Embedding finitely generated free-by-cyclic groups in {finitely generated free}-by-cyclic groups
Marco Linton
TL;DR
The paper addresses embedding finitely generated free-by-cyclic groups into finitely generated free-by-cyclic groups by refining Feighn–Handel’s invariant-graph machinery to the bi-invariant graph-triple setting and proving a tightening procedure that yields finite tight minimal $\psi$-bi-invariant triples. It then derives explicit $HNN$-presentation forms for finitely generated free-by-cyclic groups and establishes a canonical free-product decomposition of the ambient free group, with a finitely generated component playing a crucial role. Geometric consequences follow: hyperbolic free-by-cyclic groups embed quasi-convexly into hyperbolic fg free-by-$\mathbb{Z}$ groups, enabling cocompact cubulation via Hagen–Wise and Agol, and leading to virtual compact specialness of hyperbolic fg free-by-$\mathbb{Z}$ groups. The results unify a robust combinatorial framework with relative-hyperbolic geometry to yield structural, embedding, and cubulation consequences for free-by-cyclic groups.
Abstract
We refine Feighn--Handel's results on subgroups of mapping tori of free groups to the special case of free-by-cyclic groups. We use these refinements to show that any finitely generated free-by-cyclic group embeds in a {finitely generated free}-by-cyclic group. When the free-by-cyclic group is hyperbolic, it embeds in a hyperbolic {finitely generated free}-by-cyclic group as a quasi-convex subgroup. Combined with a result of Hagen--Wise, this implies that all hyperbolic free-by-cyclic groups are cocompactly cubulated.