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 as a retract. 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.
Marco Linton (Fri,) studied this question.