Towers and elementary embeddings in toral relatively hyperbolic groups

TL;DR

This paper generalizes the classification of elementary embeddings from hyperbolic groups to toral relatively hyperbolic groups, using JSJ and shortening techniques to extend existing geometric and model-theoretic results.

Contribution

It extends Perin's result on elementary embeddings from hyperbolic groups to the broader class of toral relatively hyperbolic groups, employing JSJ and shortening methods.

Findings

Elementary embeddings imply the group is a tower over the subgroup.

Generalization of Perin's theorem to toral relatively hyperbolic groups.

Application of JSJ and shortening techniques in the proof.

Abstract

In a remarkable series of papers, Zlil Sela classified the first-order theories of free groups and torsion-free hyperbolic groups using geometric structures he called towers. It was later proved by Chlo\'e Perin that if $H$ is an elementarily embedded subgroup (or elementary submodel) of a torsion-free hyperbolic group $G$, then $G$ is a tower over $H$. We prove a generalization of Perin's result to toral relatively hyperbolic groups using JSJ and shortening techniques.