On uniformly continuous endomorphisms of hyperbolic groups

Abstract

We prove a generalization of the fellow traveller property for a certain type of quasi-geodesics and use it to present three equivalent geometric formulations of the bounded reduction property and prove that it is equivalent to preservation of a coarse median. We then provide an affirmative answer to a question from Araújo and Silva as to whether every nontrivial uniformly continuous endomorphism of a hyperbolic group with respect to a visual metric satisfies a Hölder condition. We remark that these results combined with the work done by Paulin prove that every endomorphism admitting a continuous extension to the completion has a finitely generated fixed point subgroup.

20 pages; added the equivalence of the BRP and coarse-median preservation; comments are welcome. arXiv admin note: text overlap with arXiv:1405.6310 by other authors