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Foldable cubical complexes of nonpositive curvature. (English) Zbl 1055.20035
The author uses the following terminology: A ‘cubical complex’ $$X$$ is a CW-complex obtained by gluing unit Euclidean cubes along their faces via isometries, with the requirements that these cubes inject into $$X$$ and the intersection of the images of two cubes is either empty or equals the image of a cube. A cubical complex is said to be ‘dimensionally homogeneous’ if there is an integer $$n\geq 1$$ such that each cube of $$X$$ is a face of some $$n$$-cube. A ‘folding’ of $$X$$ is a combinatorial map $$f\colon X\to C$$ onto an $$n$$-cube $$C$$ such that the restriction of $$f$$ to each cube is injective. The complex $$X$$ is equipped with its natural length metric. An FCC is then a connected foldable cubical complex that is dimensionally homogeneous, geodesically complete and having nonpositive curvature in the sense of Alexandrov.
In this paper, the author proves the following Theorem 1: Let $$X$$ be an FFC of dimension $$n$$. Then $$X$$ has the structure of a graph of spaces where all the vertex and edge spaces are $$(n-1)$$-dimensional FFCs and the maps from edge spaces to vertex spaces are combinatorial immersions.
As an application, the author proves the following results: Theorem 2: Let $$X$$ be a finite FFC of dimension 3 with universal cover $$\widetilde X$$ and let $$\Gamma$$ be its group of deck transformations. Then, (1) if $$\widetilde X$$ has rank one, then $$\widetilde X$$ contains a periodic rank one geodesic in the 1-skeleton; (2) if $$\widetilde X$$ has higher rank, then $$\widetilde X$$ is isometric to the product of two CAT(0) FCCs;
Theorem 3: Let $$X$$ be a finite FCC. Then any subgroup of $$\pi_1(X)$$ either contains a free group of rank two or is virtually free Abelian.
Theorem 2 is related to the ‘rank rigidity conjecture’ of Ballmann and others. Theorem 3 is due to W. Ballmann and J. Świątkowski, who in fact introduced the notion of foldable cubical complex [see Enseign. Math., II. Sér. 45, No. 1-2, 51-81 (1999; Zbl 0989.20029)]. The author gives here a new proof of Theorem 3 which is shorter than the one by Ballmann and Świątkowski.
##### MSC:
 20F65 Geometric group theory 20E08 Groups acting on trees 20F67 Hyperbolic groups and nonpositively curved groups 53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces 53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov) 57M07 Topological methods in group theory
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