2021/05/28 by Darné, Jacques, Suciu, Alexander I.
Mathematics · #17B70 #20F14 #20F40 #20J05 #Advanced Algebra and Geometry #Advanced Topics in Algebra #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR)
paper · pdf · doi:10.48550/arxiv.2105.14129
openalex publication_date 2021/05/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.