Profinite rigidity for free-by-cyclic groups with centre

2024/09/30 by Bridson, Martin R., Piwek, Paweł · 2 citations
#20E18 #20F65 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2409.20513

Abstract

A free-by-cyclic group F_N\rtimes_ϕℤ has non-trivial centre if and only if [ϕ] has finite order in \rmOut(F_N). We establish a profinite ridigity result for such groups: if Γ₁ is a free-by-cyclic group with non-trivial centre and Γ₂ is a finitely generated free-by-cyclic group with the same finite quotients as Γ₁, then Γ₂ is isomorphic to Γ₁. 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 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.

Cited by

• The conjugacy problem in Out(Fm) when the polynomial restrictions are non-growing
• On the Farrell--Tate K-theory of Out(F_n)

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