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Torus bundles not
distinguished by TQFT invariants
Louis FunarAppendix: Louis Funar and Andrei Rapinchuk |
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Geometry & Topology 17 (2013) 2289–2344
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Abstract |
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We show that there exist arbitrarily large sets of non-homeomorphic closed oriented SOL torus bundles with the same quantum (TQFT) invariants. This follows from the arithmetic behind the conjugacy problem in and its congruence quotients, the classification of SOL (polycyclic) 3–manifold groups and an elementary study of a family of Pell equations. A key ingredient is the congruence subgroup property of modular representations, as it was established by Coste and Gannon, Bantay, Xu for various versions of TQFT, and by Ng and Schauenburg for the Drinfeld doubles of spherical fusion categories. In particular, we obtain non-isomorphic 3–manifold groups with the same pro-finite completions, answering a question of Long and Reid. On the other side we prove that two torus bundles over the circle with the same and quantum invariants are (strongly) commensurable. In the appendix (joint with Andrei Rapinchuk) we show that these examples have positive density in a suitable set of discriminants. |
Keywords
mapping class group, torus bundle, modular tensor category,
congruence subgroup, $\mathrm{SL}(2,\mathbb{Z})$, conjugacy
problem, Pell equation, rational conformal field theory
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Mathematical Subject Classification 2010
Primary: 20F36, 57M07
Secondary: 20F38, 57N05
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References |
Publication
Received: 3 March 2011
Revised: 23 April 2013
Accepted: 23 April 2013
Published: 22 August 2013
Proposed: Vaughan Jones
Seconded: Cameron Gordon, Peter Teichner
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Authors |
| Louis Funar | |
| Institut Fourier University of Grenoble I BP 74, UMR 5582 38402 Saint-Martin d’Hères cedex France |
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| http://www-fourier.ujf-grenoble.fr/~funar | |
| Louis Funar | |
| Andrei Rapinchuk | |
| Department of Mathematics University of Virginia 141 Cabell Drive Kerchof Hall, PO Box 400137 Charlottesville, VA 22904-4137 USA |
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| http://pi.math.virginia.edu/Faculty/Rapinchuk/ | |
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