Laura Monk (University of Bristol)

Friedman-Ramanujan functions in random hyperbolic geometry

Abstract:

The spectral gap of a hyperbolic surface is its smallest non-zero eigenvalue λ1; it is large if and only if the surface is well-connected (hard to cut) and has good dynamical properties (the geodesic flow mixes very fast). In the large-genus limit, the spectral gap cannot be much larger than 1/4, the bottom of the spectrum of the hyperbolic plane. I will present results obtained in an ongoing collaboration with Nalini Anantharaman, in which we aim to prove that, for any ε>0, typically, λ₁ > 1/4 - ε. In order to do so, we study random hyperbolic surfaces, sampled according to the Weil-Petersson model. I will explain how the Selberg trace formula allows us to transform the spectral gap question into a random geometry question. I will then present new results allowing to study the geometry of random surfaces, and in particular to deal with non-simple closed geodesics.