Volume Entropy Rigidity for Random Groups at Low Densities

Dongming Hua (Merrick)

REVIEW 2 major objections 4 minor 37 references

Volume Entropy Rigidity for Random Groups at Low Densities

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read For a torsion-free hyperbolic group with a translation-apparent presentation, the volume entropy has a unique normalized minimizing weight, and such presentations are generic for low-density random groups.

desk verdict A genuinely new rigidity theorem for random groups at low densities, with a small but fixable factor-of-2 error in Proposition 4.8. read the letter →

arxiv 2505.23364 v1 pith:6PS67ITG submitted 2025-05-29 math.GR math.MG

Dongming Hua (Merrick) This is my paper

classification math.GRmath.MG MSC 20F6720F6520P05

keywords volumeentropyweightedwordmetricshyperbolicgroupsrandomsmallcancellationrigiditylengthspectrumtranslation-apparentpresentation

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 that, for a large class of hyperbolic groups, assigning positive weights to the generators (normalized to sum to 1) produces exactly one weighted word metric that minimizes the volume entropy, the exponential growth rate of balls. The enabling object is a 'translation-apparent presentation': a small-cancellation presentation in which each generator appears evenly inside every relator at three separate scales. The paper shows that in such a presentation every power of a generator is a geodesic with respect to every weight, so the stable translation length of a generator equals its weight; distinct weights therefore yield non-comparable length spectra, and an external rigidity theorem converts this into strict convexity and uniqueness of the minimizer. It then proves that these presentations are generic in Gromov's density model of random groups at any density below an explicit threshold d_m, and that for such generic groups the unique minimizer is almost the uniform weight and the minimal entropy is almost the free group's entropy. The result matters because volume entropy is the natural normalizable invariant for comparing metrics on a group, and this is a setting where rigidity can be located precisely and the minimizer described explicitly.

What carries the argument

The load-bearing object is the λ-translation-apparent presentation: a symmetrized, cyclically reduced presentation satisfying the C'(λ) small-cancellation condition together with three even-distribution conditions — no relator contains a subword s^n with n≥λ|r|, no subword of length ⌈4λ|r|⌉ contains half of any generator's occurrences in the relator, and every subword of length ⌈λ|r|⌉ contains at least 1/(8m) of its letters equal to any given generator. The small-cancellation side, through the lemma of [14], shows that any word representing the identity must contain a long piece of a relator; the even-distribution side ensures that such a piece cannot have small weight, so any word avoiding long relator subwords (in particular any power of a generator) is the unique geodesic for every weight. This yields the identity ℓ_{d_w}(s)=w(s) for each generator, the direct link from weights to the length spectrum. The third even-distribution condition also feeds the weighted subword-avoidance generating functions of [28], which produce the free-group entropy bounds in Proposition 4.8 and hence Theorem C.

What would settle it

For m=2 at a density such as d=d_2/2, sample random presentations with relator length ℓ up to roughly $10^{4}$ and test whether the symmetrized presentation satisfies the three even-distribution conditions of Definition 4.1 with λ=1/16; the paper predicts an exponentially small failure probability in ℓ, so observing a failure rate that stays bounded away from zero as ℓ grows would refute the genericity claim of Theorem B.

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Extended reading notes

Core claim

The central claim is Theorem A (Section 4.5): if a torsion-free non-elementary hyperbolic group (G,S) admits a λ-translation-apparent presentation, then the volume entropy is strictly convex on the simplex of normalized weights on S, and there is a unique normalized weight minimizing it. The proof shows that in such a presentation every power of a generator s is a geodesic with respect to every weight w, so the stable translation length of s equals w(s); hence distinct weights give distinct length spectra, and the length-spectrum rigidity theorem cited as [6] upgrades this to non-rough-isometry of the two weighted metrics, which the convexity-up-to-rough-isometry result rules out for two minimizers. The genericity claim is Theorem B (Section 5.11): for each m≥2 and each density 0≤d<d_m, where d_m=(m-1)^2/(6144 $m^{2}$(2m-1)^2 ln(2m-1)), a generic random group on m letters admits a 1/16-translation-apparent presentation by symmetrizing the random relators. Finally, Theorem C (Section 5.13) states that for such generic groups, every normalized weight w satisfies h(F_m,w)-ε≤h(G,w)≤h(F_m,w), and the unique minimizer is within ε of the uniform weight, with the minimum entropy within ε of the free group's minimum.

