Orbital counting for relatively Anosov groups

Andrew Zimmer; Feng Zhu; Richard Canary; Tengren Zhang

REVIEW 1 major objections 2 minor 35 references

Orbital counting for relatively Anosov groups

T0 review · 1 major / 2 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read For relatively Anosov subgroups of semisimple Lie groups, the number of orbit points of φ-length at most R is asymptotically a constant times e^{δR}, and the counting measures equidistribute.

desk verdict New orbital-counting result for relatively Anosov groups, built on Sambarino's strategy and the authors' own period-counting preprint; correct in outline, with a fixable gap in the diagonal-control step and a real dependency on [4]. read the letter →

arxiv 2607.25773 v1 pith:CYD3ZPKE submitted 2026-07-28 math.DS math.DGmath.MG

Richard Canary , Tengren Zhang , Feng Zhu , Andrew Zimmer This is my paper · ORCID

classification math.DSmath.DGmath.MG MSC 37D4037A1722E40

keywords relativelyAnosovgroupsorbitalcountingequidistributionPatterson-SullivanmeasuresBMSmeasurecriticalexponenthigher-rankLieCartanprojection

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 aims to establish orbital counting asymptotics for relatively Anosov subgroups of semisimple Lie groups, a class that generalizes convex cocompact/geometrically finite groups to higher rank. For any linear functional φ with finite critical exponent δ, it proves that the number of group elements whose φ-length is at most R grows like a constant times e^{δR}, and that the counting measures equidistribute to a product of Patterson–Sullivan measures. The significance is that this moves counting results beyond the Anosov (convex cocompact) setting to cusped, relatively hyperbolic-like higher-rank groups, matching the rank-one geometrically finite theory. The proof reduces the orbital statement to a previously established equidistribution result for periods (loxodromic conjugacy classes), following the strategy used in the Anosov case.

What carries the argument

The load-bearing tool is the φ-Bowen–Margulis–Sullivan measure m on the flow space U^φ_Γ, together with the imported period-equidistribution theorem: sums over loxodromic γ with φ(λ_θ(γ)) ≤ T of Dirac masses at (γ^−, γ^+) converge to (1/||m||) e^{δφ(G_θ)} ¯μ⊗μ. The paper's contribution is to bridge from this period statement to orbital equidistribution using two comparison lemmas: Lemma 3.2/3.3 show that, for elements with large Cartan projection, κ_θ(γ) − λ_θ(γ) is uniformly close to the Gromov product G_θ(U_θ(γ^{−1}), U_θ(γ)), up to exponentially negligible error; Proposition 3.6 controls the non-transverse diagonal using the no-atom property of Patterson–Sullivan measures.

What would settle it

Compute the orbital counting function for a concrete relatively Anosov group—for example, a cusped Hitchin representation of a geometrically finite Fuchsian group in SL(4,R) with a functional having finite critical exponent—and check whether #{γ: φ(κ(γ))≤R} e^{−δR} converges to 1/(δ||m||). A deviation from this constant at large R, or an infinite BMS mass, would settle the claim false.

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Extended reading notes

Core claim

The central claim is Theorem 1.1/1.2: if Γ is relatively P_θ-Anosov and φ has finite critical exponent δ, then #{γ: φ(κ_θ(γ)) ≤ R} ∼ C e^{δR}, with C = 1/(δ||m||), and the normalized orbital measures ν_R converge weakly to (1/||m||) ¯μ⊗μ. The proof works in the generality of transverse subgroups for which the BMS measure has finite mass and the group is of divergence type; relative Anosov is used only to guarantee divergence type and finiteness of the BMS measure. The key technical step is converting period equidistribution to orbital equidistribution by comparing Cartan and Jordan projections through the Gromov product, and separately showing the counting measures assign negligible mass to

Load-bearing premise

The entire proof depends on the imported period-equidistribution theorem and the finiteness of the BMS measure, both taken from a companion preprint; if either fails for some relatively Anosov group, the orbital counting and equidistribution claims collapse.

Editorial extensions

If this is right

The exponential growth rate of orbit points with φ-length ≤ R is exactly e^{δR} with an explicit constant C = 1/(δ||m||), not just the exponent.
The orbital measures equidistribute to the product of Patterson–Sullivan measures; in particular, the asymptotic distribution of directions from a basepoint is governed by ¯μ⊗μ.
The results extend from relatively Anosov to any transverse group for which (Γ,φ) is of divergence type and the BMS measure has finite total mass (Theorem 3.1).
Taking f≡1 recovers the orbital counting theorem from the equidistribution theorem, so orbit counting and orbit equidistribution are equivalent in this setting.
This generalizes the known Anosov counting theorem to the relatively Anosov setting, so cusped groups in higher rank satisfy the same sharp asymptotics as geometrically finite rank-one groups.

