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arxiv: 2606.19779 · v1 · pith:MYKDNXJRnew · submitted 2026-06-18 · 🧮 math.GT · math.DS· math.GR

A global shadow lemma for relatively Morse groups in higher rank

Dongryul M. Kim , Hee Oh This is my paper

Pith reviewed 2026-06-26 15:34 UTC · model grok-4.3

classification 🧮 math.GT math.DSmath.GR
keywords Patterson-Sullivan measuresshadow lemmarelatively Morse subgroupshigher-rank semisimple Lie groupsGromov modelHausdorff measurevisual quasi-metriccuspidal regions
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The pith

Patterson-Sullivan measures for relatively Morse subgroups satisfy a uniform global shadow lemma across the entire Gromov model, including cusps.

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

The paper proves a global shadow lemma giving uniform estimates for the mass that Patterson-Sullivan measures assign to shadows, no matter where the center point sits in the Gromov model. The uniformity continues to hold when centers lie deep inside cuspidal regions, extending earlier results known only for geometrically finite real hyperbolic groups. A sympathetic reader would care because the estimates describe the distribution of group orbits near the boundary in a controlled and location-independent way. From the lemma follow uniform local estimates on the measures and sufficient conditions for the measures to coincide, up to scale, with the Hausdorff measure coming from the visual quasi-metric. The work therefore supplies a basic comparison tool for discrete actions on higher-rank spaces.

Core claim

We prove a global shadow lemma for Patterson-Sullivan measures associated to relatively Morse subgroups of higher-rank semisimple Lie groups. The estimate is uniform for shadows centered at arbitrary points in a Gromov model, including points deep in the cuspidal part. This extends the global shadow lemma of Stratmann-Velani for geometrically finite real hyperbolic groups. As applications, we obtain uniform local estimates for Patterson-Sullivan measures, and we give sufficient conditions under which these measures agree, up to scale, with the Hausdorff measure defined by the associated visual quasi-metric.

What carries the argument

The global shadow lemma, which supplies a location-independent bound on the Patterson-Sullivan measure of any shadow set in the Gromov model.

If this is right

  • Uniform local estimates for Patterson-Sullivan measures hold at every scale and location.
  • Under the stated conditions the measures coincide up to scale with the Hausdorff measure induced by the visual quasi-metric.
  • The estimates remain valid for shadows centered at arbitrary points, including those deep inside cusps.
  • The lemma applies directly to any relatively Morse subgroup of a higher-rank semisimple Lie group.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The uniformity may allow direct comparison of local dimensions of the measures at cuspidal and non-cuspidal points.
  • The same shadow-control technique could be tested on other boundary measures once an analogous Gromov model is available.
  • Agreement with Hausdorff measure would immediately give explicit dimension formulas for the limit sets of these subgroups.

Load-bearing premise

The subgroups are relatively Morse and the ambient higher-rank semisimple Lie groups admit a Gromov model in which Patterson-Sullivan measures are defined.

What would settle it

A sequence of shadows whose centers move deeper into a cusp, yet whose Patterson-Sullivan measures grow or shrink by an unbounded factor relative to the shadow size, would violate the claimed uniformity.

read the original abstract

Patterson-Sullivan measures encode the distribution of orbits of discrete group actions near the boundary. In this paper, we prove a global shadow lemma for Patterson-Sullivan measures associated to relatively Morse subgroups of higher-rank semisimple Lie groups. The estimate is uniform for shadows centered at arbitrary points in a Gromov model, including points deep in the cuspidal part. This extends the global shadow lemma of Stratmann-Velani for geometrically finite real hyperbolic groups. As applications, we obtain uniform local estimates for Patterson-Sullivan measures, and we give sufficient conditions under which these measures agree, up to scale, with the Hausdorff measure defined by the associated visual quasi-metric.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The manuscript proves a global shadow lemma for Patterson-Sullivan measures associated to relatively Morse subgroups of higher-rank semisimple Lie groups. The estimate is uniform for shadows centered at arbitrary points in a Gromov model, including points deep in the cuspidal part. This extends the global shadow lemma of Stratmann-Velani for geometrically finite real hyperbolic groups. As applications, the paper obtains uniform local estimates for Patterson-Sullivan measures and gives sufficient conditions under which these measures agree, up to scale, with the Hausdorff measure defined by the associated visual quasi-metric.

