Infinitely-ended hyperbolic groups with homeomorphic Gromov boundaries

Alexandre Martin (1) , Jacek Świątkowski (2)

1 IRMA - Institut de Recherche Mathématique Avancée
2 Mathematical Institute [Wroclaw]

We show that the Gromov boundary of the free product of two infinite hyper-bolic groups is uniquely determined up to homeomorphism by the homeomorphism types of the boundaries of its factors. We generalize this result to graphs of hyperbolic groups over finite subgroups. Finally, we give a necessary and sufficient condition for the Gromov boundaries of any two hyperbolic groups to be homeomorphic (in terms of the topology of the boundaries of factors in terminal splittings over finite subgroups).

Domaines: Théorie des groupes [math.GR]

Alexandre Martin, Jacek Świątkowski. Infinitely-ended hyperbolic groups with homeomorphic Gromov boundaries. Journal of Group Theory, 2015, ⟨10.1515/jgth-2014-0043⟩. ⟨hal-01262062⟩