Load-bearing premise

The proof assumes the length-spectrum rigidity theorem cited as [6], namely that two hyperbolic metrics on a non-elementary hyperbolic group with equal stable translation lengths for every element are necessarily roughly isometric, and the paper cites rather than reproves that external result.

Editorial extensions

If this is right

Any torsion-free hyperbolic group with a translation-apparent presentation has a unique entropy-minimizing normalized weight, and its volume entropy is strictly convex on the weight simplex.
For every m≥2 and every density d<d_m, generic random groups on m letters have unique minimizers, giving an explicit infinite family of groups with volume-entropy rigidity.
For these generic groups the minimizer is nearly uniform and the minimal entropy is nearly that of the free group, so the uniform weighting is essentially optimal relative to any perturbation.
Stability holds: whenever a marked group has a unique minimizing weight, any sequence of weights whose entropy converges to the minimum must converge to that weight.
The same uniqueness criterion applies to surface groups with standard generators, where the uniform weight is the unique minimizer even though the standard presentation is not translation-apparent.

Reading between the lines

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

Because the proof transfers along the existence of a translation-apparent presentation and a length-spectrum rigidity theorem, the same uniqueness conclusion should hold for any other class of groups (for instance relatively hyperbolic groups) in which both ingredients are available; the paper does not discuss this extension.
The explicit value of d_m is an artifact of the Chernoff-bound estimates and the arbitrary choice λ=1/16; the author notes that d_m could be improved, which suggests the true regime where uniqueness is generic may be substantially larger than the theorem states.
The near-uniformity of the minimizer is a quantitative statement about the natural generating set, indicating that for fixed high-length random groups the entropy landscape is very flat around the uniform weight; one testable consequence is that small random perturbations of the uniform weight should change the entropy by an amount of order 1/ℓ, matching the paper's error bounds.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: this paper deserves a referee, but not a desk reject. The new idea is translation-apparent presentations, and the proof that generic low-density random groups admit them is clever. Theorems A, B, C are meaningful advances. The main issue I found is a factor-of-2 mistake in the proof of Proposition 4.8. The third even-distribution condition yields |u|_w > N|u|/(16m), not N|u|/(8m) as written. This propagates to the bound on q(b), so the displayed derivation of p(b)<0 is wrong. The gap is repairable: with the corrected exponent, the paper's hypothesis 8mjl < (2m−1)^{l/(16m)−2} still implies the needed inequality for large l, so the theorem statements survive. But the proof needs editing before publication. That's the main thing. The rest of the paper is in good shape. The convexity argument in Prop 3.6 is clean, and the use of Cantrell-Tanaka's length-spectrum rigidity to get strict uniqueness from translation lengths is a natural and well-executed step. The surface-group application (Theorem 3.13) is a nice bonus. The probabilistic estimates in Section 5 are careful; the Markov-chain Chernoff bound is applied correctly, and the explicit density d_m is a welcome touch even if it is tiny. The reliance on Cantrell-Tanaka (Theorem 2.3) is heavy but legitimate: it's a published external theorem, and the paper cites it accurately. The density threshold d_m → 0 as m → ∞ is a limitation but the paper says so openly, and the few-relator d=0 case is still covered. Who is this for? Geometric group theorists working on entropy rigidity, random groups, and growth. The paper is readable and well structured. It deserves a serious referee, with the expectation that Prop 4.8 will be fixed. Recommendation: send it out, but ask the referee to check the constants in Prop 4.8 and Theorem C. I'd probably want to re-check the derivation of the corrected bound in the revision.