Significance. If the result holds, the uniform global shadow lemma would be a useful technical tool in the study of boundary dynamics and Patterson-Sullivan measures for discrete subgroups in higher-rank Lie groups. The extension beyond the classical hyperbolic case, together with the claimed uniformity deep in cusps, would support applications to local dimension estimates and comparisons between Patterson-Sullivan and Hausdorff measures on the boundary.

minor comments (1)
  1. The abstract refers to the 'Gromov model' and 'relatively Morse' hypothesis without a self-contained definition or reference to the precise statement used; adding a short paragraph recalling these notions in §1 would improve readability for readers outside the immediate subfield.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of the manuscript. The recommendation is listed as uncertain, but the report contains no specific major comments or questions to address. We are prepared to respond to any concrete concerns if they are provided.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper claims a global shadow lemma for Patterson-Sullivan measures on relatively Morse subgroups of higher-rank semisimple Lie groups, presented explicitly as an extension of the independent Stratmann-Velani result for geometrically finite real hyperbolic groups. The abstract and reader's summary indicate the result follows from the relatively Morse hypothesis together with standard existence of Patterson-Sullivan measures and the Gromov model; no self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the stated theorem. The derivation chain is therefore self-contained against external benchmarks and prior independent work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available; the paper relies on standard domain assumptions from geometric group theory and Lie group dynamics that are not detailed here.

axioms (1)
  • domain assumption Patterson-Sullivan measures exist and are well-defined for the relatively Morse subgroups in question
    Invoked implicitly as the objects to which the lemma applies (abstract).

pith-pipeline@v0.9.1-grok · 5639 in / 1142 out tokens · 26069 ms · 2026-06-26T15:34:12.731022+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

30 extracted references

  1. [1]

    Y. Benoist. Propri´ et´ es asymptotiques des groupes lin´ eaires.Geom. Funct. Anal., 7(1):1–47, 1997

    1997

  2. [2]

    Benoist and H

    Y. Benoist and H. Oh. Effective equidistribution ofS-integral points on symmetric varieties.Ann. Inst. Fourier (Grenoble), 62(5):1889–1942, 2012

    1942

  3. [3]

    Bowditch

    B. Bowditch. A topological characterisation of hyperbolic groups.J. Amer. Math. Soc., 11(3):643–667, 1998. 44 DONGRYUL M. KIM AND HEE OH

    1998

  4. [4]

    Bowditch

    B. Bowditch. Convergence groups and configuration spaces. InGeometric group theory down under (Canberra, 1996), pages 23–54. de Gruyter, Berlin, 1999

    1996

  5. [5]

    Bowditch

    B. Bowditch. Relatively hyperbolic groups.Internat. J. Algebra Comput., 22(3):1250016, 66, 2012

    2012

  6. [6]

    Bray and G

    H. Bray and G. Tiozzo. A global shadow lemma and logarithm law for geometrically finite Hilbert geometries.arXiv preprint arXiv:2111.04618, 2021

    arXiv 2021

  7. [7]

    Canary, T

    R. Canary, T. Zhang, and A. Zimmer. Cusped Hitchin representations and Anosov representations of geometrically finite Fuchsian groups.Adv. Math., 404(part B):Paper No. 108439, 67, 2022

    2022

  8. [8]

    Canary, T

    R. Canary, T. Zhang, and A. Zimmer. Patterson-Sullivan measures for relatively Anosov groups.Math. Ann., 392(2):2309–2363, 2025

    2025

  9. [9]

    T. Das, D. Simmons, and M. Urba´ nski. Tukia’s isomorphism theorem in CAT(−1) spaces.Ann. Acad. Sci. Fenn. Math., 41(2):659–680, 2016

    2016

  10. [10]

    Dey and M

    S. Dey and M. Kapovich. Patterson-Sullivan theory for Anosov subgroups.Trans. Amer. Math. Soc., 375(12):8687–8737, 2022

    2022

  11. [11]

    S. Dey, D. M. Kim, and H. Oh. Ahlfors regularity of Patterson-Sullivan measures of Anosov groups and applications.arXiv preprint arXiv:2401.12398, To appear in Compos. Math

    arXiv

  12. [12]

