Regularity of Manhattan manifolds and exact dimensionality for relatively Anosov groups

Eduardo Reyes; Tianqi Wang

Regularity of Manhattan manifolds and exact dimensionality for relatively Anosov groups

T0 review · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read For relatively Anosov groups, Patterson–Sullivan measures are exact dimensional and Manhattan manifolds are C^1.

desk verdict A serious, useful paper with a genuine gap in the proof of Theorem A and a murky novelty claim around Corollary C; both are repairable, and it deserves proper refereeing. read the letter →

arxiv 2607.13730 v1 pith:QLZNQY42 submitted 2026-07-15 math.GR math.DSmath.GT

Eduardo Reyes , Tianqi Wang This is my paper · ORCID

classification math.GRmath.DSmath.GT MSC 20F6722E4037A2537D40

keywords relativelyAnosovgroupsPatterson–SullivanmeasuresexactdimensionalityManhattanmanifoldsgrowthindicatorBowen–Margulis–SullivanGromovhyperbolicmetricssubadditiveergodictheorem

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 Patterson–Sullivan measures associated to relatively Anosov groups have a well-defined local scaling exponent: for almost every boundary point, the measure of a radius-r ball decays like r^h, and h equals the Hausdorff dimension of the measure. It also proves that the Manhattan manifold parametrization — the critical exponent of weighted group series — is C^1, yielding a C^1 and strictly concave growth indicator. Both results are obtained by dynamical methods: a first-return system on the flow space of transverse flag pairs, the shadow lemma, and the subadditive ergodic theorem. A further result establishes that for relatively Morse groups, the scalar Cartan metric is Gromov hyperbolic, so the exact-dimensionality theorem applies to the associated visual premetric.

What carries the argument

The central object is the Bowen–Margulis–Sullivan measure m^ψ on the flow space of transverse flag pairs × R, constructed from the Patterson–Sullivan measures and the Gromov product. Its finiteness and strong mixing (inherited from prior work) make it possible to pass to a first-return dynamical system on a compact section, giving an ergodic system to which the subadditive ergodic theorem and Kac's return-time lemma apply. Around this core sit the shadow lemma, which controls the Patterson–Sullivan measure of shadows by exponentials of the linear form on the Cartan projection, and the reparametrization cocycle that relates the flow time to distances in the Gromov model and to the Cartan proj

What would settle it

Look for a relatively Anosov group and a proper positive linear form of critical exponent 1 for which the Patterson–Sullivan measure of a ball of radius r around a typical boundary point fails to scale with a single power of r, or find two non-parallel vectors in the interior of the limit cone where the growth indicator is affine, contradicting strict concavity.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is a two-part regularity statement. First, if Γ is relatively θ-Anosov and ψ is a proper positive linear form with critical exponent 1, then the unique Patterson–Sullivan probability measure ν^ψ is exact dimensional with respect to any visual quasi-metric coming from a Gromov model in the cusped-space quasi-isometry class: the limit h = lim_{r→0} log ν^ψ(B(ξ,r)) / log r exists for ν^ψ-almost every ξ and equals the Hausdorff dimension of ν^ψ. Second, the Manhattan manifold parametrization Θ for any finite tuple of proper linear forms is continuously differentiable, and consequently the θ-growth indicator is C^1 and strictly concave on the interi

Load-bearing premise

The arguments inherit, without proof, the finiteness and strong mixing of the Bowen–Margulis–Sullivan measure for relatively Anosov groups; if that measure ever failed to be finite or mixing, the first-return ergodic system and both main theorems would collapse.

Editorial extensions

If this is right

For any relatively Anosov group and any proper positive linear form with critical exponent 1, the Patterson–Sullivan measure has a definite Hausdorff dimension with respect to any visual quasi-metric from the cusped Gromov model.
Exact dimensionality is obtained without Ahlfors regularity, so it covers non-symmetric linear forms where the measure is not comparable to a Hausdorff measure.
The Manhattan curves for pairs of relatively Anosov groups are C^1, without thermodynamic formalism, and the growth indicator is C^1 and strictly concave on the interior of the limit cone.
For relatively Morse groups, the scalar Cartan metric is Gromov hyperbolic, making the Dey–Kim–Oh premetric a visual quasi-metric, so the exact-dimensionality theorem applies to it.

