eigenvalues of Xn below 3 16 − ε. We conjecture that the same result holds with 3 16 replaced by 1 4 .