  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).