Reading between the lines

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

A natural testable extension would be to replace the ball {φ(κ)≤R} with weighted sums or anisotropic test functions; the equidistribution statement should carry over with the same BMS constant, though the paper does not address this.
The explicit constant C = 1/(δ||m||) ties the count to the BMS mass; computing ||m|| in explicit families (for example, cusped Borel Anosov representations) would yield concrete counting constants, which the paper leaves open.
The proof's treatment of the non-transverse diagonal suggests that counting with a functional that vanishes on some simple roots could behave non-exponentially or discontinuously; probing the boundary case of infinite critical exponent would delimit the theorem's scope.
Because the orbital result is deduced from a period-equidistribution theorem, any strengthening or relaxation of that theorem (e.g., removing the finiteness of the BMS measure) would immediately upgrade the present counting results.

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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

The paper proves orbital counting and orbital equidistribution for relatively Anosov groups — Theorems 1.1 and 1.2 — with the expected exponential rate δ and the constant 1/(δ∥m∥). The result is new and worth having: Sambarino treated Anosov groups, Kim–Oh–Pan's announced work is a different Borel-Anosov setting, and this closes the natural relatively-Anosov analogue. The strategy, honestly stated, is Sambarino's, adapted through the authors' own period-counting theorem from [4]. No new mechanism, but a legitimate extension within an established program.

What it does well: Section 3 is a coherent, readable proof. Lemmas 3.2 and 3.3 control the Cartan–Jordan comparison via the Tits representations and the Gromov product; Proposition 3.5 cleanly upgrades period equidistribution to orbital equidistribution on the transverse locus. The paper is explicit about what it imports: Theorem 2.8 is quoted from [4] and not re-derived.

Two soft spots, in proportion. First, the dependency on [4] is real and load-bearing: the central quantitative input is an overlapping-author preprint (three of four authors in common), and finiteness of the BMS measure is imported from the same circle. That is self-citation, not circularity — the strategy is standard and there are no fitted constants — but a referee needs [4] checked, or its proof summarized, before the main theorem is certified.

Second, the stress-test note holds up: Proposition 3.6 has a genuine gap. The final chain asserts ∑_n μ(V_n) ≤ (d+1)∥m∥, but the multiplicity argument in the text gives ∑_n μ(V_n) ≤ d+1, so the step requires ∥m∥ ≥ 1, which is nowhere proved and can fail — rank-one convex-cocompact groups with thin convex cores have arbitrarily small BMS mass. The condition μ(U) ≤ ∥m∥ included in the choice of r_1 does not rescue the step. The fix is simple: strengthen the μ̄ bound by an extra factor of ∥m∥. The theorem is probably right, but the diagonal control is incomplete as written, and this gap is independent of the [4] dependency.

Who this is for: higher-rank geometric group theorists and dynamical systems people working on Anosov and relatively Anosov groups. It deserves serious refereeing, with the [4] dependency addressed and Proposition 3.6 repaired. I would treat the main theorem as settled only after both.

Referee Report

1 major / 2 minor

Summary. The paper establishes orbital counting and orbital equidistribution results for relatively P_theta-Anosov subgroups of semisimple Lie groups. Theorem 1.1 states that for any linear functional phi on the theta-Cartan subspace with finite critical exponent delta, the number of group elements with phi-distance at most R grows like C e^{delta R}, and Theorem 1.2 gives convergence of the corresponding empirical measures on F_theta^2 to a multiple of the Patterson-Sullivan product measure. The proof follows Sambarino's strategy: Lemmas 3.2 and 3.3 compare the Cartan and Jordan projections using the Gromov product, Proposition 3.5 deduces equidistribution on the transverse locus from the period-equidistribution result Theorem 2.8 imported from the authors' preprint [4], Proposition 3.6 controls the non-transverse locus, and Section 3.4 assembles these ingredients. The abstract and introduction explicitly acknowledge that the main theorems are a reduction to the period-equidistribution results of [4].

