Torsion homology and cellular approximation

In this note we describe the role of the Schur multiplier in the structure of the $p$-torsion of discrete groups. More concretely, we show how the knowledge of $H_{2}G$ allows to approximate many groups by colimits of copies of finite $p$-groups. Our examples include interesting families of non-commutative infinite groups, including Burnside groups, certain solvable examples and the first Grigorchuk group. We also provide a counterexample for a conjecture of E. Farjoun.