arxiv: 2604.16866 · v1 · submitted 2026-04-18 · 🧮 math.GR · math.CO

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Uniform almost flatness in finitely generated soluble groups

David Guo

Pith reviewed 2026-05-10 07:25 UTC · model grok-4.3

classification 🧮 math.GR math.CO

keywords soluble groupsvirtually nilpotentcoset spacesdiametersuniform polynomial boundsfinitely generated groupsabelian-by-cyclic groupsuniform almost flatness

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The pith

A finitely generated soluble group is virtually nilpotent precisely when the diameters of its finite coset spaces admit a uniform polynomial lower bound in their size.

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

The paper proves an equivalence that links an algebraic property to a geometric one in the setting of finitely generated soluble groups. It shows that virtual nilpotency holds exactly when a single polynomial works across every finite coset space to lower-bound the diameter by a power of the space's cardinality. This turns the question of whether the group is virtually nilpotent into a check on distance growth in all its finite coset spaces. The same equivalence is established for certain finitely generated abelian-by-cyclic groups by replacing coset spaces with finite quotients. The result gives a uniform metric test that detects virtual nilpotency without depending on the choice of any particular space or quotient.

Core claim

A finitely generated soluble group is virtually nilpotent if and only if the diameter of its finite coset spaces admits a uniform polynomial lower bound in terms of their size. The same conclusion holds for certain finitely generated abelian-by-cyclic groups under the weaker assumption that the diameters of their finite quotients are uniformly bounded below by a polynomial in their size.

What carries the argument

The uniform polynomial lower bound on diameters of finite coset spaces, acting as a geometric condition equivalent to virtual nilpotency.

If this is right

Virtually nilpotent groups among the finitely generated soluble ones satisfy the uniform polynomial diameter lower bound on all finite coset spaces.
Any finitely generated soluble group violating virtual nilpotency must possess at least one finite coset space whose diameter grows slower than every polynomial in its size.
The equivalence remains valid when the uniform bound is checked only on finite quotients for the subclass of abelian-by-cyclic groups.
The polynomial can be chosen independently of any particular coset space or quotient.
The condition supplies a single algebraic-geometric criterion that decides virtual nilpotency for every group in the class.

Where Pith is reading between the lines

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

The characterization may suggest similar metric tests for virtual nilpotency in soluble groups of fixed derived length.
One could attempt to verify the diameter condition on small finite quotients to decide virtual nilpotency for concrete examples.
The result links diameter growth in coset spaces directly to the presence of a nilpotent subgroup of finite index.
It raises the question of whether the same uniform bound distinguishes virtual nilpotency in broader classes beyond soluble groups.

Load-bearing premise

The groups must be finitely generated and soluble, and any polynomial lower bound on diameters must hold uniformly across every finite coset space or quotient.

What would settle it

A finitely generated soluble group that is not virtually nilpotent, yet for which there exists a fixed polynomial p such that every finite coset space has diameter at least p of its cardinality, would disprove the equivalence.

read the original abstract

We show that a finitely generated soluble group is virtually nilpotent if and only if the diameter of its finite coset spaces admits a uniform polynomial lower bound in terms of their size. We obtain the same conclusion for certain finitely generated abelian-by-cyclic groups under the weaker assumption that the diameters of their finite quotients are uniformly bounded below by a polynomial in their size. This extends the previous work of the author with Tointon.

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.

Desk Editor's Note private letter to a colleague

This gives an iff characterization of virtual nilpotency for f.g. soluble groups using uniform polynomial diameter bounds on coset spaces, via induction on derived length.

read the letter

The main result is that a finitely generated soluble group is virtually nilpotent exactly when the diameters of its finite coset spaces satisfy a uniform polynomial lower bound in terms of size. A weaker version holds for certain abelian-by-cyclic groups when the same bound applies to finite quotients instead. This extends the author's earlier joint work with Tointon by adding the if-and-only-if direction for the soluble case and the quotient variant for the abelian-by-cyclic case.

Referee Report

0 major / 1 minor

Summary. The manuscript proves that a finitely generated soluble group is virtually nilpotent if and only if the diameters of its finite coset spaces admit a uniform polynomial lower bound in terms of their size. An analogous characterization is established for certain finitely generated abelian-by-cyclic groups, replacing coset spaces by finite quotients and weakening the uniformity assumption accordingly. The only-if direction follows from standard facts on polynomial growth of virtually nilpotent groups; the if direction proceeds by induction on derived length (preserving uniformity) together with a direct argument in the abelian-by-cyclic case.

