Profinite rigidity, fibering, and the figure-eight knot

TL;DR

This paper demonstrates that the figure-eight knot complement is uniquely identified by its finite quotients among 3-manifolds and characterizes certain 3-manifolds with similar finite quotients as fibered with free group fibers.

Contribution

It proves the profinite rigidity of the figure-eight knot complement and characterizes 3-manifolds with finite quotients matching free-by-cyclic groups.

Findings

The figure-eight knot complement is distinguished by its finite quotients.

Certain 3-manifolds with specific finite quotients are fibered over the circle.

Characterization of 3-manifolds with finite quotients matching free-by-cyclic groups.

Abstract

We establish results concerning the profinite completions of 3-manifold groups. In particular, we prove that the complement of the figure-eight knot $S^3-K$ is distinguished from all other compact 3-manifolds by the set of finite quotients of its fundamental group. In addition, we show that if $M$ is a compact 3-manifold with $b_1(M)=1$, and $\pi_1(M)$ has the same finite quotients as a free-by-cyclic group $F_r\rtimes\mathbb{Z}$, then $M$ has non-empty boundary, fibres over the circle with compact fibre, and $\pi_1(M)\cong F_r\rtimes_\psi\mathbb{Z}$ for some $\psi\in{\rm{Out}}(F_r)$.