Significance. If the main theorems hold, they provide the first orbital counting asymptotics for relatively Anosov groups with general linear functionals, generalizing Sambarino's results for Anosov groups. The paper is clearly structured and transparent about its dependencies: the central quantitative input is Theorem 2.8 from [4], a preprint sharing three of the four authors, and the finiteness of the BMS measure is likewise imported from [15]/[4]. Within that conditional framework, the proof is coherent and the adaptation of the Gromov-product estimates in Lemmas 3.2–3.3 is a genuine contribution. The argument has no free parameters and is not circular in the sense of using the desired conclusion as an input. However, the proof contains a local but real gap in Proposition 3.6, described below, and the overall validity of the main theorems is contingent on the correctness and publication status of [4].

major comments (1)

[Section 3.3, proof of Proposition 3.6] The final displayed chain in the proof of Proposition 3.6 is not justified as written. The text obtains sum_n ∫ f_n d(μ̄⊗μ) ≤ 2 sum_n μ̄(V_n) μ(η_{i_n}V_n) ≤ (ε/(C1(d+1))) sum_n μ(V_n), and then asserts this is ≤ (ε/C1)∥m∥. That last assertion requires sum_n μ(V_n) ≤ (d+1)∥m∥. The available justification is the multiplicity bound sum_n μ(V_n) ≤ (d+1) μ(Λ) = d+1, together with the earlier individual bound μ(U)≤∥m∥ for sets of diameter at most r_1. Individual bounds do not control the sum, and the multiplicity bound gives d+1, not (d+1)∥m∥. Since ∥m∥ is not shown to be at least 1 and can indeed be less than 1 in rank-one geometrically finite examples, the displayed inequality is a genuine gap. The fix is local: choose r_1 so that μ̄(U) ≤ ε∥m∥/(2 C1 C3(d+1)) instead of ε/(2 C1 C3(d+1)); then the chain yields the desired bound using sum_n μ(V_n) ≤ d+1, without needing ∥m∥≥1. As written, the

minor comments (2)

[Section 3.3, proof of Proposition 3.6] The text refers to 'Lemma 3.5 and 3.7' when Proposition 3.5 and Lemma 3.7 are meant. Please correct the cross-reference.
[Section 3.3, proof of Proposition 3.6] The sentence 'By Lemma 3.8, ... there exists i_n such that V_n ∩ η_{i_n} V_n is empty. In particular, V_n × η_{i_n} V_n is compact in F^{(2)}_θ' reverses the logical order. Lemma 3.8 gives directly that V_n × η_{i_n} V_n is contained in the transverse locus; the emptiness of the intersection is a consequence, not the premise. Consider rephrasing for clarity.

Circularity Check

1 steps flagged · score 4.0 of 10

The orbital-counting proof is an honest period-to-orbit reduction, but its decisive quantitative input is imported as a load-bearing self-citation from the authors' own preprint [4].

self citation load bearing [Introduction (p. 2) and Section 3.2, Theorem 2.8 (p. 10); used in Proposition 3.5]
"The crucial tool in the proof of Theorem 1.2 is an earlier equidistribution result which was developed for counting ϕ-lengths in [4]. ... Proof. By [4, Prop. 10.4 and Thm. 1.3], the flow ψ_t is mixing with respect to m, and hence the theorem follows from [4, Thm. 6.1]."

Theorem 1.2 is obtained by applying Proposition 3.5 and Proposition 3.6. Proposition 3.5's entire quantitative limit is reduced to the period-equidistribution statement of Theorem 2.8; the proof of Theorem 2.8 in this text is only a citation to [4]. Reference [4] is the same-author preprint by Blayac, Canary, Zhu, and Zimmer, so the decisive input (the mixing/equidistribution theorem and the normalization by ∥m∥) is imported from the authors' own earlier work rather than re-derived here. This is not a definitional tautology: orbital counting is not one of Theorem 2.8's assumptions, and the conversion via Lemmas 3.2/3.3 is substantive. But the central quantitative step is load-bearing self-citation, so the main theorem inherits its strength from [4].

