Żuk's criterion for Banach spaces and random groups

We prove a Banach version of Żuk's criterion for groups acting on partite (i.e., colorable) simplicial complexes. Using this new criterion, we derive a new fixed point theorem for random groups in the Gromov density model with respect to several classes of Banach spaces (spaces, Hilbertian spaces, uniformly curved spaces). In particular, we show that for every p, a group in the Gromov density model has asymptotically almost surely property and give a sharp lower bound for the growth of the conformal dimension of the boundary of such group as a function of the parameters of the density model.