We prove that the first reduced cohomology with values in a mixing $L^p$-representation, $1\<p\<\infty$, vanishes for a class of amenable groups including connected amenable Lie groups. In particular this solves for this class of amenable groups a conjecture of Gromov saying that every finitely generated amenable group has no first reduced ${\ell }^p$-cohomology. As a byproduct, we prove a conjecture by Pansu. Namely, the first reduced $L^p$-cohomology on homogeneous, closed at infinity, Riemannian manifolds vanishes. We also...

Dans cet article, nous nous intéresserons à certaines propriétés des variétés riemanniennes non compactes qui ne dépendant que de leur géométrie à l'infini; pour cela, nous utiliserons un procédé de discrétisation qui associe un graph (pondéré) à une variété.

Let $G$ be a compactly generated locally compact group and let $U$ be a compact generating set. We prove that if $G$ has polynomial growth, then ${\left(U^n\right)}_{n\in \mathbb{N}}$ is a Følner sequence and we give a polynomial estimate of the rate of decay of $\frac{\mu (U^{n+1}\setminus U^n)}{\mu \left(U^n\right)}.$ Our proof uses only two ingredients: the doubling property and a weak geodesic property that we call Property (M). As a matter of fact, the result remains true in a wide class of doubling metric measured spaces including manifolds and graphs. As an application, we obtain a $L^p$-pointwise...