On the fundamental group of certain polyhedral products

Let K   be a finite simplicial complex, and (X,A)(X,A) be a pair of spaces. The purpose of this article is to study the fundamental group of the polyhedral product denoted ZK(X,A)ZK(X,A), which denotes the moment-angle complex of Buchstaber and Panov in the case (X,A)=(D2,S1)(X,A)=(D2,S1), with extension to arbitrary pairs in [2] as given in Definition 2.2 here.For the case of a discrete group G, we give necessary and sufficient conditions on the abstract simplicial complex K   such that the polyhedral product denoted by ZK(BG̲) is an Eilenberg–Mac Lane space. The fundamental group of ZK(BG̲) is shown to depend only on the 1-skeleton of K. Further special examples of polyhedral products are also investigated.Finally, we use polyhedral products to study an extension problem related to transitively commutative groups, which are given in Definition 5.2.