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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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