  We show that there exist infinitely many pairs 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 lastly by Ng
and Schauenburg for the Drinfeld doubles of spherical fusion categories. On the
other side we prove that two torus bundles over the circle with the same
quantum invariants are (strongly) commensurable. The examples above show that
this is the best that it could be expected.