Reading between the lines

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

The axiomatic framework suggests the same two conclusions should hold for any group action with a shadow lemma and finite mixing Bowen–Margulis–Sullivan measures, such as transverse groups or Gromov–Patterson–Sullivan systems.
If the drift ratios were Hölder continuous in the parameter, the same argument might upgrade the C^1 conclusion to C^{1,α}; the paper does not claim this.
Strict concavity of the growth indicator, being projective, could yield a uniqueness/finiteness result for supporting hyperplanes at interior points of the limit cone, with rigidity consequences for the group's Cartan image.
A direct testable extension: compute the exact-dimensional exponent h for a concrete cusped Fuchsian or Hitchin representation and verify it coincides with the drift ratio predicted by Theorem A.

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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

Here's my read. The paper has two genuinely new results worth knowing: Theorem A, exact dimensionality of Patterson–Sullivan measures for relatively Anosov groups in visual metrics from Gromov models, and Theorem D, hyperbolicity of the scalar Cartan metric for relatively Morse groups. Theorem B, C^1 regularity of Manhattan manifolds, is new to this paper but independently proved by Kim–Oh–Zimmer, and the authors disclose that in Remark 1.1. The dynamical strategy — first-return maps, cocycles, Kingman, Kac — is coherent, and the axiomatic framework in Theorem 4.1 is clean. The shadow comparisons and drift-continuity arguments look serious. I found no fitted parameters or invented entities.

The proof of Theorem A has a real gap. In verifying condition (2) of Theorem 4.1, the authors choose an arbitrary compact section K of the BMS flow and assert that π(F_K) has full ν^ψ measure by Poincaré recurrence plus local product structure. That implication is false in general: in a suspension over an ergodic base, K = A×[0,ε] with A of positive but not full measure gives F_K conull in K but π(F_K)=A. Nothing in the proof forces K to project onto a full-measure subset of Λ_θ. As written, Theorem 4.1 yields exact dimensionality only on π(F_K), not on all of Λ_θ. This is load-bearing for Theorem A, but it is also repairable: one can probably choose a compact section with full projection (for instance Λ_θ×{η_0}×[0,ε] with η_0 not fixed by any parabolic), or add a Γ-ergodicity/quasi-conformal propagation argument. Either repair should be in the revision.

Second, the novelty accounting for Corollary C is murky. Strict convexity of the dual boundary is imported from [CZZ25, Cor. 1.10], and that corollary may already contain the strict concavity of the growth indicator. The authors claim strict concavity is new while also noting C^1 regularity is immediate from the same source. They need to state precisely what [CZZ25, Cor 1.10] does and does not imply.

Third, the dependence on Theorem 2.13 — finiteness and mixing of BMS measures — is inherited from Kim–Oh and BCZZ and is load-bearing. That is a legitimate dependency, but worth flagging because the whole proof structure collapses without it.

Who this is for: people working in higher-rank Teichmüller theory, Anosov dynamics, and Patterson–Sullivan theory. The results organize previously scattered cases and open counting/pressure applications. It deserves a serious referee; the right outcome is a revision, not a desk rejection. I would not lean on Theorem A in its present form until condition (2) is fixed.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper introduces no new entities or fitted parameters. It rests on a stack of recent external results: KO25, BCZZ24a, CZZ25, and KOW25. KOW25 includes the present author Wang and CRS24 includes Reyes, so parts of the setup are semi-self-citations, but they are prior theorems independent of the target claims. The original input is the first-return/cocycle framework in Section 3 and its application to exact dimensionality and Manhattan regularity.

assumptions (7)

