Endomorphisms of relatively hyperbolic groups
by
Igor Belegradek
Georgia Tech
Coauthors: Andrzej Szczepanski

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 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.

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.

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).

Date received: February 24, 2005