pith. sign in

arxiv: 2208.14975 · v2 · submitted 2022-08-31 · 🧮 math.GR

The derived series of GGS-groups

Jan Moritz Petschick This is my paper

Pith reviewed 2026-05-24 11:03 UTC · model grok-4.3

classification 🧮 math.GR
keywords GGS-groupsderived seriesrooted treesgroup indicescommutator subgroupsbranch groupsprime-regular trees
0
0 comments X

The pith

GGS-groups with non-constant defining tuples over prime-regular trees have explicitly calculable indices |G:G^{(n)}| for every derived subgroup that depend only mildly on the tuple.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes formulas for the indices of all terms in the derived series of such a GGS-group and gives a structural description of each G^{(n)}. A sympathetic reader would care because these groups act on rooted trees and serve as concrete examples where the commutator process can be tracked level by level. The result shows that the index sequence is largely insensitive to the precise choice of the non-constant tuple. This supplies a uniform picture of how the derived series descends in an entire family of groups.

Core claim

Given a GGS-group G with non-constant defining tuple over a prime-regular rooted tree, the indices |G:G^{(n)}| can be calculated and the structure of the higher derived subgroups G^{(n)} described for every natural number n; the values |G:G^{(n)}| depend only mildly on the structure of the defining tuple.

What carries the argument

The derived series G^{(n)} of the GGS-group, whose indices are obtained from the action on the prime-regular tree and the non-constant defining tuple.

If this is right

  • The index |G:G^{(n)}| is determined for every n by a mild function of the tuple.
  • Each higher derived subgroup G^{(n)} admits an explicit structural description inside the group.
  • The same index pattern applies across many different non-constant tuples.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The mild dependence suggests that the derived series behaves similarly for most choices of non-constant tuples, which may simplify comparisons among different GGS-groups.
  • The explicit indices could be used to decide membership questions or to compute the abelianization of successive quotients in these groups.

Load-bearing premise

The defining tuple is required to be non-constant and the underlying tree must be prime-regular; without these the stated index formulas need not hold.

What would settle it

An explicit computation, for a concrete non-constant tuple on a prime-regular tree, of the index |G:G^{(2)}| that differs from the formula given in the paper.

Figures

Figures reproduced from arXiv: 2208.14975 by Jan Moritz Petschick.

Figure 1
Figure 1. Figure 1: Part of the top of the subgroup lattice of a GGS-group, with some supergroups added. Passage from the left to the right side signifies the application of ψ. All indices are logarithmic. we can easily deduce the index of the second derived subgroup. Recall that logp [G′ × p . . .×G′ : Stab(1)′ ] = sym(e) by Theorem 2.10, and has that Stab(1)′ has p-logarithmic index p + 1 in G by Lemma 2.13. Thus logp |G : … view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript claims that for a GGS-group G with non-constant defining tuple acting on a prime-regular rooted tree, the indices |G : G^{(n)}| can be calculated explicitly and the structure of the derived subgroups G^{(n)} described for every n, with these indices depending only mildly on the specific form of the defining tuple.

Significance. If the explicit calculations hold, the result supplies concrete information on the derived series of GGS-groups, a family of branch groups of ongoing interest in geometric group theory. The observation of mild dependence on the tuple offers a potentially simplifying feature for further structural questions about these groups.

minor comments (2)
  1. The introduction would benefit from a brief reminder of the precise definition of a GGS-group and the meaning of 'non-constant defining tuple' before the main statements.
  2. Notation for the prime-regular tree and the action could be standardized across sections to avoid minor inconsistencies in subscript usage.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report and recommendation to accept the manuscript. No major comments were raised, so we have nothing further to address.

Circularity Check

0 steps flagged

No significant circularity; direct calculation from definitions

full rationale

The paper states it calculates |G:G^{(n)}| and describes G^{(n)} directly from the definition of GGS-groups with non-constant defining tuples over prime-regular trees. No equations, predictions, or claims reduce by construction to fitted parameters, self-citations, or renamed inputs. The result is scoped to explicit hypotheses and presented as computation from the group action, with no load-bearing self-citation chains or ansatz smuggling indicated in the abstract or context.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on the standard definition of derived subgroups and the prior definition of GGS-groups; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • standard math The derived subgroup G' is the commutator subgroup [G,G] and higher derived subgroups are defined iteratively by G^{(n+1)} = (G^{(n)})'
    This is the standard inductive definition used to form the derived series in any group.
  • domain assumption GGS-groups are defined via a non-constant tuple of elements in the automorphism group of the tree
    The abstract invokes the standard construction of GGS-groups from the literature.

