Irreducibility of a Free Group Endomorphism is a Mapping Torus Invariant

Abstract

We prove that the property of a free group endomorphism being irreducible is a group invariant of the ascending HNN extension it defines. This answers a question posed by Dowdall-Kapovich-Leininger. We further prove that being irreducible and atoroidal is a commensurability invariant. The invariance follows from an algebraic characterization of ascending HNN extensions that determines exactly when their defining endomorphisms are irreducible and atoroidal; specifically, we show that the endomorphism is irreducible and atoroidal if and only if the ascending HNN extension has no infinite index subgroups that are ascending HNN extensions.

v1: 14 pages. Comments are welcome. v2: Minor changes and updated references

Cited by in corpus (3)

• Irreducible Nonsurjective Endomorphisms of $F_n$ are Hyperbolic
• Handlebody Bundles and Polytopes
• The mimimally displaced set of an irreducible automorphism of $F_N$ is co-compact