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

Bourdon, Marc; Rémy, Bertrand

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

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.

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.

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
Mots-clés : Cohomologie $L^p$, groupe de Lie, espace symétrique, invariance par quasi-isométrie, suite spectrale, annulation et non-annulation de cohomologie, système de racines

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