Presenting the Zassenhaus Lie algebra by the Magnus Lie algebra

TL;DR

This paper demonstrates that the Zassenhaus restricted Lie algebra of certain groups can be explicitly presented and computed using the Magnus Lie algebra, with applications to various classes of groups.

Contribution

It establishes a method to present and compute the Zassenhaus Lie algebra via the Magnus Lie algebra for specific groups, extending understanding of their algebraic structure.

Findings

Zassenhaus Lie algebra can be presented by the Magnus Lie algebra.

Explicit computation is possible for torsion-free lower central series groups.

Results apply to surface groups, Artin groups, and braid groups.

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.