On two-generator subgroups of mapping torus groups
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866918068113899520 |
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| author | Andrew, Naomi Bering IV, Edgar A. Kapovich, Ilya Shalen, Peter Vidussi, Stefano |
| author_facet | Andrew, Naomi Bering IV, Edgar A. Kapovich, Ilya Shalen, Peter Vidussi, Stefano |
| contents | We prove that if $G_ϕ=\langle F, t| t x t^{-1} =ϕ(x), x\in F\rangle$ is the mapping torus group of an injective endomorphism $ϕ: F\to F$ of a free group $F$ (of possibly infinite rank), then every two-generator subgroup $H$ of $G_ϕ$ is either free or a (finitary) sub-mapping torus. As an application we show that if $ϕ\in \mathrm{Out}(F_r)$ (where $r\ge 2$) is a fully irreducible atoroidal automorphism then every two-generator subgroup of $G_ϕ$ is either free or has finite index in $G_ϕ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_08985 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On two-generator subgroups of mapping torus groups Andrew, Naomi Bering IV, Edgar A. Kapovich, Ilya Shalen, Peter Vidussi, Stefano Group Theory Primary 20F65, Secondary 20F05, 57M We prove that if $G_ϕ=\langle F, t| t x t^{-1} =ϕ(x), x\in F\rangle$ is the mapping torus group of an injective endomorphism $ϕ: F\to F$ of a free group $F$ (of possibly infinite rank), then every two-generator subgroup $H$ of $G_ϕ$ is either free or a (finitary) sub-mapping torus. As an application we show that if $ϕ\in \mathrm{Out}(F_r)$ (where $r\ge 2$) is a fully irreducible atoroidal automorphism then every two-generator subgroup of $G_ϕ$ is either free or has finite index in $G_ϕ$. |
| title | On two-generator subgroups of mapping torus groups |
| topic | Group Theory Primary 20F65, Secondary 20F05, 57M |
| url | https://arxiv.org/abs/2405.08985 |