Profinite rigidity and surface bundles over the circle

27 Jul 2017

If $M$ is a compact 3-manifold whose first betti number is 1, and $N$ is a compact 3-manifold such that $π_1N$ and $π_1M$ have the same finite quotients, then $M$ fibres over the circle if and only if $N$ does. We prove that groups of the form $F_2$ $\rtimes$ $\Bbb Z$ 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 $π_1M$ and $π_1N$ maps onto $G$ and the other does not.