Non-vanishing for group $L^p$-cohomology of solvable and semisimple Lie groups

Marc Bourdon; Bertrand Rémy

doi:10.5802/jep.232

Non-vanishing for group

L^{p}

-cohomology of solvable and semisimple Lie groups
[Non-annulation de la cohomologie

L^{p}

pour les groupes résolubles et semi-simples]

Marc Bourdon ¹ ; Bertrand Rémy ²

¹ Laboratoire Paul Painlevé, UMR 8524 CNRS / Université de Lille Cité Scientifique, Bât. M2, 59655 Villeneuve d’Ascq, France
² Unité de Mathématiques Pures et Appliquées, UMR 5669 CNRS / École normale supérieure de Lyon 46 allée d’Italie, 69364 Lyon cedex 07, France

Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 771-814.

Résumé
Abstract

Nous obtenons des résultats de non-annulation de la cohomologie $L^{p}$ pour des groupes de Lie lorsque $p$ est grand et quand le degré est égal au rang du groupe. Ces résultats s’appliquent à la fois aux groupes semi-simples et à certains groupes résolubles. En particulier, ils confirment que la question de Gromov concernant l’annulation en-dessous du rang est formulée de façon optimale. Pour obtenir ces résultats, des annulations complémentaires sont combinées à l’usage de suites spectrales. Afin de déduire le cas semi-simple du cas résoluble, nous utilisons également des comparaisons entre diverses versions de la cohomologie $L^{p}$ , et nous appliquons l’invariance par quasi-isométrie.

We obtain non-vanishing of group $L^{p}$ -cohomology of Lie groups for $p$ large and when the degree is equal to the rank of the group. This applies both to semisimple and to some suitable solvable groups. In particular, it confirms that Gromov’s question on vanishing below the rank is formulated optimally. To achieve this, some complementary vanishings are combined with the use of spectral sequences. To deduce the semisimple case from the solvable one, we also need comparison results between various theories for $L^{p}$ -cohomology, allowing the use of quasi-isometry invariance.

Reçu le : 2021-10-18
Accepté le : 2023-04-04
Publié le : 2023-05-04

DOI : 10.5802/jep.232

Classification : 20J05, 20J06, 22E15, 22E41, 53C23, 53C30, 55N35, 57T10, 57T15
Keywords: $L^p$-cohomology, Lie group, symmetric space, quasi-isometric invariance, spectral sequence, cohomology (non-)vanishing, root system
Mot clés : Cohomologie $L^p$, groupe de Lie, espace symétrique, invariance par quasi-isométrie, suite spectrale, annulation et non-annulation de cohomologie, système de racines

Affiliations des auteurs :

Marc Bourdon ¹ ; Bertrand Rémy ²

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Marc Bourdon and Bertrand R\'emy},
     title = {Non-vanishing for group $L^p$-cohomology of solvable and semisimple {Lie} groups},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {771--814},
     publisher = {\'Ecole polytechnique},
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     year = {2023},
     doi = {10.5802/jep.232},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.232/}
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Marc Bourdon; Bertrand Rémy. Non-vanishing for group $L^p$-cohomology of solvable and semisimple Lie groups. Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 771-814. doi : 10.5802/jep.232. https://jep.centre-mersenne.org/articles/10.5802/jep.232/

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