A connection between cellularization for groups and spaces via two-complexes

Let MM denote a two-dimensional Moore space (so H2(M;Z)=0), with fundamental group GG. The MM-cellular spaces are those one can build from MM by using wedges, push-outs, and telescopes (and hence all pointed homotopy colimits). The issue we address here is the characterization of the class of MM-cellular spaces by means of algebraic properties derived from the group GG. We show that the cellular type of the fundamental group and homological information does not suffice, and one is forced to study a certain universal extension.