On groups acting on nonpositively curved cubical complexes
Cubical complexes are cell complexes whose cells are cubes. The interest here is in nonpositively curved cubical complexes (those that satisfy the comparison triangle condition \(\text{CAT}(0)\)), foldable cubical complexes (those that admit a combinatorial map onto a cube), and groups acting by automorphisms on such complexes. The first part of the paper is expository and surveys many of the constructions in which nonpositively curved cubical complexes arise. The second part of the paper contains three main results about certain groups \(\Gamma\) that act by automorphisms on a simply connected foldable nonpositively curved cubical complex \(X\). The first is a strengthening of a result of \textit{G. Niblo} and \textit{L. Reeves} [Geom. Topol. 1, 1-7 (1997; Zbl 0887.20016)] and concerns embedding \(X\) into a product of trees so that the group acts without fixed points on one of the trees if it so acts on \(X\). The other two results concern the question of the existence of a subgroup of finite index of \(\Gamma\) admitting a homomorphism onto \(\mathbb{Z}^n\) where \(n=\dim X\).
scientific article; zbMATH DE number 1558733
scientific article