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We characterize those Lie groups, and algebraic groups over a local field ofcharacteristic 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 ofGromov-hyperbolic groups among those groups. In particular we prove that anynon-elementary Gromov-hyperbolic algebraic group over a non-Archimedean localfield of zero characteristic is quasi-isometric to a 3-regular tree. We alsoextend the study to semidirect products of a general locally compact group by acyclic group acting by contracting automorphisms.

Contracting automorphisms and L^p-cohomology in degree one