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Growth of Approximate Groups in Hyperbolic Groups

T0 review · 0 major / 2 minor · reviewed 2026-06-27 · grok-4.3

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Pith's one-line read In a hyperbolic group, any infinite approximate semigroup A with A² inside A times a finite set either generates a virtually cyclic subgroup or has positive exponential growth.

desk verdict The growth dichotomy for infinite approximate groups in f.g. hyperbolic groups, plus the linear-loss product bound with sharp free-group constants, looks like a clean new statement aligned with standard hyperbolic geometry. read the letter →

arxiv 2606.13632 v2 pith:76NZ3EA3 submitted 2026-06-11 math.GR math.CO

Michael Saks , Gal Yehuda This is my paper · ORCID
classification math.GRmath.CO
keywords approximategroupshyperbolicexponentialgrowthvirtuallycyclicproductfreesemigroups
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves a growth dichotomy for infinite approximate groups and semigroups inside finitely generated hyperbolic groups: if A is infinite and satisfies A² ⊆ A X for finite X, then either the subgroup generated by A is virtually cyclic or A itself expands exponentially fast in the word metric. A reader would care because this rules out any intermediate growth regime for such approximate structures, giving a clean classification that relies on the negative curvature geometry of the ambient group. The authors also supply a uniform product-growth lower bound |UV| ≥ c |U||V| / (n+k+1) for subsets of balls of radii n and k; this linear loss is shown to be optimal whenever the group contains an element of infinite order, and sharper explicit constants are obtained inside free groups.

What carries the argument

The approximate-semigroup containment A² ⊆ A X, which forces controlled expansion of products; hyperbolicity of G then converts this containment into either virtual cyclicity or exponential growth via thin-triangle geometry.

What would settle it

An explicit infinite set A inside some finitely generated hyperbolic group G such that A² ⊆ A X for finite X, ⟨A⟩ is not virtually cyclic, yet the growth rate of A in the word metric is subexponential.

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Extended reading notes

Core claim

If G is a finitely generated hyperbolic group and A ⊆ G is infinite with A² ⊆ A X for some finite X ⊆ G, then either ⟨A⟩ is virtually cyclic or A has positive exponential growth in the ambient word metric. The paper further shows that every hyperbolic group admits a constant c_{G,S} > 0 such that |UV| ≥ c_{G,S} |U||V| / (n+k+1) whenever U lies in the ball of radius n and V in the ball of radius k; the linear loss factor is optimal in order, and inside free groups the constant can be taken as 1/4 in general and (2/3 + 1/(3·4^{min(n,k)})) on spheres.

Load-bearing premise

The ambient group G must be hyperbolic, so that thin triangles or quasigeodesic properties can bound the geometry of products and separate the virtually-cyclic case from the exponential-growth case.

Editorial extensions

If this is right

  • Any infinite approximate semigroup with subexponential growth in a hyperbolic group must generate a virtually cyclic subgroup.
  • The product-growth lower bound guarantees that every approximate semigroup inside a hyperbolic group possesses a well-defined growth rate.
  • The linear loss factor (n+k+1) in the product inequality cannot be improved to a constant when the group contains an element of infinite order.
  • Inside free groups the product constants 1/4 and (2/3 + 1/(3·4^{min(n,k)})) are sharp for the respective settings.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same dichotomy may hold in broader classes of groups whose Cayley graphs satisfy thin-triangle or negative-curvature inequalities.
  • The product-growth criterion could be used to study growth rates of approximate semigroups inside quotients or subgroups of hyperbolic groups.
  • Computational checks in small hyperbolic groups such as surface groups or small triangle groups would give concrete evidence for the sharpness of the constants.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 2 minor

Summary. The manuscript proves a growth dichotomy for infinite approximate groups (and semigroups) inside finitely generated hyperbolic groups: if A is infinite and satisfies A² ⊆ AX for finite X, then either the subgroup generated by A is virtually cyclic or A has positive exponential growth rate with respect to the word metric on G. It also establishes a general product-growth lower bound |UV| ≥ c_{G,S} |U||V| / (n + k + 1) for subsets U of the ball of radius n and V of the ball of radius k, shows that the linear loss is optimal in order when G has torsion-free elements, obtains the explicit constant c = 1/4 in free groups, and proves a sharper sphere-to-sphere bound |UV| ≥ (2/3 + 1/(3·4^{min{n,k}})) |U||V| that is sharp for all n, k.

