A random cover of a compact hyperbolic surface has relative spectral gap 316-ε

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 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 Δ _X or ΔXn. We say that an eigenvalue of X_n 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 X_n below 316-ε. We conjecture that the same result holds with 316 replaced by 14.