Unravelling the dodecahedral spaces (Q1622922)

A cubing (or cubulation) of a 3-manifold is a decomposition into Euclidean cubes identified along their faces by Euclidean isometries; an NPC-cubing is a non-positively curved cubing which satisfies in addition the Gromov link condition at each vertex. If a cubing is NPC then each connected component of the canonically immersed surface, formed by the three middle squares of each cube parallel to the faces, is \(\pi_1\)-injective. If in addition the cube complex is \textit{special} (as defined by \textit{F. Haglund} and \textit{D. T. Wise} [Geom. Funct. Anal. 17, No. 5, 1551--1620 (2008; Zbl 1155.53025)]) then the 3-manifold is virtually Haken. In such a setting, but considering special cube complexes in arbitrary dimensions now, \textit{I. Agol} [Doc. Math. 18, 1045--1087 (2013; Zbl 1286.57019)] proved Waldhausen's virtual Haken conjecture from 1968 for hyperbolic 3-manifolds. The hyperbolic Weber-Seifert dodecahedral 3-manifold is obtained by identifying opposite faces of a regular hyperbolic dodecahedron, with all dihedral angles \(2\pi/5\), after a \(3\pi/5\)-twist. The dodecahedron has a natural decomposition into 20 cubes which gives an NPC-cubing for the Weber-Seifert space. The main result of the present paper states that the hyperbolic Weber-Seifert dodecahedral space admits a cover of degree 60 such that the lifted natural cubulation is special. Also, the Weber-Seifert space has a 6-sheeted irregular cover such that the canonical hypersurfaces give a very short hierarchy. The authors consider also the natural cubulation and covers of the spherical dodecahedral space (i.e., the Poincaré homology sphere). For the entire collection see [Zbl 1394.00023].