Lower central series and split extensions

TL;DR

This paper generalizes the understanding of lower central series and related Lie algebras for split group extensions, extending classical theorems to various LCS versions and applying results to surface braid groups.

Contribution

It extends Falk-Randell's theorem to other LCS variants, generalizes Bellingeri-Gervais results, and explores residual properties of semi-direct products, with applications to surface braid groups.

Findings

Generalized LCS theorems to mod-q, rational, and Zassenhaus versions.

Provided new proofs of classical residual properties theorems.

Applied results to surface braid groups, confirming residual nilpotency.

Abstract

Following Lazard, we study the $N$-series of a group $G$ and their associated graded Lie algebras. The main examples we consider are the lower central series (LCS), Stallings' rational and mod-$q$ versions, and Zassenhaus' mod-$p$ version of the LCS. We treat them as part of a general construction of the $\mathcal P$-LCS, for a property $\mathcal P$ of filtrations. We describe these $N$-series and the associated Lie algebras in the case when $G$ splits as a semi-direct product, in terms of the relevant data for the factors and the monodromy action. This allows us to generalize the well-known theorem of Falk-Randell regarding the LCS of split extensions to other versions of the LCS. In particular, we generalize the mod-$q$ version of Bellingeri-Gervais to any integer $q$, and we prove analogous results for the rational LCS and Zassenhaus' mod-$p$ LCS. We then use the same tools to study residual properties of semi-direct products, and how they interact with residual properties of the factors. We also give a new proof of a classical theorem of Gruenberg. Finally, we apply our results to surface braid groups, which naturally split as semi-direct products, allowing us to recover and generalize known results about the residual nilpotency of groups of pure braids on surfaces.