Strongly torsion generated groups

It has long been known that the integral homology of a non-trivial finite group must be non-zero in infinitely many dimensions [15]. Recent work on the Sullivan Conjecture in homotopy theory has made it possible to extend this result to non-acyclic locally finite groups. For more general groups with torsion it becomes more difficult to make such a strong statement. Nevertheless we show that when a non-perfect group is generated by torsion elements its integral homology must also be non-zero in infinitely many dimensions. Remarkably, this result is best possible in that for perfect torsion generated groups all (finite or infinite) sequences of abelian groups are shown to be attainable as homology groups. Surprisingly, it is often preferable to work with a special subclass of torsion generated groups, here called strongly torsion generated groups, which is of interest in its own right.