# Poincaré inequalities and rigidity for actions on Banach spaces

Journal of the European Mathematical Society (2015)

- Volume: 017, Issue: 3, page 689-709
- ISSN: 1435-9855

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topNowak, 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 -

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