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Poincaré inequalities, embeddings, and wild groups

Published online by Cambridge University Press:  24 August 2011

Assaf Naor
Affiliation:
Courant Institute of Mathematical Sciences, New York University, New York, NY, USA (email: [email protected])
Lior Silberman
Affiliation:
Department of Mathematics, The University of British Columbia, Vancouver, BC, Canada (email: [email protected])
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Abstract

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We present geometric conditions on a metric space (Y,dY) ensuring that, almost surely, any isometric action on Y by Gromov’s expander-based random group has a common fixed point. These geometric conditions involve uniform convexity and the validity of nonlinear Poincaré inequalities, and they are stable under natural operations such as scaling, Gromov–Hausdorff limits, and Cartesian products. We use methods from metric embedding theory to establish the validity of these conditions for a variety of classes of metric spaces, thus establishing new fixed point results for actions of Gromov’s ‘wild groups’.

Type
Research Article
Copyright
Copyright © Foundation Compositio Mathematica 2011

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