Contracting automorphisms and L^p-cohomology in degree one

TL;DR

This paper characterizes Lie and algebraic groups with vanishing first reduced L^p-cohomology for all p>1, and applies this to classify Gromov-hyperbolic groups, showing certain algebraic groups are quasi-isometric to a 3-regular tree.

Contribution

It extends Pansu's result by characterizing groups with zero first reduced L^p-cohomology and describes Gromov-hyperbolic algebraic groups over non-Archimedean fields.

Findings

Groups with zero first reduced L^p-cohomology are characterized.

Non-elementary Gromov-hyperbolic algebraic groups over non-Archimedean fields are quasi-isometric to a 3-regular tree.

Extension of the study to semidirect products with contracting automorphisms.

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.