Referee Report

2 major / 4 minor

Summary. The paper studies the rigidity of the volume entropy for weighted word metrics on hyperbolic groups. It introduces the notion of a λ-translation-apparent presentation, which combines the C′(λ) small cancellation condition with three 'even distribution' conditions on relators. The main results are: Theorem A, asserting that a torsion-free hyperbolic group admitting such a presentation has a unique normalized weight minimizing the volume entropy; Theorem B, asserting that for each m ≥ 2 there is an explicit small density d_m such that a generic random group on m letters at density d < d_m admits a 1/16-translation-apparent presentation; and Theorem C, asserting that such random groups have entropy arbitrarily close to that of the free group for every weight, that the unique minimizer is arbitrarily close to the uniform weight, and that the minimum entropy is arbitrarily close to that of the free group. The proofs combine small cancellation theory, a counting method of Myers for weighted subword avoidance, a Chernoff bound for Markov chains, and the recent convexity and length-spectrum results of Cantrell–Tanaka. The paper also proves a stability result for minimizers when uniqueness holds.

Significance. The paper makes a valuable contribution to the study of volume entropy rigidity for weighted word metrics. The notion of a translation-apparent presentation is a new and potentially useful tool, and the explicit genericity threshold in Theorem B gives a concrete family of hyperbolic groups where the minimizer is unique and almost uniform. The proof of Theorem C relies on a detailed counting argument with Myers' generating functions, and the paper provides both explicit constants and a self-contained appendix. If the technical gaps identified below are repaired, the results will constitute a solid advance. The main external dependency, the Cantrell–Tanaka length-spectrum rigidity theorem, is clearly stated and used in a standard way. The paper is well organized and the exposition is generally clear, though several proof details need correction.

major comments (2)

[Section 4.2, Proposition 4.8] The lower bound on |u|_w uses an incorrect interpretation of the third even distribution condition. Definition 4.1(iii) states that for each pair a ∈ A, #_{a±}(u) > |u|/(8m). It does not imply #_s(u) > |u|/(8m) for each individual s ∈ S. Summing the pair contributions gives |u|_w = Σ_a #_{a±}(u) w(a) > (|u|/(8m)) Σ_{a∈A} w(a) = N|u|/(16m), not N|u|/(8m). Consequently, the estimate q(b) ≤ 2mj/b^{Nl/(8m)} should read q(b) ≤ 2mj/b^{Nl/(16m)}, and the subsequent derivation of p(b) < 0 is invalid. This error propagates to Proposition 4.9 and Theorem C. A repair is to replace the sufficient condition by 8mjl < M_0^{N(l/(32m)-2)} and to choose ℓ in Theorem 5.13 accordingly; however, the hypothesis as stated, 8mjl < (2m−1)^{l/(16m)-2}, does not imply the stronger corrected condition, so the statement of Proposition 4.8 must be amended.
[Section 4.1, Proposition 4.3] The proof applies the second even distribution condition to the word u3, whose length is only known to be less than ⌈4λ|r|⌉, whereas condition (ii) is formulated for subwords of length exactly ⌈4λ|r|⌉. The argument should first extend u3 to a subword of r of that length, which is possible because u3 is a suffix of a cyclic permutation of r, and then apply condition (ii). As written, this step is unjustified. Since Proposition 4.3 underpins the proof of Theorem A via Corollary 4.4, this detail needs to be fixed.

minor comments (4)

