Endomorphisms of relatively hyperbolic groups

Abstract

We generalize some results of Paulin and Rips-Sela on endomorphisms of hyperbolic groups to relatively hyperbolic groups, and in particular prove the following. (1) If G is a finitely generated non-elementary relatively hyperbolic group with slender parabolic subgroups, and either G is not co-Hopfian or Out(G) is infinite, then G splits over a slender group. (2) If a finitely generated non-parabolic subgroup H of a non-elementary relatively hyperbolic group is not Hopfian, then H acts non-trivially on an R-tree. (3) Every non-elementary relatively hyperbolic group has a non-elementary relatively hyperbolic quotient that is Hopfian. (4) Any finitely presented group is isomorphic to a finite index subgroup of Out(H) for some Kazhdan group H. (This sharpens a result of Ollivier-Wise).

final version

References in corpus (3)

• Limit groups are CAT(0)
• Accidental parabolics and relatively hyperbolic groups
• Small cancellations over relatively hyperbolic groups and embedding theorems

Cited by in corpus (4)

• Finitely presentable, non-Hopfian groups with Kazhdan's Property (T) and infinite outer automorphism group
• On the residual finiteness of outer automorphisms of relatively hyperbolic groups
• Groups acting on tree-graded spaces and splittings of relatively hyperbolic group
• Rips construction and Kazhdan property (T)