Presenting the Zassenhaus Lie algebra by the Magnus Lie algebra
Ettore Marmo, David Riley, Thomas Weigel
TL;DR
The paper shows that the restricted Zassenhaus $ abla F_p$-Lie algebra of a group $G$ can be presented from the Magnus $ abla Z$-Lie algebra via restrictification, yielding a natural surjection $ ext{hat} heta_{G,ullet}$ from $( ext{gr}^{ abla}_ullet(G) ensor abla F_p)^{[p]}$ onto $ ext{gr}^{ ext{Z}, p}_ullet(G)$. For $ ext{gamma}$-free groups (and many important pro-$p$ cases), this map is an isomorphism, enabling explicit computation of the restricted Zassenhaus algebra from the Magnus algebra; in general, the kernel is described in terms of $x^{[p]}$-type relations. The results apply to a wide spectrum of groups, including orientable surface groups, right-angled Artin groups, pure braid groups, and fundamental groups of supersolvable hyperplane and strictly supersolvable toric arrangements, as well as to uniformly powerful pro-$p$ groups and certain mild non $ ext{gamma}$-free pro-$p$ groups. This creates a unified framework for translating computations in the Magnus Lie algebra into the restricted Zassenhaus algebra, with implications for Lie-theoretic tools like the May spectral sequence and for a broad class of group-theoretic constructions. The work thus provides both structural insight and practical computational methods for a large family of groups through the interaction of Magnus and Zassenhaus filtrations.
Abstract
It is shown that the Zassenhaus restricted $\mathbb F_p$-Lie algebra of a (pro-p) group G can be presented by the Magnus Lie algebra of G. For the class of (pro-p) groups for which the terms of the lower central series are torsion-free, the Zassenhaus restricted $\mathbb F_p$-Lie algebra can be explicitly computed from the Magnus Lie algebra. These results apply to orientable surface groups, right-angled Artin groups, pure braid groups, fundamental groups of supersolvable hyperplane arrangements and fundamental groups of strictly supersolvable toric arrangements.