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3 - Cohomologie et actions isométriques propres sur les espaces Lp

Published online by Cambridge University Press:  05 January 2016

By
Marc Bourdon
Edited by
C. S. Aravinda ,
F. T. Farrell  and
J. -F. Lafont
Marc Bourdon
Affiliation:
Université de Lille 1
C. S. Aravinda
Affiliation:
TIFR Centre for Applicable Mathematics, Bangalore, India
F. T. Farrell
Affiliation:
Tsinghua University, Beijing
J. -F. Lafont
Affiliation:
Ohio State University
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Summary

Abstract

These notes present and discuss some of the basic properties and results of the lp-cohomology of groups.We also use the lp-cohomology to study the proper isometric actions of word hyperbolic groups on Lp-spaces.

Introduction

Ce texte traite de certains aspects de la cohomologie lp des groupes, et des actions isométriques sur les espaces Lp. Pratiquement aucun résultat qu'il contient n'est original. Il comporte deux parties. La première est une relecture détaillée et commentée de certains passages du livre [32] de M. Gromov qui portent sur la cohomologie lp. Cette partie a également bénéficié des nombreuses discussions que j'ai pu avoir avec P. Pansu. Dans la seconde partie, la cohomologie lp est utilisée pour étudier les actions isométriques propres des groupes (Gromov) hyperboliques sur les espaces Lp. Chaque partie se termine par un survol de quelques résultats complémentaires et par des questions.

Cohomologielp

Le premier thème abordé est celui de la cohomologie lp des groupes de type Fini Г. Nous en présentons des définitions et résultats de base. Sont discutés en particulier:

  1. • L'invariance par quasi-isométrie de la cohomologie lp de Г (Th.3 et Def.4),

  2. • Plusieurs caractérisations de la moyennabilité en termes d'homologie et de cohomologie lp (Th.6),

  3. • L'annulation de la cohomologie lp réduite de Г lorsque le centre Г est infini (Prop.10),

  4. • Des énoncés d'annulation de la 1-cohomologie lp en présence d'un sousgroupe distingué ayant des propriétés spéciales (Prop.13 et Th.14),

  5. • Un résultat de représentation harmonique de la 1-cohomologie lp des groupes non moyennables (Cor.7).

Actions isométriques propres

La 1-cohomologie lp de Г décrit les actions isométriques de Г sur lp(Г) associées à la représentation régulière droite. Elle participe donc au second thème abordé dans ces notes, qui est celui des actions isométriques sur les espaces Lp.

Rappelons qu'une action de Г sur un Banach V est dite propre si pour toute partie bornée P ⊂ V le cardinal des g ∈ Г tels que gP ∩ P ≠ ∅ est fini.

Un groupe est dit a-T-menable s'il possède une action isométrique propre sur un Hilbert. Cette notion apparait dans [32] p.177. Elle joue un rôle de premier plan dans l’étude des groupes via leurs actions sur les espaces de Hilbert (voir notamment [16]).

Type
Chapter
Information
Geometry, Topology, and Dynamics in Negative Curvature , pp. 84 - 109
Publisher: Cambridge University Press
Print publication year: 2016

