Profinite rigidity witnessed by Dehn fillings of cusped hyperbolic 3-manifolds
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910053151277056 |
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| author | Xu, Xiaoyu |
| author_facet | Xu, Xiaoyu |
| contents | Any profinite isomorphism between two cusped finite-volume hyperbolic 3-manifolds carries profinite isomorphisms between their Dehn fillings. With this observation, we prove that some cusped finite-volume hyperbolic 3-manifolds are profinitely rigid among all compact, orientable 3-manifolds, through detecting their exceptional Dehn fillings. In addition, we improved a criteria for profinite rigidity of a hyperbolic knot complement or a hyperbolic-type satellite knot complement among compact, orientable 3-manifolds, through examining its characterising slopes. We obtain the following profinitely rigid examples: the complement of the Whitehead link, Whitehead sister link, $\frac{3}{10}$ two-bridge link; specific surgeries on one component of these links; the complement of (full) twist knots $\mathcal{K}_n$, Eudave-Muñoz knots $K(3,1,n,0)$, Pretzel knots $P(-3,3,2n+1)$, $5_2$ knot; the Berge manifold, and many more. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_05229 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Profinite rigidity witnessed by Dehn fillings of cusped hyperbolic 3-manifolds Xu, Xiaoyu Geometric Topology Group Theory 57M10, 57M50 (primary) 57K10, 57K32, 20E18 (secondary) Any profinite isomorphism between two cusped finite-volume hyperbolic 3-manifolds carries profinite isomorphisms between their Dehn fillings. With this observation, we prove that some cusped finite-volume hyperbolic 3-manifolds are profinitely rigid among all compact, orientable 3-manifolds, through detecting their exceptional Dehn fillings. In addition, we improved a criteria for profinite rigidity of a hyperbolic knot complement or a hyperbolic-type satellite knot complement among compact, orientable 3-manifolds, through examining its characterising slopes. We obtain the following profinitely rigid examples: the complement of the Whitehead link, Whitehead sister link, $\frac{3}{10}$ two-bridge link; specific surgeries on one component of these links; the complement of (full) twist knots $\mathcal{K}_n$, Eudave-Muñoz knots $K(3,1,n,0)$, Pretzel knots $P(-3,3,2n+1)$, $5_2$ knot; the Berge manifold, and many more. |
| title | Profinite rigidity witnessed by Dehn fillings of cusped hyperbolic 3-manifolds |
| topic | Geometric Topology Group Theory 57M10, 57M50 (primary) 57K10, 57K32, 20E18 (secondary) |
| url | https://arxiv.org/abs/2412.05229 |