Gromov conjectured that the L^p-cohomology of simple Lie groups vanishes below the rank.
Farb conjectured a fixed point property for actions of lattices in such groups on CAT(0) cell complexes of dimension lower than the rank.
In this talk I will give a short introduction to group cohomology and prove that vanishing of \ell^1-cohomology up to degree n implies a finite orbit for every action on an n-dimensional contrcatible complex, thus in particular establishing that Gromov's conjecture implies Farb's conjecture. I will then give some insight to the proof of Gromov's conjecture.
This talk is based on joint work with Saar Bader, Uri Bader and Roman Sauer.
