Lower central series and split extensions

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.