Foldable cubical complexes of nonpositive curvature.

*(English)*Zbl 1055.20035The 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.

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.

Reviewer: Athanase Papadopoulos (Strasbourg)

##### 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 |

**OpenURL**

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