Poincare inequalities, embeddings, and wild groups

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 Poincare 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'.