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References

[1] U., Bader, A., Furman, T., Gelander, N., Monod, Property (T) and rigidity for actions on Banach spaces, Acta Math. 198 (2007), no. 1, 57-105.Google Scholar
[2] M., Bonk, J., Heinonen, P., Koskela, Uniformizing Gromov hyperbolic spaces, Astérisque 270, 2001.Google Scholar
[3] M., Bourdon, Sur les immeubles fuchsiens et leur type de quasi-isométrie, Ergodic theory and Dynamical Systems 20 (2000), 343-364.Google Scholar
[4] M., Bourdon, Cohomologie lp et produits amalgamés, Geometriae Dedicata 107 (2004), 85-98.Google Scholar
[5] M., Bourdon, B., Kleiner, Combinatorial modulus, the Combinatorial Loewner Property, and Coxeter groups, Groups Geom. Dyn.7 (2013), no. 1, 39-107.
[6] M., Bourdon, B., Kleiner, Some applications of lp-cohomology to boundaries of Gromov hyperbolic spaces, Groups Geom. Dyn. 9 (2015), no. 2, 435-478.
[7] M., Bourdon, F., Martin, A., Valette, Vanishing and non-vanishing for the first Lp-cohomology of groups, Comment. Math. Helv. 80 (2005), 377-389.
[8] M., Bourdon, H., Pajot, Cohomologie lp et espaces de Besov, J. reine angew. Math. 558 (2003), 85-108.Google Scholar
[9] M., Bridson, A., Haefliger, Metric spaces of non-positive curvature, Grundlheren der Mathematischen Wissenschaften 319, Springer, 1999.Google Scholar
[10] K.S., Brown, Cohomology of groups, Graduate texts in Mathematics, Springer-Verlag, 1982.
[11] N., Brown, E., Guentner, Uniform embedding of bounded geometry spaces into reflexive Banach spaces, Proc. Amer. Math. Soc. 133 (2005), no.7, 2045-2050.Google Scholar
[12] Yu. A., Brudnyi, N. Ya., Krugljak, Interpolation functors and interpolation spaces, Vol. 1, North-Holland Mathematical Library, 47 (1991).Google Scholar
[13] I., Chatterji, C., Drutu, F., Haglund, Kazhdan and Haagerup properties from the median viewpoint, Adv. Math. 225 (2010), no.2, 882-921.Google Scholar
[14] I., Chavel, Isoperimetric inequalities. Differential geometric and analytic perspectives, Cambridge Tracts in Maths 145, 2001.Google Scholar
[15] J., Cheeger, M., Gromov, L2-cohomology and group cohomology, Topology 25 (1986), p 189-215.Google Scholar
[16] P-A., Cherix, M., Cowling, P., Jolissaint, P., Julg, A., Valette, Groups with the Haagerup property. Gromov's a-T-menability, Progress in Mathematics 197, Birkhäuser, 2001.Google Scholar
[17] P-A., Cherix, F., Martin, A., Valette, Spaces with measured walls, the Haagerup property and property (T), Ergod. Th. Dynam. Sys. (2004), 24, 1895-1908.Google Scholar
[18] Y., De Cornulier, R., Tessera, Contracting automorphisms and Lp-cohomology in degree one, Arkiv for Matematiks 49, no. 2, (2011), 295-324.Google Scholar
[19] Y., De Cornulier, R. Tessera, , A., Valette, Isometric group actions on Banach spaces and representations vanishing at infinity, Transform. Groups 13 (2008), no. 1, 125-147.Google Scholar
[20] M., Coornaert, Mesures de Patterson-Sullivan sur le bord d'un espace hyperbolique au sens de M. Gromov, Pacific Journal of Mathematics 159 (1993), 241-270.Google Scholar
[21] M., Coornaert, T., Delzant, A., Papadopoulos, Géométrie et théorie des groupes, les groupes hyperboliques de M. Gromov, Lecture Notes in Mathematics 1441, Springer-Verlag, 1991.
[22] J., Dymara, T., Januszkiewicz, Cohomology of buildings and their automorphism groups, Invent. Math. 150 (2002), 579-627.Google Scholar
[23] G., Elek, The lp-cohomology and the conformal dimension of hyperbolic cones, Geometriae Dedicata 68 (1997), 263-279.Google Scholar
[24] G., Elek, Coarse cohomology and lp-cohomology, K-Theory 13 (1998), no. 1, 1-22.Google Scholar
[25] D., Gaboriau, Invariants l2 de relations d’équivalence et de groupes, Inst. Hautes Etudes Sci. Publ. Math. 95 (2002), 93-150.Google Scholar
[26] V., Gerasimov, Floyd maps for relatively hyperbolic groups, Geom. Funct. Anal. 22 (2012), no. 5, 1361-1399.Google Scholar
[27] S. M., Gersten, Isoperimetric functions of groups and exotic cohomology, In: Combinatorial and Geometric Group Theory, Edinburgh 1993, London Math. Soc. Lecture Notes Ser. 204, Cambridge Univ. Press, 1995, 87-104.Google Scholar
