Banach Zuk's criterion for partite complexes with application to random groups

We prove a Banach version of Żuk's criterion for groups acting on partite 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 (L^p 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 (FL^p) 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.