Torsion homology and cellular approximation

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 us to approximate many groups by colimits of copies of p-groups. Our examples include interesting families of noncommutative infinite groups, including Burnside groups, certain solvable groups and branch groups. We also provide a counterexample for a conjecture of Emmanuel Farjoun.