# 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.