Significance. If the proofs hold, the dichotomy supplies a precise geometric alternative for approximate groups that leverages hyperbolicity in an essential way, while the product-growth criterion furnishes a concrete tool for establishing the existence of growth rates. The optimality statements and the explicit sharp constants in free groups are genuine strengths that make the results falsifiable and directly usable.

minor comments (2)
  1. [Abstract] The notation B_n and S_n for balls and spheres in the word metric is used without an explicit definition in the abstract; a sentence clarifying the generating set and the metric should be added at the first occurrence in the introduction.
  2. [Theorem 1.3] In the statement of the product-growth criterion, the dependence of c_{G,S} on the hyperbolicity constant δ and the generating set S is not quantified; a remark on whether the constant can be made effective would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation of the manuscript and for recommending acceptance. No major comments were raised in the report.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: theorems derived from definitions of hyperbolicity and approximate groups

full rationale

The paper states and proves a growth dichotomy theorem and product-growth inequalities directly from the standard definitions of hyperbolic groups (thin triangles, quasigeodesics) and approximate semigroups (A² ⊆ AX with X finite). No equations involve fitted parameters renamed as predictions, no self-definitional loops, and no load-bearing self-citations or ansatzes imported from prior work by the authors. The central claims are externally falsifiable via the geometry of hyperbolic groups and do not reduce to their inputs by construction.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

The paper is a pure-mathematics proof relying on the standard axioms of group theory and the definition of hyperbolic groups; no free parameters, invented entities, or ad-hoc axioms are visible in the abstract.

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Cite this review

Pith. "Pith review of Growth of Approximate Groups in Hyperbolic Groups." pith.science (2026). https://pith.science/paper/76NZ3EA3

@misc{pith2026260613632,
  author       = {Pith},
  title        = {Pith review of: Growth of Approximate Groups in Hyperbolic Groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/76NZ3EA3}},
  note         = {Machine review of arXiv:2606.13632}
}
abstract

We prove a growth dichotomy for infinite approximate groups, and more generally approximate semigroups, in hyperbolic groups. If \(G\) is a finitely generated hyperbolic group and \(A\subseteq G\) is infinite with \[ A^2\subseteq AX \] for some finite \(X\subseteq G\), then either \(\langle A\rangle\) is virtually cyclic, or \(A\) has positive exponential growth in the ambient word metric. We also introduce a product-growth criterion for the existence of growth rates of approximate semigroups. The criterion applies to hyperbolic groups: if \(G\) is hyperbolic with finite generating set \(S\), then there is a constant \(c_{G,S}>0\) such that \[ |UV| \geq c_{G,S}\,\frac{|U||V|}{n+k+1}, \qquad U\subseteq B_n,\; V\subseteq B_k. \] The linear loss is optimal in order whenever \(G\) contains an element of infinite order. In the free group with its standard generating set one may take \(c_{G,S}=1/4\). We also prove that, in a free group, if \(U\subseteq S_n\) and \(V\subseteq S_k\), then \[ |UV|\geq \left(\frac{2}{3}+\frac{1}{3\cdot 4^{\min\{n,k\}}}\right)|U||V|, \] and this constant is sharp for all \(n,k\).

