Vanishing of the second $L^p$-cohomology group for most semisimple groups of rank at least 3
[Annulation de la cohomologie $L^p$ en degré 2 pour la plupart des groupes semi-simples de rang au moins 3]
López Neumann, Antonio1
1Institute of Mathematics, of the Polish Academy of Sciences (IMPAN), Jana i Jędrzeja Śniadeckich 8, 00-656 Warsaw (Poland)
Annales de l'Institut Fourier, Online first, 39 p.

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 $\mathrm{SL}(4)$, for simple groups of rank $\ge 4$ that are not of exceptional type or of type $D_4$ and for all semisimple, non-simple groups of rank $\ge 3$. 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.

On montre l’annulation de la cohomologie $L^p$ en degré 2 pour la plupart des groupes semi-simples de rang au moins 3 sur des corps locaux. Plus précisément, on montre ce résultat pour $\mathrm{SL}(4)$, pour les groupes simples de rang au moins $4$ qui ne sont ni de type exceptionnel ni de type $D_4$ et pour tous les groupes semi-simples, non simples de rang au moins $3$. Nos méthodes s’appliquent lorsque $p$ est assez grand dans le cas réel et pour tout $p>1$ dans le cas non archimédien. Ce résultat suggère une réponse affirmative à la question de Gromov sur l’annulation de la cohomologie $L^p$ des groupes semi-simples dans les degrés au dessous du rang. Nos méthodes consistent à utiliser une suite spectrale à la Bourdon–Rémy, adapter une version du phénomène de Mautner à la Cornulier–Tessera et puis conclure par une étude combinatoire au cas par cas des groupes classiques simples.

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DOI : 10.5802/aif.3777
Classification : 20F67, 20G07, 22E41, 43A15
Keywords: Semisimple algebraic groups, $L^p$-cohomology, spectral sequence, Heintze groups, root systems
Mots-clés : Groupes algébriques semi-simples, cohomologie $L^p$, suite spectrale, groupes de Heintze, systèmes de racines

López Neumann, Antonio  1

1 Institute of Mathematics, of the Polish Academy of Sciences (IMPAN), Jana i Jędrzeja Śniadeckich 8, 00-656 Warsaw (Poland) 
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López Neumann, Antonio. Vanishing of the second $L^p$-cohomology group for most semisimple groups of rank at least 3. Annales de l'Institut Fourier, Online first, 39 p.

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