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arXiv · 2412.05229

Profinite rigidity witnessed by Dehn fillings of cusped hyperbolic 3-manifolds

Xiaoyu Xu
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Abstract

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.

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Xiaoyu Xu. 2026-03-16. Profinite rigidity witnessed by Dehn fillings of cusped hyperbolic 3-manifolds. https://arxiv.org/abs/2412.05229

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