Abstract

It is well known that if we take a group G, and quotient out by the normal closure of all non-trivial commutators, then the resulting group is abelian (and in some sense “universal”). However, if we instead quotient out by the normal closure of all torsion elements of G, then the resulting group need not be torsion-free. But, by taking a countably infinite “tower” of quotients of torsion elements, we do eventually come to a torsion-free group, which is universal in the same way that the abelianisation is. We give an explicit construction of a finitely presented group which is not torsion-free, and for which the first quotient by torsion elements is again not torsion-free. We extend this construction, and combine it with some classical embedding results, to construct a finitely generated (and recursively presented) group for which no finite iteration of quotients by torsion elements yields a torsion-free group. This is a joint work with Rishi Vyas.