Torus bundles not distinguished by TQFT invariants

Abstract

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 $SL\left(2,\mathbb{Z}\right)$ 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 $U\left(1\right)$ and $SU\left(2\right)$ 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.