The lower central series of a group G is defined by γ1=G and γn = [G,γn-1]. The ‘dimension series’, introduced by Magnus, is defined using the group algebra over the integers:
δn = { g : g-1 belongs to the nth power of the augmentation ideal }.
It has been, for the last 80 years, a fundamental problem of group theory to relate these two series. One always has δn ≥ γn, and a conjecture by Magnus, with false proofs by Cohn, Losey, etc., claims that they coincide; but Rips constructed an example with δ4 / γ4 cyclic of order 2. On the positive side, Sjogren showed that δn / γn is always a torsion group, of exponent bounded by a function of n. Furthermore, it was believed (and falsely proved by Gupta) that only 2-torsion may occur.
In joint work with Roman Mikhailov, we prove however that every torsion abelian group may occur as a quotient δn / γn; this proves that Sjogren’s result is essentially optimal.
Even more interestingly, we show that this problem is intimately connected to the homotopy groups πn(Sm) of spheres; more precisely, the quotient δn / γn is related to the difference between homotopy and homology. We may explicitly produce p-torsion elements starting from the order-p element in the homotopy group π2p(S2) due to Serre.
This video was produced by the Universidade de São Paulo, as part of the LieJor Online Seminar: Algebras, Representations, and Applications.
