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Zbl 1225.22019
Tessera, Romain
Vanishing of the first reduced cohomology with values in an $L^p$-representation. (English)
[J] Ann. Inst. Fourier 59, No. 2, 851-876 (2009). ISSN 0373-0956; ISSN 1777-5310/e

Summary: 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 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 $L^p$-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 $L^p$-cohomology for some $1<p<\infty$.

MSC 2000:: *22F30 Homogeneous spaces
43A07 Means on groups, etc.
22D10 Unitary representations of locally compact groups
20F65 Geometric group theory
43A85 Analysis on homogeneous spaces
53C23 Global topological methods (a la Gromov)

Keywords: reduced $L^p$-cohomology; amenable groups; Følner sequences; hyperbolic metric spaces; homogeneous Riemannian manifold

Cited in: Zbl 1257.22008 Zbl 1167.22004

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