Contracting automorphisms and L^p-cohomology in degree one

Abstract

We characterize those Lie groups, and algebraic groups over a local field of characteristic zero, whose first reduced L^p-cohomology is zero for all p>1, extending a result of Pansu. As an application, we obtain a description of Gromov-hyperbolic groups among those groups. In particular we prove that any non-elementary Gromov-hyperbolic algebraic group over a non-Archimedean local field of zero characteristic is quasi-isometric to a 3-regular tree. We also extend the study to semidirect products of a general locally compact group by a cyclic group acting by contracting automorphisms.

27 pages, no figure

References in corpus (3)

• Isometric group actions on Banach spaces and representations vanishing at infinity
• Dimension of asymptotic cones of Lie groups
• Vanishing of the first reduced cohomology with values in an $L^p$-representation

Cited by in corpus (12)

• On the quasi-isometric classification of locally compact groups
• Fixed points and amenability in non-positive curvature
• Commability and focal locally compact groups
• Quasi-isometric invariance of continuous group $L^p$-cohomology, and first applications to vanishings
• Geometric presentations of Lie groups and their Dehn functions
• Poincaré profiles of Lie groups and a coarse geometric dichotomy
• Large-scale sublinearly Lipschitz geometry of hyperbolic spaces
• De Rham $L^p$-Cohomology For Higher Rank Spaces And Groups: Critical Exponents And Rigidity
• On the l^{q,p} cohomology of Carnot groups
• Embeddings of locally compact hyperbolic groups into Lp-spaces
• On the logarithmic coarse structures of Lie groups and hyperbolic spaces
• Quasi-actions and rough Cayley graphs for locally compact groups