Typical hyperbolic surfaces have an optimal spectral gap

The first non-zero Laplace eigenvalue of a hyperbolic surface, or its spectral gap, measures how well-connected the surface is: surfaces with a large spectral gap are hard to cut in pieces, have a small diameter and fast mixing times. For large hyperbolic surfaces (of large area or large genus g, equivalently), we know that the spectral gap is asymptotically bounded above by 1/4. The aim of this talk is to present an upcoming article, joint with Nalini Anantharaman, where we prove that most hyperbolic surfaces have a near-optimal spectral gap. That is to say, we prove that, for any ε>0, the Weil-Petersson probability for a hyperbolic surface of genus g to have a spectral gap greater than 1/4-ε goes to one as g goes to infinity. This statement is analogous to Alon’s 1986 conjecture for regular graphs, proven by Friedman in 2003. I will present our approach, which shares many similarities with Friedman’s work, and relies on creating cancellations in the trace method.

 

The focus of this talk will be mostly analytic as I will present its geometric components at the GGT seminar. Both talks will be disjoint and independent, with the intention that they can be viewed either separately or together.