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

Let X be a compact connected hyperbolic surface, that is, a closed connected orientable smooth surface with a Riemannian metric of constant curvature $$-1$$ -1 . For each n∈ N n∈N , let Xₙ 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 Xn is an eigenvalue of the associated Laplacian operator Δ X ΔX or Δ _Xₙ ΔXn . We say that an eigenvalue of Xₙ Xn is new if it occurs with greater multiplicity than in X . We prove that for any ε >0 ε>0 , with probability tending to 1 as n→ ∞ n→∞ , there are no new eigenvalues of Xₙ Xn below 3/16-ε 316-ε . We conjecture that the same result holds with 3/16 316 replaced by 1/4 14 .