Profinite rigidity, fibering, and the figure-eight knot

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 $F_r⋊\mathbb{Z}$, then $M$ has non-empty boundary, fibres over the circle with compact fibre, and ${\pi }_1(M)\cong F_r{⋊}_{\psi }\mathbb{Z}$ for some $\psi \in \mathrm{Out}(F_r)$.

15 pages, no figures