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Poincaré inequalities, embeddings, and wild groups

Part of: Special aspects of infinite or finite groups Normed linear spaces and Banach spaces; Banach lattices Variational problems in infinite-dimensional spaces

Published online by Cambridge University Press:  24 August 2011

Assaf Naor  and
Lior Silberman
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’.

Keywords

Gromov’s random groupsfixed pointsPoincaré inequalities

MSC classification

Secondary: 20F65: Geometric group theory 58E40: Group actions 46B85: Embeddings of discrete metric spaces into Banach spaces; applications in topology and computer science
Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 5 , September 2011 , pp. 1546 - 1572
Copyright
Copyright © Foundation Compositio Mathematica 2011

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