Endomorphisms of relatively hyperbolic groups
Abstract
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).
Cite
@article{arxiv.math/0501321,
title = {Endomorphisms of relatively hyperbolic groups},
author = {Igor Belegradek and Andrzej Szczepanski and Oleg V. Belegradek},
journal= {arXiv preprint arXiv:math/0501321},
year = {2011}
}
Comments
final version