Geodesic
Amenable hyperbolic groups
Journal of the European Mathematical Society, Tome 17 (2015) no. 11, pp. 2903-2947 Cet article a éte moissonné depuis la source EMS Press

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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.
DOI : 10.4171/jems/575
Classification : 20-XX, 05-XX, 22-XX, 43-XX
Keywords: Gromov hyperbolic group, locally compact group, amenable group, contracting automorphisms, compacting automorphisms 
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     author = {Pierre-Emmanuel Caprace and Yves de Cornulier and Nicolas Monod and Romain Tessera},
     title = {Amenable hyperbolic groups},
     journal = {Journal of the European Mathematical Society},
     pages = {2903--2947},
     year = {2015},
     volume = {17},
     number = {11},
     doi = {10.4171/jems/575},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/575/}
}
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Pierre-Emmanuel Caprace; Yves de Cornulier; Nicolas Monod; Romain Tessera. Amenable hyperbolic groups. Journal of the European Mathematical Society, Tome 17 (2015) no. 11, pp. 2903-2947. doi: 10.4171/jems/575

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