H\"older conditions for endomorphisms of hyperbolic groups

TL;DR

This paper characterizes when endomorphisms of hyperbolic groups satisfy H"older conditions in terms of virtual injectivity and quasiconvexity, and discusses Lipschitz conditions for free group automorphisms.

Contribution

It provides a complete characterization of H"older continuity for endomorphisms of hyperbolic groups and explores Lipschitz conditions in free groups.

Findings

Endomorphisms satisfy H"older conditions iff they are virtually injective and have quasiconvex images.

In virtually free or torsion-free co-hopfian groups, H"older continuity coincides with virtual injectivity.

Lipschitz conditions are analyzed specifically for free group automorphisms.

Abstract

It is proved that an endomorphism $\varphi$ of a hyperbolic group $G$ satisfies a H\"older condition with respect to a visual metric if and only if $\varphi$ is virtually injective and $G\varphi$ is a quasiconvex subgroup of $G$. If $G$ is virtually free or torsion-free co-hopfian, then $\varphi$ is uniformly continuous if and only if it it satisfies a H\"older condition if and only if it is virtually injective. Lipschitz conditions are discussed for free group automorphisms.