Automorphisms and endomorphisms of lacunary hyperbolic groups

TL;DR

This paper investigates the automorphism and endomorphism structures of lacunary hyperbolic groups, establishing their Hopfian property, conditions for co-Hopfianity, and constructing groups with prescribed automorphism groups.

Contribution

It proves that all lacunary hyperbolic groups are Hopfian and explores conditions under which they are co-Hopfian, also constructing groups with specific automorphism group properties.

Findings

Lacunary hyperbolic groups are Hopfian.

Certain lacunary hyperbolic groups are co-Hopfian if they have the fixed point property.

Constructed groups with automorphism groups that are infinite, locally finite, and contain any given locally finite group.

Abstract

In this article we study automorphisms and endomorphisms of lacunary hyperbolic groups. We prove that every lacunary hyperbolic group is Hopfian, answering a question by Henry Wilton. In addition, we show that if a lacunary hyperbolic group has the fix point property for actions on $\mathbf R$-trees, then it is co-Hopfian and its outer automorphism group is locally finite. We also construct lacunary hyperbolic groups whose automorphism group is infinite, locally finite, and contains any locally finite group given in advance.