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...

In this paper we answer three open problems on varieties of topological groups by invoking Lie group theory. We also reprove in the present context that locally compact groups with arbitrarily small invariant identity neighborhoods can be approximated by Lie groups

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...