We characterize those Lie groups, and algebraic groups over a local field of characteristic zero, whose first reduced Lp-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 general semidirect products of a locally compact group by a cyclic group acting by contracting automorphisms.
The first author was supported by ANR quantiT JCO8_318197.
"Contracting automorphisms and Lp-cohomology in degree one." Ark. Mat. 49 (2) 295 - 324, October 2011. https://doi.org/10.1007/s11512-010-0127-z