Foldable cubical complexes of nonpositive curvature

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.