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

Bertrand Remy (1) , Marc Bourdon (2)

1 CMLS - Centre de Mathématiques Laurent Schwartz
2 LPP - Laboratoire Paul Painlevé - UMR 8524

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.

Mots clés: L^p-cohomology; topological group; Lie group; symmetric space; quasi-isometric invariance; spectral sequence; cohomology vanishing; root system

Domaines: Théorie des groupes [math.GR]

Bertrand Remy, Marc Bourdon. Quasi-isometric invariance of continuous group $L^p$-cohomology, and first applications to vanishings. 2019. ⟨hal-01742591v2⟩