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Abstract:   It was shown long ago by Huber that the first nonzero eigenvalue of the 
Laplacian on a closed hyperbolic surface cannot exceed that of the 
hyperbolic plane, asymptotically as the genus goes to infinity. Whether 
there exists a sequence of closed hyperbolic surfaces that achieves this 
bound---an old conjecture of Buser---was settled a few years ago by Hide 
and Magee. This was done by exhibiting a sequence of covering spaces of a 
fixed base surface that have good spectral properties. In this talk, I 
will discuss joint work with Magee and Puder where we show that this 
phenomenon is in fact much more prevalent: given any closed hyperbolic 
surface, not only do there exist covering spaces that have good spectral 
properties, but this is in fact the case for all but a vanishing fraction 
of its covering spaces. The proof is based on recent developments on the 
notion of strong convergence, which combines ideas from random matrix 
theory, representation theory, and combinatorial group theory.