# 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

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topTessera, 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<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 $1<p<\infty .$},

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<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 $1<p<\infty .$

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

top- M. Bourdon, Cohomologie $lp$ et produits amalgamés, Geom. Ded. 107 (2004), 85-98 Zbl1124.20025MR2110755
- M. Bourdon, F. Martin, A. Valette, Vanishing and non-vanishing of the first ${L}^{p}$-cohomology of groups, Comment. math. Helv. 80 (2005), 377-389 Zbl1139.20045MR2142247
- M. Bourdon, H. Pajot, Cohomologie ${L}^{p}$ et espaces de Besov, Journal fur die Reine und Angewandte Mathematik 558 (2003), 85-108 Zbl1044.20026MR1979183
- A. Cheeger, M. Gromov, ${L}^{2}$-cohomology and group cohomology, Topology 25 (1986), 189-215 Zbl0597.57020MR837621
- Y. de Cornulier, R. Tessera, Quasi-isometrically embedded free sub-semigroups, (2006) Zbl1184.20041
- Y. de Cornulier, R. Tessera, A. Valette, Isometric group actions on Hilbert spaces: growth of cocycles Zbl1129.22004
- Y. de Cornulier, R. Tessera, A. Valette, Isometric group actions on Banach spaces and representations vanishing at infinity, (2006) Zbl1149.22006
- P. Delorme, 1-cohomologie des représentations unitaires des groupes de Lie semi-simples et résolubles. Produits tensoriels continus de représentations, Bull. Soc. Math. France 105 (1977), 281-336 Zbl0404.22006MR578893
- E. Ghys, P. de la Harpe, Sur les groupes hyperboliques d’après Mikhael Gromov, Progress in Mathematics 83 (1990), Birkhäuser Boston Inc., Boston, MA Zbl0731.20025MR1086648
- V. M. Gol’dshteĭn, V. I. Kuz’minov, I. A. Shvedov, ${L}_{p}$-cohomology of Riemannian manifolds, Issled. Geom. Mat. Anal. 199 (1987), 101-116 Zbl0654.58030
- M. Gromov, Hyperbolic groups, Essays in group theory 8 (1987), 75-263, Math. Sci. Res. Inst. Publ., New York Zbl0634.20015MR919829
- M. Gromov, Asymptotic invariants of groups, 182 (1993), Cambridge University Press Zbl0841.20039MR1253544
- Y. Guivarc’h, Croissance polynomiale et périodes des fonctions harmoniques, Bull. Sc. Math. France 101 (1973), 333-379 Zbl0294.43003MR369608
- E. Heintze, On Homogeneous Manifolds with Negative Curvature, Math. Ann. 211 (1974), 23-34 Zbl0273.53042MR353210
- N. Holopainen, G. Soardi, A strong Liouville theorem for $p$-harmonic functions on graphs, Ann. Acad. Sci. Fenn. Math. 22 (1997), 205-226 Zbl0874.31008MR1430400
- ${\ell}^{p}$-cohomology for groups of type $F{P}_{n}$, (2006)
- F. Martin, Reduced 1-cohomology of connected locally compact groups and applications, J. Lie Theory 16 (2006), 311-328 Zbl1115.22006MR2197595
- F. Martin, A. Valette, On the first ${L}^{p}$-cohomology of discrete groups, (2006) Zbl1175.20045
- P. Pansu, Cohomologie ${L}^{p}$ des variétés à courbure négative, cas du degré 1, Rend. Semin. Mat., Torino Fasc. Spec. (1989), 95-120 Zbl0723.53023MR1086210
- P. Pansu, Métriques de Carnot-Caratheodory et quasi-isométries des espaces symmétriques de rang un, Ann. Math. 14 (1989), 177-212 Zbl0678.53042
- P. Pansu, Cohomologie ${L}^{p}$: invariance sous quasiisométries, Preprint (1995)
- P. Pansu, Cohomologie ${L}^{p}$, espaces homogènes et pincement, (1999)
- P. Pansu, Cohomologie ${L}^{p}$ en degré 1 des espaces homogènes, Preprint (2006) Zbl1197.43001
- M. J. Puls, The first ${L}^{p}$-cohomology of some finitely generated groups and $p$-harmonic functions, J. Funct. Ana. 237 (2006), 391-401 Zbl1094.43003MR2230342
- Y. Shalom, Harmonic analysis, cohomology, and the large scale geometry of amenable groups, Acta Math. 193 (2004), 119-185 Zbl1064.43004MR2096453
- R. Tessera, Asymptotic isoperimetry on groups and uniform embeddings into Banach spaces, math.GR/0603138 (2006) Zbl1274.43009
- R. Tessera, Large scale Sobolev inequalities on metric measure spaces and applications, arXiv math.MG/0702751 (2006) Zbl1194.53036

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