The profinite completion of $3$-manifold groups, fiberedness and the Thurston norm

TL;DR

This paper investigates how the profinite completion of 3-manifold groups relates to geometric properties like fiberedness and the Thurston norm, and explores its power in distinguishing knots, including torus and hyperbolic knots.

Contribution

It demonstrates that profinite completions induce Thurston norm isometries and fibered class bijections, and shows their effectiveness in distinguishing certain knots.

Findings

Profinite isomorphisms induce Thurston norm isometries.

Profinite completion distinguishes torus and figure-eight knots.

It can differentiate hyperbolic knots with specific Alexander polynomial properties.

Abstract

We show that a regular isomorphism of profinite completion of the fundamental groups of two 3-manifolds $N_1$ and $N_2$ induces an isometry of the Thurston norms and a bijection between the fibered classes. We study to what extent does the profinite completion of knot groups distinguish knots and show that it distinguishes each torus knot and the figure eight knot among all knots. We show also that it distinguishes between hyperbolic knots with cyclically commensurable complements under the assumption that their Alexander polynomials have at least one zero which is not a root of unity.