Cocompactly cubulated graph manifolds (Q2351752)

A torsion-free group is \textit{cocompactly cubulated} if it is the fundamental group of a compact nonpositively curved cube complex. A cocompactly cubulated group is \textit{compact special} if the (compact) complex admits a local isometry into the Salvetti complex of a right-angled Artin group. The authors state that the main goal of the paper is to answer Question 9.4 of \textit{M. Aschenbrenner}, \textit{S. Friedl} and \textit{H. Wilton} [\(3\)-manifold groups. EMS Series of Lectures in Mathematics. Zürich: European Mathematical Society (EMS) (2015; Zbl 1326.57001)] by characterizing graph manifolds whose fundamental group is virtually cocompactly cubulated. A graph manifold \(M\) is \textit{chargeless} if each Seifert piece disjoint from \(\partial M\) contains a horizontal surface whose boundary circles are vertical in adjacent Seifert pieces. The authors prove that the fundamental group of a chargeless graph manifold is virtually (\(3\)-dimensional) compact special and conversely, if the fundamental group of a graph manifold \(M\) is virtually cocompactly cubulated, then \(M\) is chargeless.