We prove that the profinite completion of the fundamental group of a compact 3-manifold M satisfies a Tits alternative: if a closed subgroup H does not contain a free pro-p subgroup for any p, then H is virtually soluble, ...

We completely describe the finitely generated pro-$p$ subgroups of the profinite completion of the fundamental group of an arbitrary 3-manifold. We also prove a pro-$p$ analogue of the main theorem of Bass–Serre theory for ...