Hyperbolic Immersions of Free Groups

TL;DR

This paper characterizes when the fundamental group of a mapping torus of a graph immersion is hyperbolic, linking it to the absence of Baumslag-Solitar subgroups and extending results to all injective endomorphisms of free groups.

Contribution

It provides a criterion for hyperbolicity of mapping tori of graph immersions and extends the applicability to all injective endomorphisms of free groups.

Findings

Mapping tori are hyperbolic iff no Baumslag-Solitar subgroups are present.

The theorem applies to all injective endomorphisms of F_2.

Framework established for extending the theorem to all injective endomorphisms of F_n.

Abstract

We prove that the mapping torus of a graph immersion has a word-hyperbolic fundamental group if and only if the corresponding endomorphism does not produce Baumslag-Solitar subgroups. Due to a result by Reynolds, this theorem applies to all injective endomorphisms of $F_2$ and nonsurjective fully irreducible endomorphisms of $F_n$. We also give a framework for extending the theorem to all injective endomorphisms of $F_n$.