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. If G is a nonelementary 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. If H is a nonparabolic subgroup of a relatively hyperbolic group, and if any isometric H-action on an R-tree is trivial, then H is Hopfian. If G is a nonelementary relatively hyperbolic group whose peripheral subgroups are finitely generated, then G has a nonelementary relatively hyperbolic quotient that is Hopfian. Any finitely presented group is isomorphic to a finite index subgroup of Out( H) for some group H with Kazhdan property ( T). ( This sharpens a result of Ollivier-Wise).