The study reveals that isomorphisms of profinite completions are regular in cusped hyperbolic 3-manifolds, suggesting the A-polynomial is a fundamental invariant.
We prove that any isomorphism between the profinite completions of the fundamental groups of two cusped finite-volume hyperbolic 3-manifolds is regular and peripheral regular. As an application, we show that the A-polynomial of prime knots in S³ is a profinite invariant, up to possible mirror image.
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Xiaoyu Xu (2025) studied this question.
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