We obtain non-vanishing of group -cohomology of Lie groups for 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 -cohomology, allowing the use of quasi-isometry invariance.
Nous obtenons des résultats de non-annulation de la cohomologie pour des groupes de Lie lorsque 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 , et nous appliquons l’invariance par quasi-isométrie.
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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
Marc Bourdon 1; Bertrand Rémy 2
@article{JEP_2023__10__771_0, 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}, volume = {10}, year = {2023}, doi = {10.5802/jep.232}, language = {en}, url = {https://jep.centre-mersenne.org/articles/10.5802/jep.232/} }
TY - JOUR AU - Marc Bourdon AU - Bertrand Rémy TI - Non-vanishing for group $L^p$-cohomology of solvable and semisimple Lie groups JO - Journal de l’École polytechnique — Mathématiques PY - 2023 SP - 771 EP - 814 VL - 10 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.232/ DO - 10.5802/jep.232 LA - en ID - JEP_2023__10__771_0 ER -
%0 Journal Article %A Marc Bourdon %A Bertrand Rémy %T Non-vanishing for group $L^p$-cohomology of solvable and semisimple Lie groups %J Journal de l’École polytechnique — Mathématiques %D 2023 %P 771-814 %V 10 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.232/ %R 10.5802/jep.232 %G en %F JEP_2023__10__771_0
Marc Bourdon; Bertrand Rémy. Non-vanishing for group $L^p$-cohomology of solvable and semisimple Lie groups. Journal de l’École polytechnique — Mathématiques, Volume 10 (2023), pp. 771-814. doi : 10.5802/jep.232. https://jep.centre-mersenne.org/articles/10.5802/jep.232/
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