Graphes, moyennabilité et bas du spectre de variétés topologiquement infinies

الأصل · EN

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 N is a riemannian covering and the metric of N is generic, then λ₀(M)≥ λ₀(N) with equality if and only if the deck-group is amenable. This generalizes a result of R. Brooks.