Right-Angled Spaces and Groups

The notion of a “polyhedral product” of a collection of spaces (or pairs of spaces) indexed by the vertex set I of a simplicial complex L is used to define the main examples of complexes and groups that are discussed in this book. If L is a flag complex and I indexes a collection of copies of the infinite cyclic group, then the polyhedral product is the standard classifying space for the “right-angled Artin group” (abbreviated as RAAG) associated to L. More generally, if I indexes a collection of classifying spaces BG(i), then the polyhedral product is the classifying space for the “graph product” of the \(G_i\) . The universal cover of a polyhedral product is often a \(\operatorname {CAT}(0)\) cube complex and isometry groups of these cube complexes are often graph products. For example, if the spaces that are indexed by I are cones over discrete sets, then the universal cover of the polyhedral product is a right-angled building (abbreviated as \(\operatorname {RAB}\) ). If each of the discrete sets is a group \(G_i\) , \(i\in I\) , then the relevant isometry group of the \(\operatorname {RAB}\) is the graph product of the \(G_i\) . When each of the factors is the cyclic group of order 2, the graph product is the “right-angled Coxeter group” (abbreviated as RACG) associated to L; when each factor is infinite cyclic, then the graph product is a RAAG. In these cases the right-angled buildings are called, respectively, the “Davis–Moussong complex” of the RACG or the “Deligne complex” of the RAAG. So, Sect. 3.1 is mainly concerned with graph products of discrete groups. The notion of a “wreath graph-product” also plays an important role in this chapter. By definition, it is the semidirect product of a graph product with a group of automorphisms of L. In Sect. 3.2 we discuss cubical structures induced by right-angled reflection groups on manifolds and we take a fairly deep dive into the theory of cocompact reflection groups on contractible manifolds. Here the flag complex L is a simplicial sphere (or more generally, a generalized homology sphere). The corresponding polyhedral product of intervals is also known as the “generalized moment angle manifold.” Its fundamental group is the commutator subgroup of a RACG. Further quotients lead to the notion of a “small cover” of a right-angled Coxeter orbifold. Various related ideas, such as Haken manifolds and the reflection group trick, are also explained in Sect. 3.2. In Sect. 3.3 we discuss a similar class of examples acting on \(\operatorname {CAT}(0)\) cube complexes formed by “blowing up Coxeter zonotopes” and taking the universal covers. In general the relevant groups are not Coxeter groups, rather they are “mock reflection groups.”