Bas du spectre de surfaces hyperboliques de volume infini
Abstract
This article presents some methods to control the bottom of the spectrum of the Laplacian λ0 on hyperbolic surfaces with infinite volume. Our first result bounds the λ0 of a geometrically finite surface in terms of the geometry of its convex core. We then focus on infinite type periodic hyperbolic surfaces built by gluing copies of a geometrically finite surface with boundary according to the plan of an infinite graph. We control the λ0 of the so-obtained infinite surfaces by constants coming from spectral properties of the building brick and combinatorial datae of the graph. We then use these methods to control the λ0 of two other kind of infinite type hyperbolic surfaces : those admitting a splitting into bounded pieces, and some riemannian coverings.