Torus bundles not distinguished by TQFT invariants

By L. Funar. 2011

65 citations recorded by the source corpus.

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;Z/ 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. 20F36, 57M07; 20F38, 57N05

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/75f5e132e2ebf61cd2851ebf7e9888758da2a94b

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Citations in the corpus

Absolute profinite rigidity and hyperbolic geometry (2018, unknown): “[8], [11], [13], [14], [26], [28], [50] and [51].”
Determining Fuchsian groups by their finite quotients (2014, background): “that C(Γ 1) = C(Γ 2). There also exist pairs of ﬁnitelygenerated residually free groups Γ 1 6∼= Γ 2 such that C(Γ 1) = C(Γ 2); see Remark 4.19. In addition, in the context of 3-manifold groups, Funar [21] has shown the existence of non-homeomorphic torus bundles over the circle whose fundamental gro”
Distinguishing geometries using finite quotients (2014, methodology): “Funar [9] used work of Stebe [46] to exhibit Sol manifolds that answer the Calegari– Freedman–Walker question in the negative, and more recently Hempel [21] has exhibited Seifert fibred examples.”
Profinite rigidity for Seifert fibre spaces (2015, methodology): “Funar [11] built on work of Stebe [21] to give infinite families of Sol manifolds with the same finite quotients.”
Some 3-manifold groups with the same finite quotients (2014, background): “ general π1(Mφ) and π1(Mφk ) will not be isomorphic. This is discussed in section 5. Note that these examples illustrate the necessity of the hypothesis in the above mentioned result of [LoRe]. Funar [Fun] gives examples of closed 3-manifolds with the geometry Solwhich have non-isomorphic fundamenta”