full rationale

There is no fitted-input circularity: no parameter is tuned to orbital data and the asymptotic constant 1/(δ∥m∥) is not a fitted value. There is also no renaming or ansatz-smuggling: the Gromov-product estimates and the period-to-orbit comparison are explicitly derived in this paper. The claimed derivation chain is: Theorem 2.8 (period equidistribution, imported from [4]) -> Proposition 3.5 (compact-support orbital equidistribution) -> Proposition 3.6 (control of the diagonal) -> Theorem 1.2 -> Theorem 1.1 via f≡1. The only circularity-adjacent finding is that the first and most quantitative link, Theorem 2.8, is a same-author citation. The reviewer's separate concern about Proposition 3.6 (the bound ∑ μ(V_n) ≤ (d+1)∥m∥ is not justified; only ∑ μ(V_n) ≤ d+1 follows from μ(Λ)=1) is a correctness gap, not a circularity, because it concerns a missing inequality rather than an identification of the conclusion with an input. Since the central claim still has substantial independent content beyond the cited period result, a moderate score of 4 is appropriate.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters and no new entities are introduced. The cost is transferred to five external theorems, several of which are preprints by the present authors; no machine-checked or code-level verification is provided.

assumptions (5)

standard math KAK, Jordan and Iwasawa decompositions and the restricted-root structure of semisimple Lie groups
Used throughout Section 2 to define Cartan projection, Jordan projection, and Iwasawa cocycle.
standard math Existence and properties of Tits representations (Proposition 2.3)
Cited from [11, Sec. 3], [24, Lem. 6.4], and [7, Lem. B.5]; needed in the proof of Lemma 3.2.
domain assumption Theorem 2.8: equidistribution of periods for P_θ-transverse, divergence-type groups with finite BMS mass
Imported from [4], a preprint by Blayac, Canary, Zhu, and Zimmer, where three of four authors overlap with the present paper. It is the main quantitative engine of Section 3.
domain assumption Relatively Anosov groups are of divergence type, and their Patterson–Sullivan measures have no atoms
From [8, Thm. 1.1] and [7, Thm. 1.4]; used to guarantee the BMS measure exists and to control the diagonal in Proposition 3.6.
domain assumption Finiteness of the BMS measure for relatively Anosov groups
From [15, Thm. 1.1] or [4, Thm. 1.4]; needed to normalize the limit measure in Theorems 1.1 and 1.2.

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Pith. "Pith review of Orbital counting for relatively Anosov groups." pith.science (2026). https://pith.science/paper/CYD3ZPKE

@misc{pith2026260725773,
  author       = {Pith},
  title        = {Pith review of: Orbital counting for relatively Anosov groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CYD3ZPKE}},
  note         = {Machine review of arXiv:2607.25773}
}

read the original abstract

We obtain orbital counting results for relatively Anosov groups with respect to linear functionals with finite critical exponent. Our counting results follow from an equidistribution result and rely crucially on previous equidistribution results obtained in our proof of counting results for periods. Our results generalize earlier work of Sambarino in the setting of Anosov groups.

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [4]

Counting, mixing and equidistribution for GPS systems with applications to relatively Anosov groups,

P.-L. Blayac, R. Canary, F. Zhu, and A. Zimmer, “Counting, mixing and equidistribution for GPS systems with applications to relatively Anosov groups,” preprint, arXiv:2404.09718

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2025

[3] [1]

Patterson-Sullivan theory in higher rank symmetric spaces,

P. Albuquerque, “Patterson-Sullivan theory in higher rank symmetric spaces,”G.A.F.A.9(1999), 1–28

1999

[4] [2]

Propri´ et´ es asymptotiques des groupes lin´ eaires,

Y. Benoist, “Propri´ et´ es asymptotiques des groupes lin´ eaires,”G.A.F.A.7(1997), 1–47

1997

[5] [3]

Patterson–Sullivan theory for coarse cocycles,

P.-L. Blayac, R. Canary, F. Zhu, and A. Zimmer, “Patterson–Sullivan theory for coarse cocycles,” preprint, arXiv:2404.09713

arXiv

[6] [5]

Symbolic dynamics for hyperbolic flows,

R. Bowen, “Symbolic dynamics for hyperbolic flows,”Amer. J. Math.95(1973), 429–459

1973

[7] [6]

Entropy rigidity for cusped Hitchin representations

R. Canary, T. Zhang and A. Zimmer, “Entropy rigidity for cusped Hitchin representations”.J. Topol.19(2026), e70064

2026

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Patterson–Sullivan measures for transverse groups,

R. Canary, T. Zhang and A. Zimmer, “Patterson–Sullivan measures for transverse groups,”J. Mod. Dyn.20(2024), 319–377. ORBITAL COUNTING FOR RELATIVELY ANOSOV GROUPS 19

2024