Abstract: The lower central series of a group $G$ is defined by $\gamma_1=G$ and $\gamma_n = [G,\gamma_{n-1}]$. The "dimension series", introduced by Magnus, is defined using the group algebra over the integers: $\delta_n = \{g: g-1\text{ belongs to the $n$-th 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 $\delta_n\ge\gamma_n$, and a conjecture by Magnus, with false proofs by Cohn, Losey, etc., claims that they coincide; but Rips constructed an example with $\delta_4/\gamma_4$ cyclic of order 2. On the positive side, Sjogren showed that $\delta_n/\gamma_n$ 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 the torsion in the quotients $\delta_n/\gamma_n$ can be arbitrarily specified; thus Sjogren's result is optimal.

Even more interestingly, I will show that the dimension quotient $\delta_n/gamma_n$ is related to the difference between homotopy and homology: our construction is fundamentally based on embedding the torsion of the homotopy group $\pi_n(S^2\vee S^2)$ in dimension quotients. We can even make this quite explicit on the order-$p$ element in $\pi_{2p}(S^2)$ due to Serre.