Amenable hyperbolic groups

  • Pierre-Emmanuel Caprace

    Université Catholique de Louvain, Belgium
  • Yves de Cornulier

    Université Paris-Sud, Orsay, France
  • Nicolas Monod

    Ecole Polytechnique Fédérale de Lausanne, Switzerland
  • Romain Tessera

    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.

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