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- Title
Proper isometric actions of hyperbolic groups on $L^p$-spaces.
- Authors
Nica, Bogdan
- 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 $p$.
- Subjects
ISOMETRICS (Mathematics); HYPERBOLIC groups; HYPERBOLIC spaces; AFFINE geometry; INVARIANT measures
- Publication
Compositio Mathematica, 2013, Vol 149, Issue 5, p773
- ISSN
0010-437X
- Publication type
Article
- DOI
10.1112/S0010437X12000693