On the lower central series of a large family of non-periodic GGS-groups

TL;DR

This paper determines the lower central series of a large family of non-periodic GGS-groups for an odd prime p, extending understanding beyond the Grigorchuk group and confirming a related conjecture.

Contribution

It provides the first detailed analysis of the lower central series for a broad class of non-periodic GGS-groups, including indices and width properties.

Findings

Calculated indices between consecutive lower central series terms.

Established that these groups and their profinite completions have lower central width 2.

Confirmed a conjecture about the generalized Fabrykowski-Gupta groups.

Abstract

For an odd prime $p$, we determine the lower central series of a large family of non-periodic GGS-groups, which has a density of roughly $(\frac{p-1}{p})^2$ within all GGS-groups. This means a significant extension of the knowledge regarding the lower central series of distinguished classes of branch groups, which to date was basically restricted to the Grigorchuk group. As part of our results, we obtain the indices between consecutive terms of the lower central series, and we show that these groups, as well as their profinite completions, have lower central width equal to $2$. In particular, this confirms a conjecture of Bartholdi, Eick, and Hartung about the generalised Fabrykowski-Gupta groups.