Profinite rigidity and surface bundles over the circle

TL;DR

This paper demonstrates that for certain 3-manifolds, their fundamental groups' finite quotients determine whether the manifold fibers over the circle, establishing a link between profinite completions and fibering properties.

Contribution

It proves that groups of the form F_2 semidirect Z are distinguished by their profinite completions, linking profinite rigidity to surface bundle structures over the circle.

Findings

Groups of the form F_2⋊Z are uniquely identified by their profinite completions.

Punctured torus bundles over the circle are distinguished by their finite quotients if not homeomorphic.

Fibering over the circle is characterized by the profinite properties of the fundamental group.

Abstract

If $M$ is a compact 3-manifold whose first betti number is 1, and $N$ is a compact 3-manifold such that $\pi_1N$ and $\pi_1M$ have the same finite quotients, then $M$ fibres over the circle if and only if $N$ does. We prove that groups of the form $F_2\rtimes\mathbb{Z}$ are distinguished from one another by their profinite completions. Thus, regardless of betti number, if $M$ and $N$ are punctured torus bundles over the circle and $M$ is not homeomorphic to $N$, then there is a finite group $G$ such that one of $\pi_1M$ and $\pi_1N$ maps onto $G$ and the other does not.