Appears in collection : Arithmetic and Diophantine Geometry, via Ergodic Theory and o-minimality

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. Both conjectures follow from a new cohomological vanishing result, which could be seen as a Banach version of higher property T. In my talk I will survey the subject and explain the new contribution. Based on a joint work with Saar Bader, Shaked Bader and Roman Sauer.