N ov 2 00 6 Vanishing of the first reduced cohomology with values in an L p - representation

We prove that the first reduced cohomology with values in a mixing Lp-representation, 1 < p < ∞, vanishes for a class of amenable groups including 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 lp-cohomology. As a byproduct, we prove a conjecture by Pansu. Namely, the first reduced Lp-cohomology on homogeneous, closed at infinity, Riemannian manifolds vanishes. Combining our results with those of Pansu, we obtain a new characterization of Gromov hyperbolic homogeneous manifolds: these are the ones having non-zero first reduced Lp-cohomology for some 1 < p < ∞.