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

Bourdon, Marc; Rémy, Bertrand

doi:10.5802/ahl.61

Bourdon, Marc; Rémy, Bertrand

Quasi-isometric invariance of continuous group

L^{p}

-cohomology, and first applications to vanishings

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

Metadata

DOI10.5802/ahl.61

Keywords $L^p$-cohomology, topological group, Lie group, symmetric space, quasi-isometric invariance, spectral sequence, cohomology vanishing, root system

Abstract

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.

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