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Distinguishing geometries
using finite quotients
Henry Wilton and Pavel Zalesskii |
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Geometry & Topology 21 (2017) 345–384
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Abstract |
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We prove that the profinite completion of the fundamental group of a compact –manifold satisfies a Tits alternative: if a closed subgroup does not contain a free pro- subgroup for any , then is virtually soluble, and furthermore of a very particular form. In particular, the profinite completion of the fundamental group of a closed, hyperbolic –manifold does not contain a subgroup isomorphic to . This gives a profinite characterization of hyperbolicity among irreducible –manifolds. We also characterize Seifert fibred –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- subgroup of the profinite completion of a torsion-free, hyperbolic, virtually special group is free pro-. |
Keywords
$3$–manifolds, profinite completions
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Mathematical Subject Classification 2010
Primary: 57N10
Secondary: 20E26, 57M05
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References |
Publication
Received: 17 February 2015
Revised: 19 October 2015
Accepted: 25 November 2015
Published: 10 February 2017
Proposed: Ian Agol
Seconded: Bruce Kleiner, Ronald Stern
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Authors |
| Henry Wilton | |
| DPMMS Cambridge University Centre for Mathematical Sciences Wilberforce Road Cambridge CB3 0WB United Kingdom |
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| Pavel Zalesskii | |
| Department of Mathematics University of Brasília 70910-9000 Brasília Brazil |
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