  We prove that the first reduced cohomology with values in a mixing
Lp-representation, p larger than 1, 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 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. We also prove
that a Gromov hyperbolic geodesic metric measure space with bounded geometry
admitting a bi-Lipschitz embedded 3-regular tree has non-trivial first reduced
Lp-cohomology for large enough p. Combining our results with those of Pansu, we
characterize Gromov hyperbolic homogeneous manifolds: these are the ones having
non-zero first reduced Lp-cohomology for some p larger than 1.
