On two-generator subgroups of mapping torus groups

Abstract

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

18 pages; Primary article by Naomi Andrew, Edgar A. Bering IV, Ilya Kapovich, and Stefano Vidussi with an appendix by Peter Shalen. Updated to incorporate referee's suggestions