Immersed surfaces in cubed manifolds. (English) Zbl 0935.57033

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, I. R. Aitchison and 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.


57Q35 Embeddings and immersions in PL-topology
57M50 General geometric structures on low-dimensional manifolds
57N10 Topology of general \(3\)-manifolds (MSC2010)


Zbl 0735.57005
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