Immersed surfaces in cubed manifolds (Q1293772)

Let \(M\) be a compact, connected, oriented PL \(n\)-manifold which is the underlying space of a cube complex, and assume \(M\) has nonpositive curvature in the path-length metric determined by the cubing. Such an \(M\) contains a canonical immersed hypersurface \(S\) defined by the requirement that the intersection of \(S\) with every \(n\)-cube \(C\) of \(M\) be the union of the \(n\) codimension-1 cubes which contain the center of \(C\) and are parallel to a pair of \((n-1)\)-faces of \(C\), and this immersed hypersurface is \(\pi_1\)-injective. In the main theorem of this paper, the authors prove that if every \((n-2)\)-cube of the cubing of \(M\) is incident to an even number of \((n-1)\)-cubes, then there exists a regular \(n!\)-fold covering of \(M\) to which \(S\) lifts as an embedding. The motivation for this theorem comes from the work connected with the geometrization conjecture in 3-dimensional topology. Namely, \textit{I. R. Aitchison} and \textit{J. H. Rubinstein} [Lond. Math. Soc. Lect. Note Ser. 151, 127-161 (1990; Zbl 0735.57005)] have conjectured that every nonpositively curved cubed atoroidal 3-manifold \(M\) admits a complete hyperbolic structure, and they proved that this conjecture is true if \(M\) contains an immersed \(\pi_1\)-injective surface which lifts to an embedded surface in some finite cover of \(M\). The authors give a number of applications of their theorem, mainly to 3-dimensional topology, proving many 3-manifolds virtually Haken.