Journal of the European Mathematical Society

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Volume 17, Issue 11, 2015, pp. 2903–2947
DOI: 10.4171/JEMS/575

Published online: 2015-10-29

Amenable hyperbolic groups

Pierre-Emmanuel Caprace[1], Yves de Cornulier[2], Nicolas Monod[3] and Romain Tessera[4]

(1) Université Catholique de Louvain, Belgium
(2) Université Paris-Sud, Orsay, France
(3) Ecole Polytechnique Fédérale de Lausanne, Switzerland
(4) Université Paris-Sud, Orsay, France

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.

Keywords: Gromov hyperbolic group, locally compact group, amenable group, contracting automorphisms, compacting automorphisms

Caprace Pierre-Emmanuel, de Cornulier Yves, Monod Nicolas, Tessera Romain: Amenable hyperbolic groups. J. Eur. Math. Soc. 17 (2015), 2903-2947. doi: 10.4171/JEMS/575