On symplectic fillings of virtually overtwisted torus bundles

We use Menkes JSJ-type decomposition theorem for symplectic fillings to reduce the classification of strong and exact symplectic fillings of virtually overtwisted torus bundles to the same problem for tight lens spaces. For virtually overtwisted structures on elliptic or parabolic torus bundles, this gives a complete classification. For virtually overtwisted structures on hyperbolic torus bundles, we show that every strong or exact filling arises from a filling of a tight lens space via round symplectic 1-handle attachment, and we give a condition under which distinct tight lens space fillings yield the same torus bundle filling.