Bas du spectre de surfaces hyperboliques de volume infini

TL;DR

This paper develops methods to estimate the bottom of the spectrum of the Laplacian on infinite volume hyperbolic surfaces, linking spectral bounds to geometric and combinatorial properties of the surfaces.

Contribution

It introduces new techniques to bound the spectral bottom of infinite hyperbolic surfaces using geometric, combinatorial, and construction-based approaches.

Findings

Bound on $$ for geometrically finite surfaces based on convex core geometry.

Spectral control of infinite periodic hyperbolic surfaces via building blocks and graph data.

Application of methods to surfaces with bounded splitting and Riemannian coverings.

Abstract

This article presents some methods to control the bottom of the spectrum of the Laplacian $\lambda_0$ on hyperbolic surfaces with infinite volume. Our first result bounds the $\lambda_0$ 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 $\lambda_0$ 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 $\lambda_0$ of two other kind of infinite type hyperbolic surfaces : those admitting a splitting into bounded pieces, and some riemannian coverings.