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$L^p$-cohomology for
higher-rank spaces and groups: critical exponents and
rigidity
Marc Bourdon and Bertrand Rémy |
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Geometry & Topology 29 (2025) 4799–4839
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DOI: 10.2140/gt.2025.29.4799
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Abstract |
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We initiate the investigation of critical exponents (in degree equal to the rank) for the vanishing of -cohomology of higher-rank Lie groups and related manifolds. We deal with the rank 2 case and exhibit such phenomena for and for a family of 5-dimensional solvable Lie groups. We use the critical exponents to compare the groups up to quasi-isometry. This leads us to exhibit a continuum of quasi-isometry classes of rank 2 irreducible solvable Lie groups of nonpositive curvature. In the course of the proof, we provide a detailed description of the -cohomology of the real and complex hyperbolic spaces. It is then combined with a spectral sequence argument, to derive our higher-rank results. |
Keywords
$L^p$-cohomology, Lie group, symmetric space,
quasi-isometric invariance, spectral sequence, cohomology
(non)vanishing
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Mathematical Subject Classification
Primary: 20J05, 20J06, 22E15, 22E41, 53C35
Secondary: 57T10, 57T15
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References |
Publication
Received: 3 January 2024
Revised: 29 March 2025
Accepted: 18 May 2025
Published: 31 December 2025
Proposed: David Fisher
Seconded: Mladen Bestvina, Martin R Bridson
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Authors |
| Marc Bourdon | |
| Laboratoire Paul Painlevé UMR 8524 CNRS / Université de Lille Villeneuve d’Ascq France |
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| Bertrand Rémy | |
| Unité de Mathématiques Pures et
Appliaquées UMR 5669 CNRS / École normale supérieure de Lyon Lyon France |
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| © 2025 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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