Geodesic
Poincaré inequalities and rigidity for actions on Banach spaces
Journal of the European Mathematical Society, Tome 17 (2015) no. 3, pp. 689-709
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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 H1(G,π)=0 for every isometric representation π of G on X. The condition is expressed in terms of p-Poincar\'{e} constants and we provide examples of groups, which satisfy such conditions and for which H1(G,π) vanishes for every isometric representation π on an Lp​ 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.
DOI : 10.4171/jems/514
Classification : 22-XX, 46-XX
Keywords: Poincaré inequality, Kazhdan’s property (T); affine isometric action 
@article{JEMS_2015_17_3_a6,
     author = {Piotr W. Nowak},
     title = {Poincar\'e inequalities and rigidity for actions on {Banach} spaces},
     journal = {Journal of the European Mathematical Society},
     pages = {689--709},
     year = {2015},
     volume = {17},
     number = {3},
     doi = {10.4171/jems/514},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/514/}
}
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%A Piotr W. Nowak
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%J Journal of the European Mathematical Society
%D 2015
%P 689-709
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Piotr W. Nowak. Poincaré inequalities and rigidity for actions on Banach spaces. Journal of the European Mathematical Society, Tome 17 (2015) no. 3, pp. 689-709. doi: 10.4171/jems/514

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