Modul: MAT760 Ergodic Theory and Dynamical Systems Seminar

## e-Seminar: The spectral gap of a random hyperbolic surface

Talk by Dr. Michael Magee

**Date:** 18.05.20 **Time:** 13.45 - 14.45 **Room:** Y27H28

On a compact hyperbolic surface, the Laplacian has a spectral gap between 0 and the next smallest eigenvalue if and only if the surface is connected. The size of the spectral gap measures both how highly connected the surface is, and the rate of exponential mixing of the geodesic flow on the surface. There is an analogous concept of spectral gap for graphs, with analogous connections to connectivity and dynamics. Motivated by theorems about the spectral gap of random regular graphs, we proved that for any r>0, a random cover of a fixed compact connected hyperbolic surface has no new eigenvalues below 3/16 - r, with probability tending to 1 as the covering degree tends to infinity. The number 3/16 is, mysteriously, the same spectral gap that Selberg obtained for congruence modular curves. The talk is intended to be accessible to graduate students and is based on joint works with Frédéric Naud and Doron Puder.