[Lemma 5.7] In the proof, the line 'the distribution of uj is the uniform distribution on reduced words of length ⌈ℓ/16⌉' should read '⌈ℓ/4⌉'.
[Lemma 5.10] The cited formula for the number of cyclically reduced words, (2m−1)^ℓ + m + (−1)^ℓ(m−1) ≥ (2m−1)^ℓ, appears to be incorrect: for m=2, ℓ=2 it would give 15, exceeding the total number of reduced words of length 2. Only the exponential rate matters for the subsequent argument, so the proof can be repaired by using the asymptotic count c_m(2m−1)^ℓ, but the displayed inequality should be corrected.
[Theorem 5.13] The bound j ≤ |R_ℓ^*| ≤ ℓ(2m−1)^{dℓ} omits the factor coming from symmetrization; each relator gives at most 2ℓ cyclic permutations and inverses, so the correct upper bound is |R_ℓ^*| ≤ 2ℓ(2m−1)^{dℓ}. The argument is asymptotic, so this does not change the main conclusion, but the displayed inequality should be fixed.
[Throughout] There are numerous typos and spacing errors in the text, such as 'W e', 'a', and 'A T' in the title, which should be corrected in a revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the derivation is self-contained and the cited external results are not self-citations.

full rationale

The paper's main chain is: define translation-apparent presentations, prove that such presentations force the generator translation lengths to equal their weights, invoke the external length-spectrum rigidity theorem of Cantrell-Tanaka to conclude uniqueness of the entropy minimizer, and then prove genericity of translation-apparent presentations for random groups via a Chernoff bound for Markov chains. None of these steps reduces to its own input. The condition 'translation-apparent' is not defined in terms of the minimizer or the entropy; it is an explicit small-cancellation plus letter-frequency hypothesis, and the equality of translation length with weight is proved in Proposition 4.3 and Corollary 4.4 rather than assumed. The constants lambda = 1/16 and d_m are chosen to make the probabilistic estimates work; they are not fitted to force the desired minimizer, and the minimizer is not prescribed by the construction. The most load-bearing external result, Theorem 2.3 from Cantrell-Tanaka, is a genuine external theorem, not a self-citation, so it does not constitute circularity under the stated rules. The skeptical concern about a factor-of-2 in Proposition 4.8 is a correctness or repairability issue in the written estimate, not a circularity: an erroneous intermediate bound does not make the conclusion equivalent to the hypotheses. Overall, the central claims carry independent content and no self-referential reduction is present.

Assumptions & free parameters 2 free parameters · 8 assumptions · 0 invented entities

The paper does not introduce new physical or mathematical entities; it introduces a new property of group presentations (translation-apparent), which is a condition, not an entity. The free parameters are hand-chosen proof constants, not fitted to data. The axioms are standard results from small cancellation theory, random group theory, and Markov chain analysis, plus external rigidity theorems cited from the literature.

free parameters (2)

lambda (small-cancellation parameter) = 1/16
Chosen by hand (Remark 5.14); defines translation-apparent presentations, the density threshold d_m, and all derived constants. The authors state this choice is arbitrary and unoptimized.
delta (Chernoff slack in Lemma 5.4) = 1/(4m)
Hand-picked slack in the Markov chain Chernoff bound; enters the constant C_m and therefore the density threshold d_m. Any sufficiently small delta would work, so this is a proof constant rather than a fitted physical parameter.

assumptions (8)

