Near optimal spectral gaps for hyperbolic surfaces

We prove that if X is a finite area non-compact hyperbolic surface, then for any ε >0, with probability tending to one as n→ ∞, a uniformly random degree n Riemannian cover of X has no eigenvalues of the Laplacian in [0,1/4-ε) other than those of X, and with the same multiplicities. As a result, using a compactification procedure due to Buser, Burger, and Dodziuk, we settle in the affirmative the question of whether there exists a sequence of closed hyperbolic surfaces with genera tending to infinity and first non-zero eigenvalue of the Laplacian tending to 1/4.