Uniform growth in small cancellation groups
Pith reviewed 2026-05-24 01:20 UTC · model grok-4.3
The pith
The class of groups with uniform exponential growth acting acylindrically on hyperbolic spaces is closed under geometric small cancellation quotients.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The class of groups of uniform exponential growth acting acylindrically on a hyperbolic space is closed under taking certain geometric small cancellation quotients. This yields a finitely generated acylindrically hyperbolic group that has uniform exponential growth but has arbitrarily large torsion balls. The uniform exponential growth rate of a classical C''(λ)-small cancellation group, for sufficiently small λ, is bounded from below by a universal positive constant. A similar result holds for uniform entropy-cardinality estimates, which yields an explicit upper bound on the isomorphism class of marked δ-hyperbolic C''(λ)-small cancellation groups of uniformly bounded entropy in terms of δ.
What carries the argument
Geometric small cancellation quotients that preserve the acylindrical action on a hyperbolic space together with the uniform exponential growth property.
If this is right
- There exists a finitely generated acylindrically hyperbolic group with uniform exponential growth whose balls contain torsion elements of arbitrarily large order.
- For sufficiently small λ the uniform exponential growth rate of any classical C''(λ)-small cancellation group is bounded below by a fixed positive constant independent of the group.
- Uniform entropy-cardinality estimates admit a similar universal lower bound for the same class of small cancellation groups.
- The set of isomorphism classes of marked δ-hyperbolic C''(λ)-small cancellation groups with uniformly bounded entropy is finite and explicitly bounded in size by a function of δ and the entropy bound.
Where Pith is reading between the lines
- The closure property implies that any counterexample to uniform exponential growth among acylindrically hyperbolic groups cannot arise from geometric small cancellation quotients of groups that already have the growth property.
- The same preservation mechanism might apply to other natural classes of quotients or to actions on other spaces if analogous control on acylindricity can be established.
- Explicit presentations of small cancellation groups could be checked computationally to obtain concrete numerical lower bounds that match or exceed the universal constant proved to exist.
Load-bearing premise
The quotients must be geometric small cancellation quotients of groups that already act acylindrically on a hyperbolic space and have uniform exponential growth, with those two properties preserved by the quotient construction.
What would settle it
A concrete geometric small cancellation quotient of an acylindrically hyperbolic group with uniform exponential growth in which the quotient group fails to have uniform exponential growth.
read the original abstract
An open question asks whether every group acting acylindrically on a hyperbolic space has uniform exponential growth. We prove that the class of groups of uniform uniform exponential growth acting acylindrically on a hyperbolic space is closed under taking certain geometric small cancellation quotients. There are two consequences: firstly, there is a finitely generated acylindrically hyperbolic group that has uniform exponential growth but has arbitrarily large torsion balls. Secondly, the uniform uniform exponential growth rate of a classical $C''(\lambda)$-small cancellation group, for sufficiently small $\lambda$, is bounded from below by a universal positive constant. We give a similar result for uniform entropy-cardinality estimates. This yields an explicit upper bound on the isomorphism class of marked $\delta$-hyperbolic $C''(\lambda)$-small cancellation groups of uniformly bounded entropy in terms of $\delta$ and the entropy bound.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript proves that the class of groups with uniform exponential growth acting acylindrically on a hyperbolic space is closed under certain geometric small cancellation quotients. Two consequences are derived: the existence of a finitely generated acylindrically hyperbolic group with uniform exponential growth but with torsion balls of arbitrarily large radius, and a universal positive lower bound on the uniform exponential growth rate of classical C''(λ)-small cancellation groups for sufficiently small λ. An analogous result for uniform entropy-cardinality estimates is given, which yields an explicit upper bound (in terms of δ and the entropy bound) on the number of isomorphism classes of marked δ-hyperbolic C''(λ)-small cancellation groups of uniformly bounded entropy.
Significance. If the central closure result holds, the work supplies new examples separating uniform exponential growth from bounded torsion in the acylindrically hyperbolic setting and supplies quantitative lower bounds and finiteness statements for small-cancellation groups. These are concrete contributions to the study of uniform growth and acylindrical hyperbolicity that build directly on existing techniques in the area.
minor comments (2)
- [Abstract] Abstract, line 2: the repeated phrase 'uniform uniform exponential growth' is evidently a typographical error and should read 'uniform exponential growth'.
- [Introduction] The abstract states two consequences and a third result on entropy-cardinality estimates, but the manuscript would benefit from a short roadmap paragraph at the end of the introduction that explicitly indicates which sections contain the proof of the main closure theorem and which contain the applications.
Simulated Author's Rebuttal
We thank the referee for their positive assessment of the manuscript, the clear summary of its contributions, and the recommendation for minor revision. No major comments appear in the report, so we have no specific points requiring rebuttal or clarification at this stage.
Circularity Check
No significant circularity identified
full rationale
The derivation establishes that a class of groups with uniform exponential growth and acylindrical actions on hyperbolic spaces is closed under geometric small cancellation quotients. This closure is presented as a theorem proved from standard properties of acylindrical hyperbolicity and small cancellation quotients, without reducing to self-definitions, fitted parameters renamed as predictions, or load-bearing self-citations. The two consequences (existence of a group with uniform growth but large torsion balls, and a universal lower bound on growth rate for C''(λ) groups) follow directly from the closure property applied to known base cases like free groups. No equation or step in the abstract or described structure equates a result to its own inputs by construction. The argument is self-contained against external benchmarks in geometric group theory.
Axiom & Free-Parameter Ledger
Reference graph
Works this paper leans on
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work page internal anchor Pith review Pith/arXiv arXiv doi:10.1090/memo/1156 2000
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