Cocompact cubulations of mixed 3-manifolds

Abstract

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 ${\pi }_{1}M$ 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.