Cocompact cubulations of mixed 3-manifolds (Q2418873)

Summary: In this paper, we complete the classification of which compact 3-manifolds have a virtually compact special fundamental group by addressing the case of mixed 3-manifolds. A compact aspherical 3-manifold \(M\) is mixed if its JSJ decomposition has at least one JSJ torus and at least one hyperbolic block. We show \(\pi_1M\) is virtually compact special if and only if \(M\) is chargeless, i.e. each interior Seifert fibered block has a trivial Euler number relative to the fibers of adjacent blocks.