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Works this paper leans on

22 extracted references · 1 canonical work pages

  1. [1]

    Bj¨ orklund and T

    M. Bj¨ orklund and T. Hartnick. Approximate lattices.Duke Math. J., 167(15):2903–2964, 2018

    2018

  2. [2]

    Breuillard, B

    E. Breuillard, B. Green, and T. Tao. The structure of approximate groups.Publications math´ ematiques de l’IH´ES, 116:115–221, 2012

    2012

  3. [3]

    Chatterji

    I. Chatterji. Introduction to the rapid decay property, 2017

    2017

  4. [4]

    Coornaert

    M. Coornaert. Mesures de patterson–sullivan sur le bord d’un espace hyperbolique au sens de gromov.Pacific J. Math., 159(2):241–270, 1993

    1993

  5. [5]

    Cordes, T

    M. Cordes, T. Hartnick, V. Toni´ c, and S. Machado.Foundations of geometric approximate group theory. Memoirs of the American Mathematical Society. American Mathematical Society,

  6. [6]

    De Bruijn and P

    N. De Bruijn and P. Erd¨ os. Some linear and some quadratic recursion formulas. i.Indag. Math.(NS), 54(5):374–382, 1951

    1951

  7. [7]

    Haagerup

    U. Haagerup. An example of a non nuclearC ∗-algebra, which has the metric approximation property.Invent. Math., 50(3):279–293, 1979

    1979

  8. [8]

    Hrushovski

    E. Hrushovski. Stable group theory and approximate subgroups.J. Amer. Math. Soc., 25(1):189–243, 2012

    2012

Show all 22 references
  1. [9]

    Jolissaint

    P. Jolissaint. Rapidly decreasing functions in reducedC ∗-algebras of groups.Trans. Amer. Math. Soc., 317(1):167–196, 1990

    1990

  2. [10]

    Kapovich and N

    I. Kapovich and N. Benakli. Boundaries of hyperbolic groups.arXiv preprint math/0202286, 2002

    2002 arXiv

  3. [11]

    Louvaris, D

    M. Louvaris, D. T. Wise, and G. Yehuda. Density of growth-rates of subgroups of a free group and the non-backtracking spectrum of the configuration model, 2024

    2024

  4. [12]

    Louvaris, D

    M. Louvaris, D. T. Wise, and G. Yehuda. Subgroups of a free group with every growth rate, 2025

    2025

  5. [13]

    S. Machado. Approximate lattices and meyer sets in nilpotent lie groups.Discrete Analysis, 2020:Paper No. 1, 18 pp., 2020

    2020

  6. [14]

    S. Machado. Good models, infinite approximate subgroups and approximate lattices, 2020

    2020

  7. [15]

    S. Machado. Infinite approximate subgroups of soluble lie groups.Math. Ann., 382(1–2):285– 301, 2022

    2022

  8. [16]

    S. Machado. Approximate lattices in higher-rank semi-simple groups.Geom. Funct. Anal., 33(4):1101–1140, 2023

    2023

  9. [17]

    S. Machado. The structure of approximate lattices in linear groups, 2023

    2023

  10. [18]

    B. Nica. On the degree of rapid decay.Proc. Amer. Math. Soc., 138(7):2341–2347, 2010

    2010

  11. [19]

    B. Nica. Norms of spherical averaging operators for some geometric group actions, 2024

    2024

  12. [20]

    M. Sapir. The rapid decay property and centroids in groups.Journal of Topology and Analysis, 7(03):513–541, 2015

    2015

  13. [21]

    Schesler

    E. Schesler. The relative exponential growth rate of subgroups of acylindrically hyperbolic groups.Journal of Group Theory, 25(2):293–326, 2022

    2022

  14. [22]

    V¨ ais¨ al¨ a

    J. V¨ ais¨ al¨ a. Gromov hyperbolic spaces.Expositiones Mathematicae, 23(3):187–231, 2005. GROWTH OF APPROXIMATE GROUPS IN HYPERBOLIC GROUPS 54 Dept of Math, Rutgers University, New Brunswick, USA Email address:saks@math.rutgers.edu Dept of Math, Yale, New Haven, USA Email add...

    2005

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