Free-by-cyclic groups are conjugacy separable
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915946605576192 |
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| author | Dahmani, François Hughes, Sam Kudlinska, Monika Touikan, Nicholas |
| author_facet | Dahmani, François Hughes, Sam Kudlinska, Monika Touikan, Nicholas |
| contents | 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 $α:G\twoheadrightarrow Q$ such that $α(g)$ is not conjugate to $α(h)$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_22346 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Free-by-cyclic groups are conjugacy separable Dahmani, François Hughes, Sam Kudlinska, Monika Touikan, Nicholas Group Theory 20E26, 20E06, 20E08, 20F65 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 $α:G\twoheadrightarrow Q$ such that $α(g)$ is not conjugate to $α(h)$. 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. |
| title | Free-by-cyclic groups are conjugacy separable |
| topic | Group Theory 20E26, 20E06, 20E08, 20F65 |
| url | https://arxiv.org/abs/2509.22346 |