Towers and elementary embeddings in toral relatively hyperbolic groups

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 Chloé Perin that if [Formula: see text] is an elementarily embedded subgroup (or elementary submodel) of a torsion-free hyperbolic group [Formula: see text], then [Formula: see text] is a tower over [Formula: see text]. We prove a generalization of Perin’s result to toral relatively hyperbolic groups using JSJ and shortening techniques.