On two-generator subgroups of mapping torus groups

TL;DR

This paper investigates the structure of two-generator subgroups within mapping torus groups derived from free group automorphisms, revealing they are either free or closely related to the original group, with implications for automorphisms.

Contribution

It establishes that two-generator subgroups of these groups are either free or sub-mapping tori, extending understanding of subgroup structures in mapping torus groups.

Findings

Two-generator subgroups are either free or sub-mapping tori.

Fully irreducible atoroidal automorphisms lead to subgroups being either free or of finite index.

Results apply to free groups of possibly infinite rank.

Abstract

We prove that if $G_\phi=\langle F, t| t x t^{-1} =\phi(x), x\in F\rangle$ is the mapping torus group of an injective endomorphism $\phi: F\to F$ of a free group $F$ (of possibly infinite rank), then every two-generator subgroup $H$ of $G_\phi$ is either free or a (finitary) sub-mapping torus. As an application we show that if $\phi\in \mathrm{Out}(F_r)$ (where $r\ge 2$) is a fully irreducible atoroidal automorphism then every two-generator subgroup of $G_\phi$ is either free or has finite index in $G_\phi$.