Poincar\'e inequalities, embeddings, and wild groups

TL;DR

This paper establishes geometric conditions involving Poincaré inequalities and convexity that guarantee fixed points for isometric group actions on certain metric spaces, advancing understanding of wild group actions.

Contribution

It introduces new geometric criteria ensuring fixed points for wild group actions, connecting metric embedding theory with group action fixed point properties.

Findings

Conditions are stable under scaling and limits.

Valid for various classes of metric spaces.

Guarantees fixed points for Gromov's wild groups actions.

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 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".