Quasi-isometric invariance of continuous group $L^p$-cohomology, and first applications to vanishings

Bourdon, Marc; Rémy, Bertrand

Annales Henri Lebesgue, Tome 3 (2020), pp. 1291-1326.

Nous prouvons que la cohomologie $L^p$ continue des groupes localement compacts à base dénombrable d’ouverts est un invariant de quasi-isométrie. En guise d’applications, nous obtenons des résultats partiels dans le sens d’une réponse positive à une question posée par M. Gromov, suggérant un comportement classique de la cohomologie $L^p$ continue des groupes de Lie réels simples. Outre l’invariance par quasi-isométrie, les ingrédients utilisés sont un argument de suite spectrale et des résultats d’annulation pour les espaces hyperboliques dus à P. Pansu. Dans les cas de groupes de Lie les mieux adaptés, nous obtenons la moitié des annulations attendues.

We show that the continuous $L^p$-cohomology of locally compact second countable groups is a quasi-isometric invariant. As an application, we prove partial results supporting a positive answer to a question asked by M. Gromov, suggesting a classical behaviour of continuous $L^p$-cohomology of simple real Lie groups. In addition to quasi-isometric invariance, the ingredients are a spectral sequence argument and Pansu’s vanishing results for real hyperbolic spaces. In the best adapted cases of simple Lie groups, we obtain nearly half of the relevant vanishings.

Classification : 20J05, 20J06, 22E15, 22E41, 53C35, 55B35, 57T10, 57T15
Mots-clés : $L^p$-cohomology, topological group, Lie group, symmetric space, quasi-isometric invariance, spectral sequence, cohomology vanishing, root system

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