standard math Kingman's subadditive ergodic theorem, Kac's lemma, and Poincaré recurrence apply to the first-return systems.
Used in Sections 3, 4, and 6 to convert return-time and cocycle asymptotics into the limits λ_α, λ_β, and τ^j_v.
standard math Legendre duality between the dual growth domain and the growth indicator (Quint [Qui03], Sambarino [Sam14]).
Used in Section 6.4 to pass from strict convexity and C^1-regularity of BD^θ_Γ to strict concavity and C^1-regularity of ψ^θ_Γ.
domain assumption Γ is relatively θ-Anosov and ψ is proper, positive, tangent with δ_ψ(Γ)=1.
This is the standing hypothesis in Theorems A and B; it guarantees existence/uniqueness of PS measures by CZZ25 and the flow-space construction.
domain assumption The Bowen–Margulis–Sullivan measure m^ψ is finite and strongly mixing (Theorem 2.13, quoted from Kim–Oh [KO25] and Blayac–Canary–Zhu–Zimmer [BCZZ24a]).
Load-bearing: without it the compact section K with finite measure and the ergodicity of the first-return system fail.
domain assumption The shadow lemma for relatively Anosov groups gives two-sided exponential bounds on ν^ψ of shadows (Lemmas 2.9 and 2.10, quoted from KO25).
This is the bridge between group-element weights α(γ) and measure of shadow sets in Theorem 4.1.
domain assumption For Theorem D, Γ is relatively θ-Morse and φ is symmetric and proper.
The Morse condition is used to prove coarse additivity of the scalar Cartan metric near geodesics; without it Theorem D is not claimed.
domain assumption [CZZ25, Cor. 1.10] gives δ_{(1-t)ψ0+tψ1}(Γ)<1 for distinct tangent forms ψ0, ψ1.
Used in Lemma 6.16 to prove strict convexity of the boundary set BD^θ_Γ, the key step for strict concavity.

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Pith. "Pith review of Regularity of Manhattan manifolds and exact dimensionality for relatively Anosov groups." pith.science (2026). https://pith.science/paper/QLZNQY42

@misc{pith2026260713730,
  author       = {Pith},
  title        = {Pith review of: Regularity of Manhattan manifolds and exact dimensionality for relatively Anosov groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QLZNQY42}},
  note         = {Machine review of arXiv:2607.13730}
}

abstract

We establish several results about Patterson--Sullivan measures for relatively Anosov groups. First, we prove that these measures are exact dimensional with respect to visual metrics induced by Gromov models in the Groves--Manning quasi-isometry class. Under the additional assumption that the group is relatively Morse, we show that the associated scalar Cartan metric is Gromov hyperbolic and that the corresponding boundary premetric is a visual metric to which the exact-dimensionality theorem applies. Second, we prove that their Manhattan manifolds are $C^1$-regular, from which we deduce that the growth indicator is $C^1$-regular and strictly concave on the interior of the limit cone. This extends the case of Anosov representations by Kim--Oh--Wang. Our methods are dynamical, and we exploit the fact due to Kim--Oh and Blayac--Canary--Zhu--Zimmer that Bowen--Margulis--Sullivan measures for relatively Anosov groups are finite and mixing.

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Reference graph

Works this paper leans on

12 extracted references · 6 linked inside Pith

[5]

The joint translation spectrum and Manhattan manifolds.arXiv preprint arXiv:2411.06375,

[CRS24] Stephen Cantrell, Eduardo Reyes, and Cagri Sert. The joint translation spectrum and Manhattan manifolds.arXiv preprint arXiv:2411.06375,

arXiv
[6]

The Manhattan curve, ergodic theory of topo- logical flows and rigidity.Geom

[CT25] Stephen Cantrell and Ryokichi Tanaka. The Manhattan curve, ergodic theory of topo- logical flows and rigidity.Geom. Topol., 29(4):1851–1907,

1907
[7]

Kim, and Hee Oh

[DKO24] Subhadip Dey, Dongryul M. Kim, and Hee Oh. Ahlfors regularity of Patterson–Sullivan measures of Anosov groups and applications.arXiv preprint arXiv:2401.12398,

arXiv
[9]

Kim and Hee Oh

[KO26] Dongryul M. Kim and Hee Oh. A global shadow lemma for relatively Morse groups in higher rank.arXiv preprint arXiv:2606.19779,

arXiv
[12]

Relatively Anosov representations via flows II: Exam- ples.J

[ZZ24b] Feng Zhu and Andrew Zimmer. Relatively Anosov representations via flows II: Exam- ples.J. Lond. Math. Soc. (2024), 109(6):e12949, 79,

2024
[1993]

A geometric corre- spondence for reparameterizations of geodesic flows.arXiv preprint arXiv:2605.02585,

[CMGR26] Stephen Cantrell, D ´ ıdac Mart ´ ınez-Granado, and Eduardo Reyes. A geometric corre- spondence for reparameterizations of geodesic flows.arXiv preprint arXiv:2605.02585,

arXiv
[2010]

Horoboundary and rigidity of filling geodesic currents.arXiv preprint arXiv:2601.02059,