pith-pipeline@v0.9.0 · 5570 in / 1372 out tokens · 30935 ms · 2026-05-24T11:03:35.150530+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

  1. [1]

    2, 619–648

    Theofanis Alexoudas, Benjamin Klopsch, and Anitha Thil laisundaram, Maximal subgroups of multi-edge spinal groups , Groups, Geometry, and Dynamics 10 (2016), no. 2, 619–648. Cited on p. 1

    work page 2016

  2. [2]

    11, 4923–4964

    Laurent Bartholdi and Zoran ˇSuni´k, On the word and period growth of some groups of tree automorphisms, Communications in Algebra 29 (2001), no. 11, 4923–4964. Cited on p. 1

    work page 2001

  3. [3]

    Di Domenico, G

    E. Di Domenico, G. A. Fern´ andez-Alcober, and N. Gavioli , GGS-groups over primary trees: Branch structures, Monatshefte f¨ ur Mathematik (2022). Cited on p. 5

    work page 2022

  4. [4]

    01, 257–284

    Anna Erschler, Iterated identities and iterational depth of groups , Journal of Modern Dynamics 9 (2015), no. 01, 257–284. Cited on p. 5

    work page 2015

  5. [5]

    New Series 49 (1985), no

    Jacek Fabrykowski and Narain Gupta, On groups with sub-exponential growth functions , The Journal of the Indian Mathematical Society. New Series 49 (1985), no. 3-4, 249–256. Cited on p. 1

    work page 1985

  6. [6]

    4, 1993–2017

    Gustavo Fern´ andez-Alcober and Amaia Zugadi-Reizabal, GGS-groups: Order of congruence quo- tients and Hausdorff dimension , Transactions of the American Mathematical Society 366 (2013), no. 4, 1993–2017. Cited on p. 1, 2, 4, 7, 10, 14

    work page 2013

  7. [7]

    Gustavo A. Fern´ andez-Alcober, Alejandra Garrido, and Jone Uria-Albizuri, On the congru- ence subgroup property for GGS-groups , Proceedings of the American Mathematical Society 145 (2017), no. 8, 3311–3322. Cited on p. 1, 2

    work page 2017

  8. [8]

    3, 385–388

    Narain Gupta and Said Sidki, On the Burnside problem for periodic groups , Mathematische Zeitschrift 182 (1983), no. 3, 385–388. Cited on p. 1 20 J. M. PETSCHICK

    work page 1983

  9. [9]

    A. W. Ingleton, The Rank of Circulant Matrices , Journal of the London Mathematical Society 31 (1956), no. 4, 445–460. Cited on p. 7

    work page 1956

  10. [10]

    Kronecker, Vorlesungen ¨ uber Zahlentheorie, Vorlesungen ¨ uber allgemeine Arithmetik, vol

    L. Kronecker, Vorlesungen ¨ uber Zahlentheorie, Vorlesungen ¨ uber allgemeine Arithmetik, vol. 1, B. G. Teubner, Leipzig, 1901. Cited on p. 7

    work page 1901

  11. [11]

    3, 347–358

    Jan Moritz Petschick, On conjugacy of GGS-groups , Journal of Group Theory 22 (2019), no. 3, 347–358. Cited on p. 1, 9, 10

    work page 2019

  12. [12]

    Cited on p

    , Two periodicity conditions for spinal groups , December 2021. Cited on p. 1

    work page 2021

  13. [13]

    1, 213–236

    Zoran ˇSuni´k, Hausdorff dimension in a family of self-similar groups , Geometriae Dedicata 124 (2007), no. 1, 213–236. Cited on p. 10

    work page 2007

  14. [14]

    4, 1319–1333

    Ana Cristina Vieira, On the lower central series and the derived series of the Gupt a-Sidki 3-group, Communications in Algebra 26 (1998), no. 4, 1319–1333. Cited on p. 1

    work page 1998

  15. [15]

    Taras Vovkivsky, Infinite torsion groups arising as generalizations of the se cond Grigorchuk group, Algebra (Moscow), de Gruyter, Berlin, 2000, pp. 357–377. C ited on p. 1 Jan Moritz Petschick: Mathematisches Institut, Heinrich- Heine-Universit¨at, 40225 D¨usseldorf, Germany Email address : jan.petschick@hhu.de

    work page 2000