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| Format: | Preprint |
| Published: |
2023
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| Online Access: | https://arxiv.org/abs/2308.12185 |
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| _version_ | 1866909358064926720 |
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| author | Wykowski, Julian |
| author_facet | Wykowski, Julian |
| contents | We investigate accessible subgroups of a profinite group $G$, i.e. subgroups $H$ appearing as vertex groups in a graph of profinite groups decomposition of $G$ with finite edge groups. We prove that any accessible subgroup $H \leq G$ arises as the kernel of a continuous derivation of $G$ in a free module over its completed group algebra. This allows us to deduce splittings of an abstract group from splittings of its profinite completion. We prove that any finitely generated subgroup $Δ$ of a finitely generated virtually free group $Γ$ whose closure is a factor in a profinite amalgamated product $\widehatΓ = \overlineΔ \amalg_K L$ along a finite $K$ must be a factor in an amalgamated product $Γ= Δ\ast_χΛ$ along some $χ\cong K$. This extends previous results of Parzanchevski--Puder, Wilton and Garrido--Jaikin-Zapirain on free factors. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_12185 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Profinite Subgroup Accessibility and Recognition of Amalgamated Factors Wykowski, Julian Group Theory 20E18 20E08 We investigate accessible subgroups of a profinite group $G$, i.e. subgroups $H$ appearing as vertex groups in a graph of profinite groups decomposition of $G$ with finite edge groups. We prove that any accessible subgroup $H \leq G$ arises as the kernel of a continuous derivation of $G$ in a free module over its completed group algebra. This allows us to deduce splittings of an abstract group from splittings of its profinite completion. We prove that any finitely generated subgroup $Δ$ of a finitely generated virtually free group $Γ$ whose closure is a factor in a profinite amalgamated product $\widehatΓ = \overlineΔ \amalg_K L$ along a finite $K$ must be a factor in an amalgamated product $Γ= Δ\ast_χΛ$ along some $χ\cong K$. This extends previous results of Parzanchevski--Puder, Wilton and Garrido--Jaikin-Zapirain on free factors. |
| title | Profinite Subgroup Accessibility and Recognition of Amalgamated Factors |
| topic | Group Theory 20E18 20E08 |
| url | https://arxiv.org/abs/2308.12185 |