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.   (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). 
Igor Belegradek (Georgia Tech), Andrzej Szczepanski (University of   Gdansk), Oleg V. Belegradek (Istanbul Bilgi University)