We describe unicorn paths in the arc graph and show that they form 1-slim triangles and are invariant under taking subpaths. We deduce that all arc graphs are 7-hyperbolic. Considering the same paths in the arc and curve graph, this also shows that all curve graphs are 17-hyperbolic, including closed surfaces.
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...
We prove that the boundary of a right-angled hyperbolic building is a universal Menger space. As a consequence, the 3-dimensional universal Menger space is the boundary of some Gromov-hyperbolic group.
The cogrowth exponent of a group controls the random walk spectrum. We prove that for a generic group (in the density model) this exponent is arbitrarily close to that of a free group. Moreover, this exponent is stable under random quotients of torsion-free hyperbolic groups.
In this article, we discuss the quasiconformal structure of boundaries of right-angled hyperbolic buildings using combinatorial tools. In particular, we exhibit some examples of buildings of dimension 3 and 4 whose boundaries satisfy the combinatorial Loewner property. This property is a weak version of the Loewner property. This is motivated by the fact that the quasiconformal structure of the boundary led to many results of rigidity in hyperbolic spaces since G.D.Mostow. In the case of buildings...
Traitant la série de Poincaré d’un groupe discret d’isométries en courbure négative comme un noyau de Green, on établit une théorie du potentiel assez comparable à la théorie classique pour affirmer un parallèle entre densités conformes à la Patterson-Sullivan et densités harmoniques, et notamment définir une frontière de Martin où les densités ergodiques forment la partie minimale, et enfin l’identifier géométriquement sous hypothèse d’hyperbolicité.