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Proper isometric actions of hyperbolic groups on $L^p$-spaces

Part of: Special aspects of infinite or finite groups Analysis on metric spaces Connections with homological algebra and category theory Abstract harmonic analysis

Published online by Cambridge University Press:  26 February 2013

Bogdan Nica
Bogdan Nica*
Affiliation:
Mathematisches Institut, Georg-August Universität Göttingen, Bunsenstrasse 3–5, D-37073 Göttingen, Germany (email: [email protected])
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Abstract

We show that every non-elementary hyperbolic group $\G $ admits a proper affine isometric action on $L^p(\bd \G \times \bd \G )$, where $\bd \G $ denotes the boundary of $\G $ and $p$ is large enough. Our construction involves a $\G $-invariant measure on $\bd \G \times \bd \G $ analogous to the Bowen–Margulis measure from the ${\rm CAT}(-1)$ setting, as well as a geometric, Busemann-type cocycle. We also deduce that $\G $ admits a proper affine isometric action on the first $\ell ^p$-cohomology group $H^1_{(p)}(\G )$ for large enough $p$.

Keywords

isometric actions on Banach spaceshyperbolic groupsBowen–Margulis measure$\ell ^p$-cohomology

MSC classification

Secondary: 20F67: Hyperbolic groups and nonpositively curved groups 20J06: Cohomology of groups 30L10: Quasiconformal mappings in metric spaces 43A15: $L^p$-spaces and other function spaces on groups, semigroups, etc.
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 5 , May 2013 , pp. 773 - 792
Copyright
Copyright © 2013 The Author(s) 

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