Strongly torsion generated groups

Abstract

It has long been known that the integral homology of a non-trivial finite group must be non-zero in infinitely many dimensions 17. Recent work on the Sullivan conjecture in homotopy theory has made it possible to extend this result to locally finite groups. For more general groups with torsion it becomes more difficult to make such a strong statement. Nevertheless we prove 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 below to be attainable as higher homology groups.