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

2023/02/18 by Neumann, Antonio López
#20F67 #20G07 #22E41 #43A15 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2302.09307

Abstract

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 D₄ 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.

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