Amenable hyperbolic groups
Pierre-Emmanuel Caprace
Université Catholique de Louvain, BelgiumYves de Cornulier
Université Paris-Sud, Orsay, FranceNicolas Monod
Ecole Polytechnique Fédérale de Lausanne, SwitzerlandRomain Tessera
Université Paris-Sud, Orsay, France
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 semiregular 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 nonuniform lattice is very restricted.
Cite this article
Pierre-Emmanuel Caprace, Yves de Cornulier, Nicolas Monod, Romain Tessera, Amenable hyperbolic groups. J. Eur. Math. Soc. 17 (2015), no. 11, pp. 2903–2947
DOI 10.4171/JEMS/575