Elementary embeddings in torsion-free hyperbolic groups

TL;DR

This paper characterizes elementary embeddings in torsion-free hyperbolic groups using hyperbolic towers and shows that subgroups elementarily embedded in free groups are free factors.

Contribution

It provides a description of elementary embeddings in hyperbolic groups and establishes that such subgroups in free groups are necessarily free factors.

Findings

Elementary embeddings are described via hyperbolic towers.

Subgroups elementarily embedded in free groups are free factors.

The results connect model theory with geometric group theory.

Abstract

We consider embeddings in a torsion-free hyperbolic group which are elementary in the sense of first-order logic. We give a description of these embeddings in terms of Sela's hyperbolic towers. We deduce as a corollary that subgroups elementarily embedded in finitely generated free groups are free factors.