Conjugation-free groups, lower central series and line arrangements

TL;DR

This paper studies the lower central series of conjugation-free groups, revealing how their ranks depend on cyclic relations and showing their associated graded Lie algebra decomposes into local components.

Contribution

It explicitly determines the ranks of lower central series quotients for conjugation-free groups using line arrangement techniques, a novel approach in this context.

Findings

Ranks depend only on the number and length of cyclic relations

The associated graded Lie algebra decomposes into local components

Provides explicit formulas for these ranks

Abstract

The quotients $G_k/G_{k+1}$ of the lower central series of a finitely presented group $G$ are an important invariant of this group. In this work we investigate the ranks of these quotients in the case of a certain class of conjugation-free groups, which are groups generated by $x_1,...,x_n$, and having only cyclic relations: $$ x_{i_t} x_{i_{t-1}} ... x_{i_1} = x_{i_{t-1}} ... x_{i_1} x_{i_t} = ... = x_{i_1} x_{i_t} ... x_{i_2}.$$ Using tools from group theory and from the theory of line arrangements we explicitly find these ranks, which depend only at the number and length of these cyclic relations. It follows that for these groups the associated graded Lie algebra $gr(G)$ decomposes, in any degree, as a direct product of local components.