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 non-zero first reduced $L^p$-cohomology for some $1<p<\infty.$
- Publication:
-
arXiv Mathematics e-prints
- Pub Date:
- October 2006
- DOI:
- 10.48550/arXiv.math/0611001
- arXiv:
- arXiv:math/0611001
- Bibcode:
- 2006math.....11001T
- Keywords:
-
- Mathematics - Geometric Topology;
- Mathematics - Group Theory;
- 20F65;
- 22F30
- E-Print:
- 20 pages, correction: minor changes (introduction)