Colimits of abelian groups

In this paper we study the colimit N2(G)N2(G) of abelian subgroups of a discrete group G  . This group is the fundamental group of a subspace B(2,G)B(2,G) of the classifying space BG as introduced in [1]. We describe N2(G)N2(G) for certain groups, and apply our results to study the homotopy type of the space B(2,G)B(2,G). We give a list of classes of groups for which B(2,G)B(2,G) is not an Eilenberg–Maclane space of type K(π,1)K(π,1).