standard math Greendlinger's Lemma (Theorem 2.7)
Used in Proposition 4.3 to force a large common subword with a relator; requires C'(lambda), lambda <= 1/6, satisfied by lambda = 1/16.
domain assumption Cantrell-Tanaka convexity and length-spectrum rigidity (Theorems 2.2, 2.3)
Main external input: Manhattan curve strict convexity up to rough similarity, and rough isometry iff translation lengths agree. Used for existence of rough-unique minimizer and Theorem A.
standard math Gromov's lemma on high powers generating free subgroups (Lemma 3.1)
Used to prove existence of minimizers (Prop 3.3) by bounding entropy below by free group entropy.
standard math Balacheff-Merlin entropy formula for weighted free groups (Lemma 3.2)
Gives exact entropy for F_k with arbitrary weights; used to compute M0 and free-group comparisons.
standard math Myers' generating-function system for weighted subword avoidance (Theorem A.1)
Framework for counting lambda-reduced words; the paper provides a corrected proof in Appendix A, so it is self-contained.
standard math Lezaud's Chernoff bound for reversible Markov chains (Theorem 5.1)
Used with spectral gap estimate to show even distribution is generic; requires irreducibility and reversibility, satisfied by the free-group walk.
standard math Sinclair-Jerrum Cheeger bound for the spectral gap (Theorem 5.3)
Used to lower-bound the spectral gap epsilon(M) by (m-1)^2/(2(2m-1)^2), yielding the exponential rate C_m.
domain assumption Known genericity of hyperbolicity and C'(2d+epsilon) for random groups at density d<1/2
Used in Theorem 5.11 to get C'(1/16); since d < d_m < 1/32, can choose epsilon with 2d + epsilon < 1/16.

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Cite this review

Pith. "Pith review of Volume Entropy Rigidity for Random Groups at Low Densities." pith.science (2026). https://pith.science/paper/6PS67ITG

@misc{pith2026250523364,
  author       = {Pith},
  title        = {Pith review of: Volume Entropy Rigidity for Random Groups at Low Densities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6PS67ITG}},
  note         = {Machine review of arXiv:2505.23364}
}

read the original abstract

We study the rigidity of the volume entropy for weighted word metrics on hyperbolic groups, building on a recent convexity result due to Cantrell-Tanaka. Using ideas from small cancellation theory, we give conditions under which a hyperbolic group admits a unique normalized weight minimizing the entropy. Moreover, we show that these conditions are generic for random groups at small densities, and that the unique minimizer of such a generic group is arbitrarily close to the uniform weight.

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Ivan Babenko and St´ ephane Sabourau, Minimal volume entropy and ﬁber growth , Journal de l’´Ecole polytechnique — Math´ ematiques 12 (2025), 481–521 (en)

work page 2025

[2] [2]

06, 2429–2434

Florent Balacheﬀ and Louis Merlin, A curvature-free Log(2k − 1) theorem, Proceedings of the American Mathematical Society 151 (2023), no. 06, 2429–2434

work page 2023

[3] [3]

5, 731–799

G´ erard Besson, Gilles Courtois, and Sylvestre Gallot, Entropies et rigidit´ e des espaces lo- calement sym´ etriques de courbure strictement., Geometric and functional analysis 5 (1995), no. 5, 731–799

work page 1995

[4] [4]

2, 123–148

James W Cannon, The combinatorial structure of cocompact discrete hyperbo lic groups , Ge- ometriae Dedicata 16 (1984), no. 2, 123–148

work page 1984

[5] [5]

Stephen Cantrell and Eduardo Reyes, Marked length spectrum rigidity from rigidity on sub- sets, arXiv:2304.13209 [math.GT], 2024

work page Pith review arXiv 2024

[6] [6]

Stephen Cantrell and Ryokichi Tanaka, The manhattan curve, ergodic theory of topological ﬂows and rigidity , Geometry and Topology (2024), In Press

work page 2024

[7] [7]

4, 1615–1681

Chris Connell, Xianzhe Dai, Jes´ us N´ u˜ nez-Zimbr´ on, Raquel Perales, Pablo Su´ arez-Serrato, and Guofang W ei, Maximal volume entropy rigidity for spaces , Journal of the London Mathemat- ical Society 104 (2021), no. 4, 1615–1681

work page 2021

[8] [8]

Chris Connell, Xianzhe Dai, Jes´ us N´ u˜ nez-Zimbr´ on, Raquel Perales, Pablo Su´ arez-Serrato, and Guofang W ei, Volume entropy and rigidity for rcd-spaces , arXiv:2411.04327 [math.DG], 2024

arXiv 2024