## 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.

### MSC:

 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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