Poincaré inequalities, embeddings, and wild groups

Details Poincaré inequalities, embeddings, and wild groups Poincaré inequalities, embeddings, and wild groups

2010-05-21

arxiv.org/abs/1005.4084v3

Mathematics

Group Theory

Minor changes to address comments of a referee. To appear in Compositio Mathematica

Scientific paper

We present geometric conditions on a metric space $(Y,d_Y)$ 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 Poincar\'e 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".

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