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Distinguishing geometries using finite quotients

Henry Wilton and Pavel Zalesskii

Geometry & Topology 21 (2017) 345–384

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 ̂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.

$3$–manifolds, profinite completions
Mathematical Subject Classification 2010
Primary: 57N10
Secondary: 20E26, 57M05
Received: 17 February 2015
Revised: 19 October 2015
Accepted: 25 November 2015
Published: 10 February 2017
Proposed: Ian Agol
Seconded: Bruce Kleiner, Ronald Stern
Henry Wilton
Cambridge University
Centre for Mathematical Sciences
Wilberforce Road
United Kingdom
Pavel Zalesskii
Department of Mathematics
University of Brasília
70910-9000 Brasília