Lower central series and split extensions

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 and Stallings’ rational and mod-p versions of this series. Building on the work of Massuyeau and Guaschi–Pereiro, we describe these N-series and Lie algebras in the case when G splits as a semidirect product, in terms of the relevant data for the factors and the monodromy action. As applications, we recover a well-known theorem of Falk–Randell regarding split extensions with trivial monodromy on abelianization and its mod-p version due to Bellingeri–Gervais and prove an analogous result for the rational lower central series of split extensions with trivial monodromy on torsion-free abelianization.

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