Title: Random covers of hyperbolic surfaces
Abstract: The spectral gap of the Laplacian on a hyperbolic surface measures how well the surface is connected. It was shown long ago by Huber that the spectral gap of such a 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 finite covering spaces of a fixed base surface that have good spectral properties. In this talk, I will discuss joint work with Magee and van Handel where we show that this phenomenon is in fact much more prevalent: given any closed hyperbolic surface, not only do there exist finite covers that have good spectral properties, but this is in fact the case for all but a vanishing fraction of its finite covers. The proof hinges on new developments on the notion of strong convergence in random matrix theory. This talk will be aimed to a general math audience.