On regularity of profinite isomorphisms between cusped hyperbolic 3-manifolds and the $A$-polynomial

Xiaoyu Xu

On regularity of profinite isomorphisms between cusped hyperbolic 3-manifolds and the $A$-polynomial

Geometric Topology 2025-10-30 v2 Group Theory

Authors: Xiaoyu Xu

Abstract

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 $S^3$ is a profinite invariant, up to possible mirror image.

Keywords

profinite completion cusped hyperbolic 3-manifold A-polynomial profinite isomorphism regularity peripheral regularity prime knot

Cite

@article{arxiv.2506.21105,
  title  = {On regularity of profinite isomorphisms between cusped hyperbolic 3-manifolds and the $A$-polynomial},
  author = {Xiaoyu Xu},
  journal= {arXiv preprint arXiv:2506.21105},
  year   = {2025}
}

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46 pages

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