2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/165761This article presents some methods to control the bottom of the spectrum of the Laplacian $λ_0$ on hyperbolic surfaces with infinite volume. Our first result bounds the $λ_0$ 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 $λ_0$ 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 $λ_0$ of two other kind of infinite type hyperbolic surfaces : those admitting a splitting into bounded pieces, and some riemannian coverings.Differential Geometrytext