Graphes, moyennabilit\'e et bas du spectre de vari\'et\'es topologiquement infinies

Abstract

From a graph $G$ with constant valency $v$ and a (non-compact) manifold $C$ with $v$ boundary components, we build a $G$-periodic manifold $M$. This process gives a class of topologically infinite manifolds which generalizes periodic manifolds and includes all riemannian coverings with finitely generated deck-group. Our main result is that, when the first eigenfunction of $C$ extends to $M$, the bottom of the spectrum of $M$ is equal to $C$'s if and only if the graph $G$ is amenable. When $G$ is not amenable, we control explicitly the gap between these bottom of the spectrum. In particular, we show that if $p : M\ra N$ is a riemannian covering and the metric of $N$ is generic, then $\lambda_0(M)\geq \lambda_0(N)$ with equality if and only if the deck-group is amenable. This generalizes a result of R. Brooks.