Contracting automorphisms and $L^p$-cohomology in degree one

International audienceWe 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 general semidirect products of a locally compact group by a cyclic group acting by contracting automorphisms