| Résume | Gromov conjectured that the L^p-cohomology of simple 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-dim contractible 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. |
