Irreducibility of a Free Group Endomorphism is a Mapping Torus Invariant

TL;DR

This paper demonstrates that the irreducibility and atoroidality of free group endomorphisms are invariants of their associated ascending HNN extensions, providing algebraic characterizations and answering a previously posed question.

Contribution

It establishes that irreducibility and atoroidality are invariants of the group structure of ascending HNN extensions, with an algebraic criterion for these properties.

Findings

Irreducibility is a group invariant of the ascending HNN extension.

Atoroidality is also a commensurability invariant.

An algebraic characterization of when endomorphisms are irreducible and atoroidal.

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.