Vanishing of the second L p -cohomology group for most semisimple groups of rank at least 3

We show vanishing of the second L p -cohomology group for most semisimple algebraic groups of rank at least 3 over local fields. More precisely, we show this result for SL ( 4 ) , for simple groups of rank ≥ 4 that are not of exceptional type or of type D 4 and for all semisimple, non-simple groups of rank ≥ 3 . Our methods work for large values of p in the real case and for all p > 1 in the non-Archimedean case. This result points towards a positive answer to Gromov’s question on vanishing of L p -cohomology of semisimple groups for all p > 1 in degrees below the rank. The methods consist in using a spectral sequence à la Bourdon–Rémy, adapting a version of Mautner’s phenomenon from Cornulier–Tessera and concluding thanks to a combinatorial case-by-case study of classical simple groups.