Free-by-cyclic groups are conjugacy separable

Abstract

We show that all finitely generated free-by-cyclic groups are conjugacy separable: if a finitely generated group $G$ surjects onto $\mathbb{Z}$ with free kernel, then for every pair of non-conjugate elements $g,h\in G$, there exists a finite quotient $\alpha:G\twoheadrightarrow Q$ such that $\alpha(g)$ is not conjugate to $\alpha(h)$. This resolves Question 19.41 of the Kourovka Notebook. We apply this to prove that the outer automorphism group of a finitely generated free-by-cyclic group is residually finite. Along the way we prove that if the monodromy of a {finitely generated free}-by-cyclic group is polynomially growing, then the double cosets of a cyclic subgroup are separable. Our approach combines vertex fillings in graph-of-groups decompositions, and Dehn fillings in relatively hyperbolic groups, according to the different geometric regimes in free-by-cyclic groups.