Co-Hopfianity is not a profinite property
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866908892902981632 |
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| author | Baik, Hyungryul Jang, Wonyong |
| author_facet | Baik, Hyungryul Jang, Wonyong |
| contents | We exhibit two finitely generated residually finite groups $G$ and $H$ with isomorphic profinite completions $\widehat{G} \cong \widehat{H}$, such that $G$ is co-Hopfian while $H$ is not. The construction utilizes Wise's residually finite version of the Rips construction applied to a finitely presented acyclic group $U$ with trivial profinite completion and a strong universality property. A key feature of our approach is the construction of $H$ as a preimage subgroup of $G$ which is conjugate to a proper subgroup of itself. This renders the non-co-Hopfianity of $H$ immediate without requiring a detailed structural analysis of the Rips kernel. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_04197 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Co-Hopfianity is not a profinite property Baik, Hyungryul Jang, Wonyong Group Theory Geometric Topology Primary: 20E18, 20F67, Secondary: 20E06, 20E26 We exhibit two finitely generated residually finite groups $G$ and $H$ with isomorphic profinite completions $\widehat{G} \cong \widehat{H}$, such that $G$ is co-Hopfian while $H$ is not. The construction utilizes Wise's residually finite version of the Rips construction applied to a finitely presented acyclic group $U$ with trivial profinite completion and a strong universality property. A key feature of our approach is the construction of $H$ as a preimage subgroup of $G$ which is conjugate to a proper subgroup of itself. This renders the non-co-Hopfianity of $H$ immediate without requiring a detailed structural analysis of the Rips kernel. |
| title | Co-Hopfianity is not a profinite property |
| topic | Group Theory Geometric Topology Primary: 20E18, 20F67, Secondary: 20E06, 20E26 |
| url | https://arxiv.org/abs/2603.04197 |