Graphes, moyennabilité et bas du spectre de variétés topologiquement infinies

Details Graphes, moyennabilité et bas du spectre de variétés topologiquement infinies Graphes, moyennabilité et bas du spectre de variétés topologiquement infinies

2010-01-14

arxiv.org/abs/1001.2501v1

Mathematics

Differential Geometry

37 pages, 4 figures, language = French, submitted

Scientific paper

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.

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