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

2024/12/06 by Xiaoyu Xu, Xu, Xiaoyu · 1 citation
Computer Science · Mathematics · #20E18 (secondary) #57K32 #57M10 #57M50 (primary) 57K10 #Computational Geometry and Mesh Generation #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2412.05229

openalex publication_date 2024/12/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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, (3)/(10) two-bridge link; specific surgeries on one component of these links; the complement of (full) twist knots K_n, Eudave-Muñoz knots K(3,1,n,0), Pretzel knots P(-3,3,2n+1), 5₂ knot; the Berge manifold, and many more.

Cited by

• Turaev--Viro invariants and profinite completions of surface bundles
• On regularity of profinite isomorphisms between cusped hyperbolic 3-manifolds and the A-polynomial

Related

• Rigidity in hyperbolic Dehn filling
• Profinite rigidity and geometric convergence
• Toward the profinite rigidity of hyperbolic 3-manifolds
• Word hyperbolic Dehn surgery
• On the number of hyperbolic 3-manifolds of a given volume