  We prove that if $X$ is a finite area non-compact hyperbolic surface, then
for any $\epsilon>0$, with probability tending to one as $n\to\infty$, a
uniformly random degree $n$ Riemannian cover of $X$ has no eigenvalues of the
Laplacian in $[0,\frac{1}{4}-\epsilon)$ 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 exist a
sequence of closed hyperbolic surfaces with genera tending to infinity and
first non-zero eigenvalue of the Laplacian tending to $\frac{1}{4}$.
