Fixed point properties and cohomology of Banach representations of arithmetic groups

We study fixed point theorems for actions of lattices of semisimple groups. They are deduced from vanishing results for the group cohomology of $L^p$-representations. We show that for lattices in simple groups of higher rank, the cohomology with $L^p$-coefficients vanishes below the rank whenever there are no invariant vectors. As a corollary of the vanishing for $L^1$-coefficients, we obtain that every action on an acyclic simplicial complex of dimension lower than the rank has a finite orbit. This in particular proves a conjecture by Farb. The $L^p$-vanishing below the rank proves a conjecture by Gromov regarding $L^p$-cohomology of symmetric spaces.