Vanishing of the first reduced cohomology with values in an $L^p$representation
Abstract
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 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. Combining our results with those of Pansu, we obtain a new characterization of Gromov hyperbolic homogeneous manifolds: these are the ones having nonzero first reduced $L^p$cohomology for some $1<p<\infty.$
 Publication:

arXiv Mathematics eprints
 Pub Date:
 October 2006
 arXiv:
 arXiv:math/0611001
 Bibcode:
 2006math.....11001T
 Keywords:

 Mathematics  Geometric Topology;
 Mathematics  Group Theory;
 20F65;
 22F30
 EPrint:
 20 pages, correction: minor changes (introduction)