Vanishing of the first reduced cohomology with values in an L p -representation
Annales de l'Institut Fourier, Volume 59 (2009) no. 2, pp. 851-876.

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 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<.

Nous prouvons que pour une classe de groupes moyennables, incluant tous les groupes moyennables de Lie connexes, la cohomologie réduite en degré 1 à valeurs dans une représentation mélangeante sur un espace L p , pour p>1, est nulle. En particulier, cela démontre pour cette classe de groupes moyennables une conjecture de Gromov s’appliquant à tous les groupes de type fini moyennables. Nous obtenons également la version “de Lie” de cette conjecture, qui avait été formulée par Pansu. Nous montrons par ailleurs qu’un espace métrique hyperbolique possédant un arbre 3-regulier quasi-isométriquement plongé a un premier groupe de cohomologie L p réduite non trivial pour p assez grand. Finalement, en combinant nos résultats avec ceux de Pansu, nous obtenons une caractérisation des variétés riemanniennes homogènes hyperboliques au sens de Gromov : ce sont celles qui possèdent de la cohomologie L p réduite en degré 1 pour p assez grand.

Received:
Revised:
Accepted:
DOI: 10.5802/aif.2449
Classification: 20F65,  22F30
Keywords: Reduced L p -cohomology, amenable groups, Folner sequences, hyperbolic metric spaces, homogeneous Riemannian manifold
Tessera, Romain 1

1 Vanderbilt University Department of Mathematics Stevenson Center Nashville, TN 37240 (USA)
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Tessera, Romain. Vanishing of the first reduced cohomology with values in an $L^p$-representation. Annales de l'Institut Fourier, Volume 59 (2009) no. 2, pp. 851-876. doi : 10.5802/aif.2449. https://aif.centre-mersenne.org/articles/10.5802/aif.2449/

[1] Bourdon, M. Cohomologie lp et produits amalgamés, Geom. Ded., Volume 107 (2004) no. 1, pp. 85-98 | Article | MR: 2110755 | Zbl: 1124.20025

[2] Bourdon, M.; Martin, F.; Valette, A. Vanishing and non-vanishing of the first L p -cohomology of groups, Comment. math. Helv., Volume 80 (2005), pp. 377-389 | Article | MR: 2142247 | Zbl: 1139.20045

[3] Bourdon, M.; Pajot, H. Cohomologie L p et espaces de Besov, Journal fur die Reine und Angewandte Mathematik, Volume 558 (2003), pp. 85-108 | Article | MR: 1979183 | Zbl: 1044.20026

[4] Cheeger, A.; Gromov, M. L 2 -cohomology and group cohomology, Topology, Volume 25 (1986), pp. 189-215 | Article | MR: 837621 | Zbl: 0597.57020

[5] de Cornulier, Y.; Tessera, R. Quasi-isometrically embedded free sub-semigroups, 2006 (To appear in Geometry and Topology)

[6] de Cornulier, Y.; Tessera, R.; Valette, A. Isometric group actions on Hilbert spaces: growth of cocycles (To appear in GAFA) | Zbl: 1129.22004

[7] de Cornulier, Y.; Tessera, R.; Valette, A. Isometric group actions on Banach spaces and representations vanishing at infinity, 2006 (To appear in Transformation Groups) | Zbl: 1149.22006

[8] Delorme, P. 1-cohomologie des représentations unitaires des groupes de Lie semi-simples et résolubles. Produits tensoriels continus de représentations, Bull. Soc. Math. France, Volume 105 (1977), pp. 281-336 | Numdam | MR: 578893 | Zbl: 0404.22006

[9] Ghys, E.; de la Harpe, P. Sur les groupes hyperboliques d’après Mikhael Gromov, Progress in Mathematics, Volume 83 (1990) | MR: 1086648 | Zbl: 0731.20025

[10] Gol’dshteĭn, V. M.; Kuz’minov, V. I.; Shvedov, I. A. L p -cohomology of Riemannian manifolds, Issled. Geom. Mat. Anal., Volume 199 (1987) no. 7, pp. 101-116 ((Russian) Trudy Inst. Mat. (Novosibirsk)) | MR: 905262 | Zbl: 0654.58030

[11] Gromov, M. Hyperbolic groups, Essays in group theory, Volume 8 (1987), pp. 75-263 | MR: 919829 | Zbl: 0634.20015

[12] Gromov, M. Asymptotic invariants of groups Volume 182, Cambridge University Press, 1993 | MR: 1253544

[13] Guivarc’h, Y. Croissance polynomiale et périodes des fonctions harmoniques, Bull. Sc. Math. France, Volume 101 (1973), pp. 333-379 | EuDML: 87214 | Numdam | MR: 369608 | Zbl: 0294.43003

[14] Heintze, E. On Homogeneous Manifolds with Negative Curvature, Math. Ann., Volume 211 (1974), pp. 23-34 | Article | EuDML: 162631 | MR: 353210 | Zbl: 0273.53042

[15] Holopainen, N.; Soardi, G. A strong Liouville theorem for p-harmonic functions on graphs, Ann. Acad. Sci. Fenn. Math., Volume 22 (1997) no. 1, pp. 205-226 | EuDML: 226694 | MR: 1430400 | Zbl: 0874.31008

[16] p -cohomology for groups of type FP n , 2006 (math.FA/0511002)

[17] Martin, F. Reduced 1-cohomology of connected locally compact groups and applications, J. Lie Theory, Volume 16 (2006), pp. 311-328 | MR: 2197595 | Zbl: 1115.22006

[18] Martin, F.; Valette, A. On the first L p -cohomology of discrete groups (2006) (Preprint) | MR: 2294249 | Zbl: 1175.20045

[19] Pansu, P. Cohomologie L p des variétés à courbure négative, cas du degré 1, Rend. Semin. Mat., Torino Fasc. Spec. (1989), pp. 95-120 | MR: 1086210 | Zbl: 0723.53023

[20] Pansu, P. Métriques de Carnot-Caratheodory et quasi-isométries des espaces symmétriques de rang un, Ann. Math., Volume 14 (1989), pp. 177-212 | MR: 979599 | Zbl: 0678.53042

[21] Pansu, P. Cohomologie L p : invariance sous quasiisométries, Preprint, 1995

[22] Pansu, P. Cohomologie L p , espaces homogènes et pincement (1999) (Unpublished manuscript)

[23] Pansu, P. Cohomologie L p en degré 1 des espaces homogènes, Preprint, 2006 | MR: 2322503 | Zbl: pre05181216

[24] Puls, M. J. The first L p -cohomology of some finitely generated groups and p-harmonic functions, J. Funct. Ana., Volume 237 (2006), pp. 391-401 | Article | MR: 2230342 | Zbl: 1094.43003

[25] Shalom, Y. Harmonic analysis, cohomology, and the large scale geometry of amenable groups, Acta Math., Volume 193 (2004), pp. 119-185 | Article | MR: 2096453 | Zbl: 1064.43004

[26] Tessera, R. Asymptotic isoperimetry on groups and uniform embeddings into Banach spaces, math.GR/0603138, 2006

[27] Tessera, R. Large scale Sobolev inequalities on metric measure spaces and applications, arXiv math.MG/0702751, 2006 | MR: 2490163 | Zbl: 1194.53036

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