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.