[JMG26] Meenakshy Jyothis and D ´ ıdac Mart ´ ınez-Granado. Horoboundary and rigidity of filling geodesic currents.arXiv preprint arXiv:2601.02059,

arXiv
[2019]

Finiteness of Bowen–Margulis–Sullivan measure for Gromov–Patterson– Sullivan systems.arXiv preprint arXiv:2604.03982,

[Wen26] Rou Wen. Finiteness of Bowen–Margulis–Sullivan measure for Gromov–Patterson– Sullivan systems.arXiv preprint arXiv:2604.03982,

arXiv

Show all 12 references

[2022]

Counting, mixing and equidistribution for GPS systems with applications to relatively Anosov groups.arXiv preprint arXiv:2404.09718,

[BCZZ24a] Pierre-Louis Blayac, Richard Canary, Feng Zhu, and Andrew Zimmer. Counting, mixing and equidistribution for GPS systems with applications to relatively Anosov groups.arXiv preprint arXiv:2404.09718,

arXiv
[2024]

Patterson– Sullivan theory for coarse cocycles.arXiv preprint arXiv:2404.09713,

[BCZZ24b] Pierre-Louis Blayac, Richard Canary, Feng Zhu, and Andrew Zimmer. Patterson– Sullivan theory for coarse cocycles.arXiv preprint arXiv:2404.09713,

arXiv
[2025]

Intersection, the Manhattan curve, and Patterson-Sullivan theory in rank 2.Int

[Bur93] Marc Burger. Intersection, the Manhattan curve, and Patterson-Sullivan theory in rank 2.Int. Math. Res. Not. IMRN, 1993(7):217–225,

1993
[2026]

Kim and Andrew Zimmer

[KZ25] Dongryul M. Kim and Andrew Zimmer. Rigidity for Patterson–Sullivan systems with applications to random walks and entropy rigidity.arXiv preprint arXiv:2505.16556,

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

[1] [5]

The joint translation spectrum and Manhattan manifolds.arXiv preprint arXiv:2411.06375,

[CRS24] Stephen Cantrell, Eduardo Reyes, and Cagri Sert. The joint translation spectrum and Manhattan manifolds.arXiv preprint arXiv:2411.06375,

arXiv

[2] [6]

The Manhattan curve, ergodic theory of topo- logical flows and rigidity.Geom

[CT25] Stephen Cantrell and Ryokichi Tanaka. The Manhattan curve, ergodic theory of topo- logical flows and rigidity.Geom. Topol., 29(4):1851–1907,

1907

[3] [7]

Kim, and Hee Oh

[DKO24] Subhadip Dey, Dongryul M. Kim, and Hee Oh. Ahlfors regularity of Patterson–Sullivan measures of Anosov groups and applications.arXiv preprint arXiv:2401.12398,

arXiv

[4] [9]

Kim and Hee Oh

[KO26] Dongryul M. Kim and Hee Oh. A global shadow lemma for relatively Morse groups in higher rank.arXiv preprint arXiv:2606.19779,

arXiv

[5] [12]

Relatively Anosov representations via flows II: Exam- ples.J

[ZZ24b] Feng Zhu and Andrew Zimmer. Relatively Anosov representations via flows II: Exam- ples.J. Lond. Math. Soc. (2024), 109(6):e12949, 79,

2024

[6] [1993]

A geometric corre- spondence for reparameterizations of geodesic flows.arXiv preprint arXiv:2605.02585,

[CMGR26] Stephen Cantrell, D ´ ıdac Mart ´ ınez-Granado, and Eduardo Reyes. A geometric corre- spondence for reparameterizations of geodesic flows.arXiv preprint arXiv:2605.02585,

arXiv

[7] [2010]

Horoboundary and rigidity of filling geodesic currents.arXiv preprint arXiv:2601.02059,

[JMG26] Meenakshy Jyothis and D ´ ıdac Mart ´ ınez-Granado. Horoboundary and rigidity of filling geodesic currents.arXiv preprint arXiv:2601.02059,

arXiv

[8] [2019]

Finiteness of Bowen–Margulis–Sullivan measure for Gromov–Patterson– Sullivan systems.arXiv preprint arXiv:2604.03982,

[Wen26] Rou Wen. Finiteness of Bowen–Margulis–Sullivan measure for Gromov–Patterson– Sullivan systems.arXiv preprint arXiv:2604.03982,

arXiv