The dynamics and geometry of free group endomorphisms

TL;DR

This paper characterizes when ascending HNN extensions of free groups are word-hyperbolic, linking their hyperbolicity to the absence of Baumslag-Solitar subgroups, and develops a structure theorem for injective endomorphisms.

Contribution

It introduces a canonical structure theorem for injective nonsurjective endomorphisms of free groups, enabling new insights into their dynamics and hyperbolic properties.

Findings

Ascending HNN extensions of free groups are hyperbolic iff they lack Baumslag-Solitar subgroups.

Develops a structure theorem for injective nonsurjective endomorphisms of free groups.

Extends hyperbolicity criteria to HNN extensions over free factors.

Abstract

We prove that ascending HNN extensions of free groups are word-hyperbolic if and only if they have no Baumslag-Solitar subgroups. This extends the theorem of Brinkmann that free-by-cyclic groups are word-hyperbolic if and only if they have no free abelian subgroups of rank 2. The paper is split into two independent parts: 1) We study the dynamics of injective nonsurjective endomorphisms of free groups. We prove a canonical structure theorem that initializes the development of improved relative train tracks for endomorphisms; this structure theorem is of independent interest since it makes many open questions about injective endomorphisms tractable. 2) As an application of the structure theorem, we are able to (relatively) combine Brinkmann's theorem with our previous work and obtain the main result stated above. In the final section, we further extend the result to HNN extensions of free groups over free factors.