Profinite rigidity and surface bundles over the circle

This is the authors' abstract. We don't add key points for this paper.

If M is a compact 3-manifold whose first betti number is 1, and N is a compact 3-manifold such that π 1 N and π 1 M 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 ⋊ 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 1 M and π 1 N maps onto G and the other does not.