Poincaré inequalities, embeddings, and wild groups

Abstract

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 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".

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

References in corpus (1)

• Metric structures in L_1: Dimension, snowflakes, and average distortion

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