Profinite almost rigidity in 3-manifolds
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866908515871752192 |
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| author | Xu, Xiaoyu |
| author_facet | Xu, Xiaoyu |
| contents | We prove that any compact, orientable 3-manifold with empty or toral boundary is profinitely almost rigid among all compact, orientable 3-manifolds. In other words, the profinite completion of its fundamental group determines its homeomorphism type to finitely many possibilities. Moreover, the profinite completion of the fundamental group of a mixed 3-manifold, together with the peripheral structure, uniquely determines the homeomorphism type of its Seifert part, i.e. the maximal graph manifold components in the JSJ-decomposition. On the other hand, without assigning the peripheral structure, the profinite completion of a mixed 3-manifold group may not uniquely determine the fundamental group of its Seifert part. The proof is based on JSJ-decomposition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_16002 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Profinite almost rigidity in 3-manifolds Xu, Xiaoyu Geometric Topology Group Theory 57M50, 20E18 (Primary) 57M05, 57M10 (Secondary) We prove that any compact, orientable 3-manifold with empty or toral boundary is profinitely almost rigid among all compact, orientable 3-manifolds. In other words, the profinite completion of its fundamental group determines its homeomorphism type to finitely many possibilities. Moreover, the profinite completion of the fundamental group of a mixed 3-manifold, together with the peripheral structure, uniquely determines the homeomorphism type of its Seifert part, i.e. the maximal graph manifold components in the JSJ-decomposition. On the other hand, without assigning the peripheral structure, the profinite completion of a mixed 3-manifold group may not uniquely determine the fundamental group of its Seifert part. The proof is based on JSJ-decomposition. |
| title | Profinite almost rigidity in 3-manifolds |
| topic | Geometric Topology Group Theory 57M50, 20E18 (Primary) 57M05, 57M10 (Secondary) |
| url | https://arxiv.org/abs/2410.16002 |