Distinguishing geometries using finite quotients

TL;DR

This paper characterizes 3-manifold geometries using properties of their profinite fundamental groups, distinguishing hyperbolic and Seifert fibered types through algebraic subgroup structures.

Contribution

It provides a profinite group-based characterization of hyperbolic and Seifert fibered 3-manifolds, advancing the understanding of 3-manifold geometry via algebraic properties.

Findings

Profinite completion of hyperbolic 3-manifold groups lacks $\

Profinite completion of Seifert fibered 3-manifold groups has a non-trivial procyclic normal subgroup.

Every finitely generated pro-$p$ subgroup of a hyperbolic, virtually special group is free pro-$p$.

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 $\hat{\mathbb{Z}}^2$. 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 non-trivial 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-$p$.