#### Volume 4, issue 1 (2004)

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Foldable cubical complexes of nonpositive curvature

### Xiangdong Xie

Algebraic & Geometric Topology 4 (2004) 603–622
 arXiv: math.MG/0409067
##### Abstract

We study finite foldable cubical complexes of nonpositive curvature (in the sense of A D Alexandrov). We show that such a complex $X$ admits a graph of spaces decomposition. It is also shown that when $dimX=3$, $X$ contains a closed rank one geodesic in the $1$–skeleton unless the universal cover of $X$ is isometric to the product of two CAT(0) cubical complexes.

##### Keywords
rank one geodesic, cubical complex, nonpositive curvature
##### Mathematical Subject Classification 2000
Primary: 20F65, 20F67
Secondary: 53C20