1 results
Proper isometric actions of hyperbolic groups on
$L^p$-spaces
- Part of
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- Journal:
- Compositio Mathematica / Volume 149 / Issue 5 / May 2013
- Published online by Cambridge University Press:
- 26 February 2013, pp. 773-792
- Print publication:
- May 2013
-
- Article
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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 
$p$.
