On Two-Generator Subgroups of Mapping Torus Groups

Ilya Kapovitch (CUNY)

Abstract: Motivated by the results of Jaco and Shalen about 3-manifold groups, we prove that if F is a free group (of possibly infinite rank), $\phi: F\to F$ is an injective endomorphism of $\phi$ and $G_\phi=\langle F,t| t x t^{-1} =\phi(x), x\in F\rangle$ is the mapping torus group of $\phi$ then every two-generator subgroup of $G_\phi$ is either free or a “finitary sub-mapping torus” (and hence isomorphic to the mapping torus of an injective endomorphism of a finite rank free group). For a fully irreducible automorphism $\phi$ of a finite rank free group $F_r$ this result implies that every two-generator subgroup of the free-by-cyclic group $G_\phi$ is either free, free abelian, a Klein bottle group or a subgroup of finite index in $G_\phi$; and if $\phi\in Out(F_r)$ is fully irreducible and atoroidal then every two-generator subgroup of $G_\phi$ is either free or of finite index in $G_\phi$. This talk is based on joint paper with Naomi Andrew, Edgar A. Bering IV and Stefano Vidussi, with an appendix by Peter Shalen.