Filtrations of free groups arising from the lower central series

2016/01/29 by Michael Chapman, Chapman, Michael, Ido Efrat +1
Mathematics · #20F40 #20H25 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Group Theory (math.GR) #Number Theory (math.NT) #Primary 20F14 #Secondary 20E05

paper · pdf · doi:10.48550/arxiv.1601.08006

openalex publication_date 2016/01/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We make a systematic study of filtrations of a free group F defined as products of powers of the lower central series of F. Under some assumptions on the exponents, we characterize these filtrations in terms of the group algebra, the Magnus algebra of non-commutative power series, and linear representations by upper-triangular unipotent matrices. These characterizations generalize classical results of Grun, Magnus, Witt, and Zassenhaus from the 1930's, as well as later results on the lower p-central filtration and the p-Zassenhaus filtrations. We derive alternative recursive definitions of such filtrations, extending results of Lazard. Finally, we relate these filtrations to Massey products in group cohomology.

Related

• Filtrations of free groups as intersections
• The lower p-central series of a free profinite group and the shuffle algebra
• The Zassenhaus filtration, Massey Products, and Representations of Profinite Groups
• Lower central series and split extensions
• Generalizing Magnus' characterization of free groups to some free products