Uploaded June 2023 | Updated September 2026, 3 days ago

Day 6 Session 1

Israel Institute for Advanced Studies Live Stream

$Doron Puder (IAS & TAU): Random covers of hyperbolic surfaces The 25th Midrasha Mathematicae on Groups, Expanders and Codes Celebrating Alex Lubotzkys 70th birthday. Day 4, Session 5 Speaker: Doron Puder (IAS & TAU) Title: Random covers of hyperbolic surfaces Abstract: The spectral gap of the Laplacian on a hyperbolic surface measures how well the surface is connected. It was shown long ago by Huber that the spectral gap of such a surface cannot exceed that of the hyperbolic plane, asymptotically as the genus goes to infinity. Whether there exists a sequence of closed hyperbolic surfaces that achieves this bound an old conjecture of Buser was settled a few years ago by Hide and Magee. This was done by exhibiting a sequence of finite covering spaces of a fixed base surface that have good spectral properties. In this talk, I will discuss joint work with Magee and van Handel where we show that this phenomenon is in fact much more prevalent: given any closed hyperbolic surface, not only do there exist finite covers that have good spectral properties, but this is in fact the case for all but a vanishing fraction of its finite covers. The proof hinges on new developments on the notion of strong convergence in random matrix theory. I plan to give a gentle introduction to this line of works.$

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