Distinguishing geometries using finite quotients

Abstract

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, and furthermore of a very particular form. In particular, the profinite completion of the fundamental group of a closed, hyperbolic $3$–manifold does not contain a subgroup isomorphic to ${\widehat{\mathbb{Z}}}^2$. This gives a profinite characterization of hyperbolicity among irreducible $3$–manifolds. We also characterize Seifert fibred $3$–manifolds as precisely those for which the profinite completion of the fundamental group has a nontrivial procyclic normal subgroup. Our techniques also apply to hyperbolic, virtually special groups, in the sense of Haglund and Wise. Finally, we prove that every finitely generated pro-$p$ subgroup of the profinite completion of a torsion-free, hyperbolic, virtually special group is free pro-$p$.