The derived series of GGS-groups

TL;DR

This paper calculates the indices of higher derived subgroups in GGS-groups over prime-regular rooted trees and describes their structure, showing these indices depend only mildly on the defining tuple.

Contribution

It provides explicit calculations of derived subgroup indices and structural descriptions for GGS-groups with non-constant defining tuples.

Findings

Indices |G:G^{(n)}| depend mildly on the defining tuple.

Structural descriptions of higher derived subgroups G^{(n)}.

Explicit formulas for subgroup indices for all n.

Abstract

Given a GGS-group $G$ with non-constant defining tuple over a prime-regular rooted tree, we calculate the indices $|G:G^{(n)}|$ and describe the structure of the higher derived subgroups $G^{(n)}$ for all $n \in \mathbb{N}$. We find that the values $|G:G^{(n)}|$ depend only mildly on the structure of the defining tuple.