[28] E., Ghys, P., De LaHarpe, (Eds), Sur les groupes hyperboliques, d'après Gromov, Progress in Mathematics 83, Birkhäuser, 1990.Google Scholar
[29] V., Goldstein, V., Kuzminov, I., Shvedov, The Kuenneth formula for Lp cohomology of warped products, Sib. Math. J. 32 (1991), 749-760.Google Scholar
[30] A., Gournay, Boundary values, random walks and lp-cohomology in degree one, Groups, Geom. Dyn., to appear.
[31] M., Gromov, Hyperbolic groups, Essays in Group theory, Ed. S.M., Gersten, Springer 1987, 72-263.Google Scholar
[32] M., Gromov, Asymptotic invariants for infinite groups, London Mathematical Society Lecture Note Series 182, Eds G.A. Niblo and M.A. Roller, 1993.
[33] F., Haglund, F., Paulin, Simplicité de groupes d'automorphismes d'espaces à courbure négative, Geom. Topol. Monograph 1 (1998), 181-248.Google Scholar
[34] J., Heinonen, Lectures on analysis on metric spaces, Universitext, Springer, 2001.Google Scholar
[35] G., Kasparov, G., Yu, The coarse geometric Novikov conjecture and uniform convexity, Adv. Math. 206 (2006), no. 1, 1-56.Google Scholar
[36] H., Kesten, Full Banach mean values on countable groups, Math. Scand. 7, (1959), 146-156.Google Scholar
[37] V., Lafforgue, Propriété (T) renforcée banachique et transformation de Fourier rapide, J. Topol. Anal. 1(2009), no. 3, 191-206.Google Scholar
[38] W., Lueck, L2-Betti numbers of mapping tori and groups, Topology, 33 (1994), 203-214.Google Scholar
[39] J., Mackay, J., Tyson, Conformal dimension. Theory and application, University Lecture Series, 54, American Mathematical Society, Providence, RI, 2010.Google Scholar
[40] B., Nica, Proper isometric actions of hyperbolic groups on Lp-spaces, Compos. Math. 149 (2013), no. 5, 773-792.Google Scholar
[41] P., Nowak, Group actions on Banach spaces, In Handbook of Group Actions; vol. II, 121-149. L., Ji, A., Papadopoulos, S.-T., Yau eds., ALM 32, International Press, Somerville; Higher Education Press, Beijing, 2015.
[42] P., Pansu, Cohomologie Lp des variétés à courbure négative, cas du degré un, PDE and Geometry 1988, Rend. Sem. Mat. Torino, Fasc. Spez. (1989), 95-120.Google Scholar
[43] P., Pansu, Cohomologie Lp : invariance sous quasiisométries, Preprint Université Paris-Sud (1995).Google Scholar
[44] P., Pansu, Cohomologie Lp, espaces homogènes et pincement, Preprint Université Paris-Sud (1999).Google Scholar
[45] P., Pansu, Cohomologie Lp en degré 1 des espaces homogènes, Potential Anal. 27, (2007), 151-165.Google Scholar
[46] P., Pansu, Cohomologie Lp et pincement, Comment. Math. Helv. 83 (2008), no.2, 327-357.Google Scholar
[47] P., Pansu, Lp-cohomology of symmetric spaces, Geometry, analysis and discrete groups, 305-326, Adv. Lect. Math. (ALM), 6, (2008).Google Scholar
[48] P., Pansu, Dimension conforme et sphère à l'infini des variétés à courbure négative, Annales Academiae Scientiarum Fennicae, Series A.I. Mathematica 14 (1990), 177-212.Google Scholar
[49] F., Paulin, Un groupe hyperbolique est déterminé par son bord, Journal of the London Math. Soc. 54 (1996), 50-74.Google Scholar
[50] M., Puls, The first Lp-cohomology of some groups with one end, Arch. Math. (Basel) 88 (2007), no. 6, 500-506.Google Scholar
[51] M. S., Raghunathan, Discrete subgroups of Lie groups, Ergebnisse der mathematik und ihrer Grenzgebiete 68, Springer-Verlag, 1972.Google Scholar
[52] A., Reznikov, Analytic Topology of Groups, Actions, Strings and Varieties, Geometry and dynamics of groups and spaces, 3-93, Prog. Math., 265, Birkauser, Basel, 2008.Google Scholar
[53] W., Rudin, Functional Analysis, McGraw-Hill, 1973.Google Scholar
[54] Y., Shalom, Rigidity, unitary representations of semisimple groups, and fondamental groups of manifolds with rank one transformation group,Ann. of Math. (2) 152 (2000), 113-182.Google Scholar
[55] R., Tessera, Vanishing of the first reduced cohomology with values in an Lp representation, Ann. Inst. Fourier (Grenoble) 59, no.2, 851-876.
[56] R., Tessera, Isoperimetric profile and random walks on locally compact solvable groups, Rev. Mat. Iberoam. 29 (2013), no. 2, 715-737.
[57] G., Yu, Hyperbolic groups admit proper affine isometric actions on lp-spaces, GAFA, 15 (2005), 1144-1151.Google Scholar

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