Distinguishing geometries using finite quotients
By Henry Wilton, Pavel A. Zalesskii. 2014
77 citations recorded by the source corpus.
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-$p$.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/2a470f3c89023de98f4c1a46407d7f4d5ec1e95a
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In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Profinite rigidity for Seifert fibre spaces (2015, background): “The theorems of [WZ14] are stated for closed manifolds, so we will be a little more circumspect about asserting profinite rigidity among all 3-manifolds.”
- Absolute profinite rigidity and hyperbolic geometry (2018, unknown): “[8], [11], [13], [14], [26], [28], [50] and [51].”