# Special Groups and Dynamics Seminar - Part 1

## Spectral Gap of the Laplacian for Random Hyperbolic Surfaces

Although there are several ways to ''choose a compact hyperbolic surface at random'', putting the Weil-Petersson probability measure on the moduli space of hyperbolic surfaces of a given topology is certainly the most natural. The work of M. Mirzakhani has made possible the study of this probabilistic model, providing exact formulas for certain integrals, as well as their asymptotic behaviour in the limit of large genus.

I will be interested in the spectral gap $\lambda_1$ of the laplacian for a random compact hyperbolic surface, in the limit of large genus $g$ : in joint work with Laura Monk, we show that asymptotically almost surely, $\lambda_1 > 1/4 -\epsilon$ for any $\epsilon >0$.

The proof relies on the trace method. We use asymptotic expansions in powers of $g^{-1}$ for volume functions giving the distribution of the length spectrum, and prove that the coefficients possess the ``Friedman-Ramanujan property" (a notion introduced by J. Friedman in his proof of the Alon conjecture for random regular graphs).