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

TL;DR

This paper investigates the spectral properties of certain infinite topological manifolds constructed from graphs and manifolds, establishing a link between the spectrum's bottom and the graph's amenability, generalizing previous results.

Contribution

It introduces a new class of topologically infinite manifolds built from graphs and manifolds, and characterizes their spectral bottom in relation to the graph's amenability, extending Brooks' theorem.

Findings

The bottom of the spectrum equals that of the base manifold if and only if the graph is amenable.

Explicit bounds are provided for the spectral gap when the graph is non-amenable.

In Riemannian coverings, the spectral bottom inequality holds with equality only for amenable deck groups.

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.