Amenable hyperbolic groups

TL;DR

This paper characterizes non-elementary Gromov-hyperbolic amenable locally compact groups, linking them to mapping tori of automorphisms, and describes their structure, including rank one Lie groups and automorphism groups of semi-regular trees.

Contribution

It provides a complete classification of hyperbolic amenable locally compact groups and describes their structure in relation to automorphism groups and Lie groups.

Findings

Hyperbolic amenable groups are mapping tori of automorphisms.

Such groups are either rank one Lie groups or automorphism groups of semi-regular trees.

The class of hyperbolic groups with certain lattice properties is highly restricted.

Abstract

We give a complete characterization of the locally compact groups that are non-elementary Gromov-hyperbolic and amenable. They coincide with the class of mapping tori of discrete or continuous one-parameter groups of compacting automorphisms. We moreover give a description of all Gromov-hyperbolic locally compact groups with a cocompact amenable subgroup: modulo a compact normal subgroup, these turn out to be either rank one simple Lie groups, or automorphism groups of semi-regular trees acting doubly transitively on the set of ends. As an application, we show that the class of hyperbolic locally compact groups with a cusp-uniform non-uniform lattice, is very restricted.