Vanishing of the first reduced cohomology with values in an L p -representation

Romain Tessera[1]

  • [1] Vanderbilt University Department of Mathematics Stevenson Center Nashville, TN 37240 (USA)

Annales de l’institut Fourier (2009)

  • Volume: 59, Issue: 2, page 851-876
  • ISSN: 0373-0956

Abstract

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We prove that the first reduced cohomology with values in a mixing L p -representation, 1 < p < ∞ , 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 ℓ 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 < ∞ .

How to cite

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Tessera, Romain. "Vanishing of the first reduced cohomology with values in an $L^p$-representation." Annales de l’institut Fourier 59.2 (2009): 851-876. <http://eudml.org/doc/10413>.

@article{Tessera2009,
abstract = {We prove that the first reduced cohomology with values in a mixing $L^p$-representation, $1&lt;p&lt;\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 $1&lt;p&lt;\infty .$},
affiliation = {Vanderbilt University Department of Mathematics Stevenson Center Nashville, TN 37240 (USA)},
author = {Tessera, Romain},
journal = {Annales de l’institut Fourier},
keywords = {Reduced $L^p$-cohomology; amenable groups; Folner sequences; hyperbolic metric spaces; homogeneous Riemannian manifold; reduced -cohomology; Følner sequences},
language = {eng},
number = {2},
pages = {851-876},
publisher = {Association des Annales de l’institut Fourier},
title = {Vanishing of the first reduced cohomology with values in an $L^p$-representation},
url = {http://eudml.org/doc/10413},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Tessera, Romain
TI - Vanishing of the first reduced cohomology with values in an $L^p$-representation
JO - Annales de l’institut Fourier
PY - 2009
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 2
SP - 851
EP - 876
AB - We prove that the first reduced cohomology with values in a mixing $L^p$-representation, $1&lt;p&lt;\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 $1&lt;p&lt;\infty .$
LA - eng
KW - Reduced $L^p$-cohomology; amenable groups; Folner sequences; hyperbolic metric spaces; homogeneous Riemannian manifold; reduced -cohomology; Følner sequences
UR - http://eudml.org/doc/10413
ER -

References

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