  We prove a Banach version of \.Zuk'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 $(F L^p)$ 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.