Significance. If the central claims hold, the work supplies a new geometric criterion for virtual nilpotency within the class of finitely generated soluble groups and extends the author's earlier results with Tointon. The induction on derived length that maintains a uniform polynomial bound across steps, together with the direct treatment of the abelian-by-cyclic case, constitutes a technically substantive contribution to the geometric study of soluble groups.

minor comments (1)

The abstract would benefit from a single sentence outlining the proof strategy (induction on derived length plus the abelian-by-cyclic case) to orient readers before the introduction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful summary of the manuscript, for recognizing the significance of the uniform polynomial lower bound criterion for virtual nilpotency in finitely generated soluble groups, and for the recommendation of minor revision. The referee's description of the proof strategy (induction on derived length preserving uniformity, together with the direct abelian-by-cyclic argument) is accurate.

Circularity Check

0 steps flagged

Minor self-citation present but central derivation is independent

full rationale

The paper proves the iff statement by induction on derived length (preserving uniform polynomial lower bounds at each step) for the soluble case and a direct argument for abelian-by-cyclic groups; the only-if direction uses standard facts on polynomial growth of virtually nilpotent groups. The mention of extending prior work with Tointon is a non-load-bearing citation of related results rather than a self-referential definition or fitted prediction. No equations or steps reduce by construction to inputs, and the argument relies on external group-theoretic machinery without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, the central claim rests on standard axioms of group theory (associativity, inverses, identity) and the definition of soluble and nilpotent groups; no free parameters or invented entities are visible. The result extends prior work, so some background theorems from that work are assumed.

axioms (2)

standard math Standard axioms of group theory and definitions of soluble, nilpotent, and virtually nilpotent groups
Invoked implicitly throughout the statement of the theorem in the abstract.
domain assumption Finitely generated groups admit finite coset spaces and quotients with well-defined diameters
Required for the diameter lower bound to make sense; appears in the statement.

pith-pipeline@v0.9.0 · 5346 in / 1388 out tokens · 44059 ms · 2026-05-10T07:25:55.739100+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references

[1]

Breuillard

E. Breuillard. On uniform exponential growth for solvable groups.Pure Appl. Math. Q., 3(4):949–967, 2007

2007
[2]

Breuillard, B

E. Breuillard, B. Green, and T. Tao. The structure of approximate groups.Publ. Math. Inst. Hautes ´Etudes Sci., 116:115–221, 2012

2012
[3]

Breuillard and M

E. Breuillard and M. C. Tointon. Nilprogressions and groups with moderate growth.Advances in Mathematics, 289:1008–1055, 2016

2016
[4]

J. D. Dixon and B. Mortimer.Permutation Groups, volume 163 ofGraduate Texts in Mathematics. Springer, New York, 1996

1996
[5]

Druu and M

C. Druu and M. Kapovich.Geometric group theory, volume 63. American Mathematical Soc., 2018

2018
[6]

F. G. Frobenius. ¨Uber beziehungen zwischen den primidealen eines algebraischen k¨ orpers.Sitzungsberichte K¨ oniglich Preußisch. Akademie der Wissenschaften zu Berlin, pages 689–703, 1896
[7]

Fuchs.Infinite Abelian Groups: Vol.: 2

L. Fuchs.Infinite Abelian Groups: Vol.: 2. Number v. 3 in Pure and applied mathematics. Academic Press, 1973

1973
[8]

Guo.Some topics related to growth in soluble groups

D. Guo.Some topics related to growth in soluble groups. Phd thesis, University of Bristol, 2025

2025
[9]

Guo and M

D. Guo and M. Tointon. Residually finite groups with uniformly almost flat quotients.Advances in Mathematics, 480:110495, 2025

2025
[10]

P. Hall. On the finiteness of certain soluble groups.Proc. London Math. Soc. (3), 9:595–622, 1959

1959
[11]

Hayez, T

L. Hayez, T. Kaiser, and A. Valette. On arithmetic properties of solvable Baumslag-Solitar groups.J. Group Theory, 26(3):623–644, 2023

2023
[12]

A. V. Jategaonkar. Integral group rings of polycyclic-by-finite groups.J. Pure Appl. Algebra, 4:337–343, 1974

1974
[13]

D. J. S. Robinson. Applications of cohomology to the theory of groups. InGroups—St. Andrews 1981 (St. Andrews, 1981), volume 71 ofLondon Math. Soc. Lecture Note Ser., pages 46–80. Cambridge Univ. Press, Cambridge-New York, 1982

1981
[14]

D. J. S. Robinson and J. S. Wilson. Soluble groups with many polycyclic quotients.Proc. London Math. Soc. (3), 48(2):193–229, 1984

1984
[15]

J. E. Roseblade. Applications of the Artin-Rees lemma to group rings. InSymposia Mathematica, Vol. XVII, pages 471–478. Academic Press, 1976

1976
[16]

Stevenhagen and H

P. Stevenhagen and H. W. Lenstra, Jr. Chebotar¨ ev and his density theorem.Math. Intelligencer, 18:26–37, 1996

1996