#### Volume 21, issue 1 (2017)

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Distinguishing geometries using finite quotients

### Henry Wilton and Pavel Zalesskii

Geometry & Topology 21 (2017) 345–384
##### 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 ${\stackrel{̂}{ℤ}}^{2}$. 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-$p$.

##### Keywords
$3$–manifolds, profinite completions
##### Mathematical Subject Classification 2010
Primary: 57N10
Secondary: 20E26, 57M05
##### 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
##### Authors
 Henry Wilton DPMMS Cambridge University Centre for Mathematical Sciences Wilberforce Road Cambridge CB3 0WB United Kingdom Pavel Zalesskii Department of Mathematics University of Brasília 70910-9000 Brasília Brazil