Automorphisms and endomorphisms of lacunary hyperbolic groups

Rémi Coulon (1) , Vincent Guirardel (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

In this article we study automorphisms and endomorphisms of lacunary hyperbolic groups. We show that if a lacunary hyperbolic group has the fix point property for actions on $\mathbb R$-trees, then its outer automorphism group is locally finite. We construct lacunary hyperbolic groups whose automorphism group is infinite, locally finite, and contains any locally finite group given in advance. We also study the Hopf and co-Hopf property for this class of groups.

Domaines: Théorie des groupes [math.GR]

Rémi Coulon, Vincent Guirardel. Automorphisms and endomorphisms of lacunary hyperbolic groups. Groups, Geometry, and Dynamics, 2019, 13 (1), pp.131-148. ⟨10.4171/GGD/488⟩. ⟨hal-01327448⟩