Three quite separate phenomena will be related:a) Gromov conjectured that the Lp-cohomology of simple groups vanishes below the rank.b) Farb conjectured a fixed point property for actions of lattices in such groups on CAT(0) cell complexes of dimension lower than the rank.c) Recent work of Loewe-Frazcyk and Bader-Sauer suggest that there is a waist inequality for locally symmetric spaces up to the rank.Both a) and b) follow from a new cohomological vanishing result, which could be seen as a Banach version of higher property T. This result provides further evidence for c) albeit does not prove it. Based on a joint work with Uri Bader, Saar Bader, Shaked Bader.