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Torus bundles not distinguished by TQFT invariants

Louis Funar

Appendix: Louis Funar and Andrei Rapinchuk

Geometry & Topology 17 (2013) 2289–2344

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 SL(2, ) 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 U(1) and SU(2) 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.

mapping class group, torus bundle, modular tensor category, congruence subgroup, $\mathrm{SL}(2,\mathbb{Z})$, conjugacy problem, Pell equation, rational conformal field theory
Mathematical Subject Classification 2010
Primary: 20F36, 57M07
Secondary: 20F38, 57N05
Received: 3 March 2011
Revised: 23 April 2013
Accepted: 23 April 2013
Published: 22 August 2013
Proposed: Vaughan Jones
Seconded: Cameron Gordon, Peter Teichner
Louis Funar
Institut Fourier
University of Grenoble I
BP 74, UMR 5582
38402 Saint-Martin d’Hères cedex
Louis Funar
Andrei Rapinchuk
Department of Mathematics
University of Virginia
141 Cabell Drive
Kerchof Hall, PO Box 400137
Charlottesville, VA 22904-4137