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Algebra Comput."],"published-print":{"date-parts":[[2008,2]]},"abstract":"<jats:p> We generalize some results of Paulin and Rips-Sela on endomorphisms of hyperbolic groups to relatively hyperbolic groups, and in particular prove the following. <\/jats:p><jats:p> \u2022 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. <\/jats:p><jats:p> \u2022 If H is a nonparabolic subgroup of a relatively hyperbolic group, and if any isometric H-action on an \u211d-tree is trivial, then H is Hopfian. <\/jats:p><jats:p> \u2022 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. <\/jats:p><jats:p> \u2022 Any finitely presented group is isomorphic to a finite index subgroup of Out (H) for some group H with Kazhdan property (T). 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