Profinite rigidity for free-by-cyclic groups with centre

Abstract

A free-by-cyclic group $F_N{⋊}_{\varphi }\mathbb{Z}$ has non-trivial centre if and only if $[\varphi ]$ has finite order in $\mathrm{Out}(F_N)$. We establish a profinite ridigity 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 $\mathbf{fsc}(G)$ 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.

29 pages, 6 figures. Comments welcome!