Endomorphisms of Relatively Hyperbolic Groups

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 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. • If H is a non-parabolic 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 non-elementary relatively hyperbolic group whose peripheral subgroups are finitely generated, then G has a non-elementary 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).