| Résume | We study the spectrum of the Laplacian for finite-area hyperbolic surfaces, focusing on the spectral gap, i.e. the smallest non-zero element of the spectrum. The spectral gap contains a lot of geometric and dynamical information about the surface. It provides control over its diameter and Cheeger constant, governs the rate of mixing of the geodesic flow and provides error terms in geodesic counting. For closed hyperbolic surfaces of large genus, 1/4 is the asymptotically optimal spectral gap. Recently, the use of probabilistic methods has led to great progress on questions around the spectral gap of large volume hyperbolic surfaces. I will explain some recent results and open problems in this area. Based on joint works with Michael Magee and with Bram Petri. |
