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

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

2012-02-13

arxiv.org/abs/1202.2597v1

Mathematics

Group Theory

17 pages

Scientific paper

According to a result of G. Yu, any hyperbolic group $\G$ admits a proper isometric action on $\ell^p(\G\times \G)$ for large enough $p$. We show that any 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 $\G$, for large enough $p$. As a key ingredient in our construction, we introduce a $\G$-invariant measure on $\bd\G\times \bd\G$ analogous to the Bowen - Margulis measure from the CAT$(-1)$ setting. 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 $p$.

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