Cubulating rhombus groups. (Q355385)

A square complex is a 2-complex obtained by attaching squares to a graph. Such a complex is said to be nonpositively curved if the link of each vertex does not contain any cycle of length \(\leq 3\). In the paper under review, the authors study a 2-dimensional generalization of nonpositively curved square complexes, namely, piecewise Euclidean complexes that are built from parallelograms instead of squares. The link condition corresponds here to a \(2\pi\) lower bound on the angle sum of corners associated to any cycle in each link. By scaling the parallelograms, the authors reduce the question to the case where parallelograms are rhombi, and the complexes are called rhombi complexes.NEWLINENEWLINE The main result is that fundamental group of a nonpositively curved rhombus complex acts properly and freely on a CAT(0) cube complex. The question of whether the fundamental group acts freely on a finite-dimensional CAT(0) cube complex is open. The authors describe some partial results about this question. They also describe walls in these new complexes and the dual CAT(0) cube complex. They relate the Sageev dual cube complexes to the de Bruin approach to Penrose tilings.