Belegradek, IgorSzczepanski, AndrzejBelegradek, Oleg V.2024-07-182024-07-1820080218-19671793-6500https://doi.org/10.1142/S0218196708004305https://hdl.handle.net/11411/7926We 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).eninfo:eu-repo/semantics/openAccessRelatively HyperbolicAutomorphism GroupHopfianCo-HopfianProperty (T)SplittingActions On TreesHopf PropertyDegenerationsFinitenessSubgroupsRigiditySpacesArticle2-s2.0-4424910449010.1142/S02181967080043051101Q29718Q3WOS:000254194500005