Profinite rigidity for free-by-cyclic groups with centre
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| Format: | Preprint |
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2024
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| _version_ | 1866915399192281088 |
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| author | Bridson, Martin R. Piwek, Paweł |
| author_facet | Bridson, Martin R. Piwek, Paweł |
| contents | A free-by-cyclic group $F_N\rtimes_ϕ\mathbb{Z}$ has non-trivial centre if and only if $[ϕ]$ has finite order in ${\rm{Out}}(F_N)$. We establish a profinite ridigity result for such groups: if $Γ_1$ is a free-by-cyclic group with non-trivial centre and $Γ_2$ is a finitely generated free-by-cyclic group with the same finite quotients as $Γ_1$, then $Γ_2$ is isomorphic to $Γ_1$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_20513 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Profinite rigidity for free-by-cyclic groups with centre Bridson, Martin R. Piwek, Paweł Group Theory 20E18, 20F65 A free-by-cyclic group $F_N\rtimes_ϕ\mathbb{Z}$ has non-trivial centre if and only if $[ϕ]$ has finite order in ${\rm{Out}}(F_N)$. We establish a profinite ridigity result for such groups: if $Γ_1$ is a free-by-cyclic group with non-trivial centre and $Γ_2$ is a finitely generated free-by-cyclic group with the same finite quotients as $Γ_1$, then $Γ_2$ is isomorphic to $Γ_1$. 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. |
| title | Profinite rigidity for free-by-cyclic groups with centre |
| topic | Group Theory 20E18, 20F65 |
| url | https://arxiv.org/abs/2409.20513 |