Abstract

It has been, for the last 80 years, a fundamental problem of group theory to relate the lower central series and the dimension series introduced by Magnus. One always has that the nth term of the dimension series contains the nth lower central subgroup, and a conjecture by Magnus, with false proofs by Cohn, Losey, etc., claims that they coincide; but Rips constructed an example with different fourth terms.On the positive side, Sjogren showed that the quotient of the nth dimension group by the nth lower central subgroup is always a torsion group, of exponent bounded by a function of $n$. Furthermore, it was believed (and falsely proven by Gupta) that only $2$-torsion may occur.

In joint work with Roman Mikhailov, we prove however that for every prime $p$ there is a group with $p$-torsion in some such quotient.

Even more interestingly, I will show that these quotients are related to the difference between homotopy and homology: our construction is fundamentally based on the order-$p$ element in the homotopy group $\pi_{2p}(S^2)$ due to Serre.