Dimension series and homotopy groups of spheres

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 every torsion abelian group may occur as a quotient $\delta_n/\gamma_n$; this proves that Sjogren's result is essentially optimal.

Even more interestingly, we show that this problem is intimately connected to the homotopy groups $\pi_n^(S^m)$ of spheres; more precisely, the quotient $\delta_n/\gamma_n$ 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 $\pi_{2p}(S^2)$ due to Serre.

group theory

Audience: researchers in the topic

GOThIC - Ischia Online Group Theory Conference

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