On uniformly continuous endomorphisms of hyperbolic groups

TL;DR

This paper explores the properties of uniformly continuous endomorphisms in hyperbolic groups, establishing geometric equivalences and conditions for fixed points, advancing understanding of their algebraic and geometric structure.

Contribution

It generalizes the fellow traveller property, provides geometric formulations of the bounded reduction property, and answers a key question about H"older conditions for endomorphisms.

Findings

Equivalent geometric formulations of the bounded reduction property.

Every nontrivial uniformly continuous endomorphism satisfies a H"older condition.

Endomorphisms with continuous extensions have finitely generated fixed point subgroups.

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 Ara\'ujo and Silva as to whether every nontrivial uniformly continuous endomorphism of a hyperbolic group with respect to a visual metric satisfies a H\"older 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.