# Poincaré inequalities, embeddings, and wild groups - Mathematics > Group Theory

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Abstract: We present geometric conditions on a metric space $Y,d Y$ ensuring thatalmost surely, any isometric action on $Y$ by Gromov-s expander-based randomgroup has a common fixed point. These geometric conditions involve uniformconvexity and the validity of nonlinear Poincar\-e inequalities, and they arestable under natural operations such as scaling, Gromov-Hausdorff limits, andCartesian products. We use methods from metric embedding theory to establishthe validity of these conditions for a variety of classes of metric spaces,thus establishing new fixed point results for actions of Gromov-s -wildgroups-.

Author: ** Assaf Naor, Lior Silberman**

Source: https://arxiv.org/