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. • 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).

Transformations on tensor products and the torsionless property in abelian groups: Corrigendum

Professor Paul L. Sperry, when reviewing the paper [1] for Mathematical Reviews, has drawn the author's attention to an error in the formulation of Theorem 4.The hypotheses on the abelian groups Ci, i ∈ I, in Theorem 4 were incorrectly stated. (They implied, for one thing, that the slender group A is divisible, an absurdity.) The correct formulation is as follows.