On endomorphisms of automatic groups

TL;DR

This paper explores geometric properties of endomorphisms in automatic groups, establishing conditions for bounded reduction properties, and analyzing quasiconvexity of images and equalizers, especially in hyperbolic groups.

Contribution

It introduces two geometric bounded reduction properties, proves their equivalence in hyperbolic groups, and demonstrates quasiconvexity results for endomorphism images and equalizers.

Findings

Two geometric versions of bounded reduction property are equivalent in hyperbolic groups.

Endomorphisms with finite kernel and quasiconvex images satisfy a synchronous bounded reduction property.

The equalizer of two endomorphisms is quasiconvex under certain conditions.

Abstract

We propose two geometric versions of the bounded reduction property and find conditions for them to coincide. In particular, for the natural automatic structure on a hyperbolic group, the two notions are equivalent. We study endomorphisms with $L$-quasiconvex image and prove that those with finite kernel satisfy a synchronous version of the bounded reduction property. Finally, we use these techniques to prove $L$-quasiconvexity of the equalizer of two endomorphisms under certain (strict) conditions.