Poincaré inequalities and rigidity for actions on Banach spaces

Nowak, Piotr. "Poincaré inequalities and rigidity for actions on Banach spaces." Journal of the European Mathematical Society 017.3 (2015): 689-709. <http://eudml.org/doc/277293>.

@article{Nowak2015,
abstract = {The aim of this paper is to extend the framework of the spectral method for proving property (T) to the class of reflexive Banach spaces and present a condition implying that every affine isometric action of a given group $G$ on a reflexive Banach space $X$ has a fixed point. This last property is a strong version of Kazhdan’s property (T) and is equivalent to the fact that $H^1(G,\pi )=0$ for every isometric representation $\pi $ of $G$ on $X$. The condition is expressed in terms of $p$-Poincaré constants and we provide examples of groups, which satisfy such conditions and for which $H^1(G,\pi )$ vanishes for every isometric representation $\pi $ on an $L_p$ space for some $p>2$. Our methods allow to estimate such a $p$ explicitly and yield several interesting applications. In particular, we obtain quantitative estimates for vanishing of 1-cohomology with coefficients in uniformly bounded representations on a Hilbert space. We also give lower bounds on the conformal dimension of the boundary of a hyperbolic group in the Gromov density model.},
author = {Nowak, Piotr},
journal = {Journal of the European Mathematical Society},
keywords = {Poincaré inequality; Kazhdan’s property (T); affine isometric action; 1-cohomology; Poincaré inequality; Kazhdan's property (T); affine isometric action; 1-cohomology},
language = {eng},
number = {3},
pages = {689-709},
publisher = {European Mathematical Society Publishing House},
title = {Poincaré inequalities and rigidity for actions on Banach spaces},
url = {http://eudml.org/doc/277293},
volume = {017},
year = {2015},
}

TY - JOUR
AU - Nowak, Piotr
TI - Poincaré inequalities and rigidity for actions on Banach spaces
JO - Journal of the European Mathematical Society
PY - 2015
PB - European Mathematical Society Publishing House
VL - 017
IS - 3
SP - 689
EP - 709
AB - The aim of this paper is to extend the framework of the spectral method for proving property (T) to the class of reflexive Banach spaces and present a condition implying that every affine isometric action of a given group $G$ on a reflexive Banach space $X$ has a fixed point. This last property is a strong version of Kazhdan’s property (T) and is equivalent to the fact that $H^1(G,\pi )=0$ for every isometric representation $\pi $ of $G$ on $X$. The condition is expressed in terms of $p$-Poincaré constants and we provide examples of groups, which satisfy such conditions and for which $H^1(G,\pi )$ vanishes for every isometric representation $\pi $ on an $L_p$ space for some $p>2$. Our methods allow to estimate such a $p$ explicitly and yield several interesting applications. In particular, we obtain quantitative estimates for vanishing of 1-cohomology with coefficients in uniformly bounded representations on a Hilbert space. We also give lower bounds on the conformal dimension of the boundary of a hyperbolic group in the Gromov density model.
LA - eng
KW - Poincaré inequality; Kazhdan’s property (T); affine isometric action; 1-cohomology; Poincaré inequality; Kazhdan's property (T); affine isometric action; 1-cohomology
UR - http://eudml.org/doc/277293
ER -