---
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.'
address: |
    Department of Mathematical Sciences\
    University of Arkansas\
    Fayetteville, AR *Web address: <https://mutanguha.com/>*
author:
- Jean Pierre Mutanguha
bibliography:
- 'refs.bib'
title: '*Irreducibility of a free group endomorphism is a mapping torus invariant*'
---

Introduction
============

Suppose S is a hyperbolic surface of finite type and $f:S \to S$ is a [*pseudo-Anosov*]{} homeomorphism, then the [*mapping torus*]{} $M_f$ is a complete finite-volume hyperbolic 3-manifold; this is Thurston’s hyperbolization theorem for 3-manifolds that fiber over a circle [@Thu82]. It is remarkable fact since the hypothesis is a dynamical statement about surface homeomorphisms but the conclusion is a geometric statement about 3-manifolds. In particular, since the converse 