Effective embedding of residually hyperbolic groups into direct products of extensions of centralizers

TL;DR

This paper develops algorithms to embed residually hyperbolic groups into direct products of groups formed by extensions of centralizers, providing effective tools for understanding their structure.

Contribution

It introduces an algorithmic method to embed residually hyperbolic groups into direct products of extensions of centralizers, advancing the understanding of their algebraic structure.

Findings

Constructed a finite collection of homomorphisms with at least one injective

Provided an effective embedding of residually hyperbolic groups into direct products

Developed an algorithmic diagram for homomorphisms into hyperbolic groups

Abstract

For any torsion-free hyperbolic group $\Gamma$ and any group $G$ that is fully residually $\Gamma$, we construct algorithmically a finite collection of homomorphisms from $G$ to groups obtained from $\Gamma$ by extensions of centralizers, at least one of which is injective. When $G$ is residually $\Gamma$, this gives a effective embedding of $G$ into a direct product of such groups. We also give an algorithmic construction of a diagram encoding the set of homomorphisms from a given finitely presented group to $\Gamma$.