    Falconer.Fractal geometry

    K. Falconer.Fractal geometry. John Wiley & Sons, Ltd., Chichester, third edition,

  13. [13]

    Mathematical foundations and applications

  14. [14]

    Kapovich and B

    M. Kapovich and B. Leeb. Relativizing characterizations of Anosov subgroups, I. Groups Geom. Dyn., 17(3):1005–1071, 2023. With an appendix by Gregory A. Soifer

    2023

  15. [15]

    Kapovich, B

    M. Kapovich, B. Leeb, and J. Porti. A Morse lemma for quasigeodesics in symmetric spaces and Euclidean buildings.Geom. Topol., 22(7):3827–3923, 2018

    2018

  16. [16]

    D. M. Kim. Conformal measure rigidity and ergodicity of horospherical foliations. arXiv preprint arXiv:2404.13727, 2024

    arXiv 2024

  17. [17]

    D. M. Kim and H. Oh. Relatively Anosov groups: finiteness, measure of maximal entropy, and reparameterization.J. Reine Angew. Math., 826:91–142, 2025

    2025

  18. [18]

    D. M. Kim, H. Oh, and Y. Wang. Properly discontinuous actions, growth indicators, and conformal measures for transverse subgroups.Math. Ann., 393(2):2391–2450, 2025

    2025

  19. [19]

    Lee and H

    M. Lee and H. Oh. Invariant measures for horospherical actions and Anosov groups. Int. Math. Res. Not. IMRN, (19):16226–16295, 2023

    2023

  20. [20]

    Papageorgiou

    E. Papageorgiou. Surjectivity of convolution operators on harmonicN Agroups.J. Geom. Anal., 35(1):Paper No. 7, 31, 2025

    2025

  21. [21]

    Patterson

    S. Patterson. The limit set of a Fuchsian group.Acta Math., 136(3-4):241–273, 1976

    1976

  22. [22]

    J.-F. Quint. Mesures de Patterson-Sullivan en rang sup´ erieur.Geom. Funct. Anal., 12(4):776–809, 2002

    2002

  23. [23]

    Rouvi` ere

    F. Rouvi` ere. Espaces de Damek-Ricci, g´ eom´ etrie et analyse. InAnalyse sur les groupes de Lie et th´ eorie des repr´ esentations (K´ enitra, 1999), volume 7 ofS´ emin. Congr., pages 45–100. Soc. Math. France, Paris, 2003

    1999

  24. [24]

    Stratmann and S

    B. Stratmann and S. L. Velani. The Patterson measure for geometrically finite groups with parabolic elements, new and old.Proc. London Math. Soc. (3), 71(1):197–220, 1995

    1995

  25. [25]

    Sullivan

    D. Sullivan. The density at infinity of a discrete group of hyperbolic motions.Inst. Hautes ´Etudes Sci. Publ. Math., (50):171–202, 1979

    1979

  26. [26]

    Sullivan

    D. Sullivan. Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups.Acta Math., 153(3-4):259–277, 1984

    1984

  27. [27]

    P. Tukia. On isomorphisms of geometrically finite M¨ obius groups.Inst. Hautes´Etudes Sci. Publ. Math., (61):171–214, 1985

    1985

  28. [28]

    A. Yaman. A topological characterisation of relatively hyperbolic groups.J. Reine Angew. Math., 566:41–89, 2004. A GLOBAL SHADOW LEMMA IN HIGHER RANK 45

    2004

  29. [29]

    Zhu and A

    F. Zhu and A. Zimmer. Relatively Anosov representations via flows I: theory.Preprint, arXiv:2207.14737, 2022. To appear in Groups Geom. Dyn

    arXiv 2022

  30. [30]

    Zhu and A

    F. Zhu and A. Zimmer. Relatively Anosov representations via flows II: Examples.J. Lond. Math. Soc. (2), 109(6):Paper No. e12949, 61, 2024. Department of Mathematics, Yale University, New Haven, CT 06511 Email address:dongryul.kim97@gmail.com Department of Mathematics, Yale University, New Haven, CT 06511 Email address:hee.oh@yale.edu

    2024