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

TL;DR

This paper proves that isomorphisms between profinite completions of cusped hyperbolic 3-manifold groups are regular and peripheral, leading to the $A$-polynomial being a profinite invariant for prime knots, up to mirror image.

Contribution

It establishes the regularity of profinite isomorphisms between fundamental groups of cusped hyperbolic 3-manifolds and shows the $A$-polynomial is a profinite invariant.

Findings

Profinite isomorphisms are regular and peripheral regular.

The $A$-polynomial is a profinite invariant for prime knots.

Proves a strong link between profinite completions and geometric invariants.

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.