The derived series of GGS groups

J. Moritz Petschick

Vol. 343 (2026), No. 1, 211–230

DOI: 10.2140/pjm.2026.343.211

Abstract

We describe the structure and indices of the members of the derived series of a GGS group $G$ defined over the $p$-adic tree. The values $|G:G^{(n)}|$ exhibit a very limited dependence on the defining vector of the group. Furthermore, we establish that the derived and Frattini series of a GGS group defined by a nonconstant vector are identical.

Keywords

derived series, GGS group, branch group, self-similar group, circulant matrix, automorphism group of regular rooted trees

Mathematical Subject Classification

Primary: 20E08

Secondary: 20F14, 20E15

Milestones

Received: 2 June 2023

Revised: 15 March 2026

Accepted: 16 March 2026

Published: 29 April 2026