GGS-groups over primary trees: Branch structures

TL;DR

This paper investigates the branch structures of GGS-groups over primary trees, establishing conditions under which these groups are weakly or strongly regular branch, extending prior results for specific cases.

Contribution

It extends known results by proving that most GGS-groups over primary trees are weakly or regularly branch over certain subgroups, with exceptions for specific parameters.

Findings

Most GGS-groups over primary trees are weakly regular branch over G''.

In most cases, these groups are regular branch over γ_3(G).

GGS-groups generated by a constant vector are not branch.

Abstract

We study branch structures in Grigorchuk-Gupta-Sidki groups (GGS-groups) over primary trees, that is, regular rooted trees of degree $p^n$ for a prime $p$. Apart from a small set of exceptions for $p=2$, we prove that all these groups are weakly regular branch over $G''$. Furthermore, in most cases they are actually regular branch over $\gamma_3(G)$. This is a significant extension of previously known results regarding periodic GGS-groups over primary trees and general GGS-groups in the case $n=1$. We also show that, as in the case $n=1$, a GGS-group generated by a constant vector is not branch.