  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".
