Geometric Analysis and Topology Seminar

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Random covers of hyperbolic surfaces

Speaker: Ramon van Handel, Princeton University

Location: Warren Weaver Hall 1302

Date: Friday, September 19, 2025, 11 a.m.

Synopsis:

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 finite 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 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.