Profinite rigidity for free-by-cyclic groups with centre

A free-by-cyclic group $F_N{⋊}_{\varphi }Z$ has non-trivial centre if and only if $\left[\varphi \right]$ has finite order in $\mathrm{Out}\left(F_N\right)$. We establish a profinite rigidity result for such groups: if ${\Gamma }_1$ is a free-by-cyclic group with non-trivial centre and ${\Gamma }_2$ is a finitely generated free-by-cyclic group with the same finite quotients as ${\Gamma }_1$, then ${\Gamma }_2$ is isomorphic to ${\Gamma }_1$. One-relator groups with centre are similarly rigid. We prove that finitely generated free-by-(finite cyclic) groups are profinitely rigid in the same sense; the proof revolves around a finite poset $\mathrm{fsc}\left(G\right)$ that carries information about the centralisers of finite subgroups of $G$, it is a complete invariant for these groups. These results provide contrasts with the lack of profinite rigidity among surface-by-cyclic groups and (free abelian)-by-cyclic groups, as well as general virtually free groups.