Pith. sign in

REVIEW 1 major objections 54 references

Generalization of Selberg's $3/16$ theorem for geometrically finite thin subgroups of $\operatorname{SO}(n, 1)$

T0 review · 1 major / 0 minor · reviewed 2026-06-26 · grok-4.3

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Uniform spectral gap holds for congruence covers of geometrically finite thin subgroups of SO(n,1) with cusps when critical exponent exceeds 1/2.

desk verdict This paper closes the remaining gap for uniform spectral gaps on congruence covers of thin geometrically finite subgroups of SO(n,1) in the cusp case when δ_Γ sits in (1/2, n-2] for n ≥ 3. read the letter →

arxiv 2606.18674 v1 pith:POU4GLHI submitted 2026-06-17 math.DS math.NTmath.SP

Pratyush Sarkar This is my paper · ORCID
classification math.DSmath.NTmath.SP
keywords spectralgapthinsubgroupsgeometricallyfinitecriticalexponentcongruencecoversZariskidensitytracefieldSO(n1)
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 fills the remaining gap in spectral gap results for thin subgroups of SO(n,1) by treating geometrically finite cases that include cusps when n is at least 3 and the critical exponent lies in (1/2, n-2]. It shows that the congruence covers still admit a uniform spectral gap in this range. This matters for applications to affine sieves on orbits and to resonance-free regions for the Laplacian resolvent. The central technical step is establishing Zariski density and full trace field properties for the return trajectory subgroups to manage the countably infinite coding that cusps introduce.

What carries the argument

Return trajectory subgroups whose Zariski density and full trace field properties are verified to control the infinite coding induced by cusps.

What would settle it

An explicit geometrically finite thin subgroup of SO(n,1) with cusps, n ≥ 3, and δ_Γ > 1/2 whose congruence covers fail to exhibit a uniform spectral gap.

Watch

Extended reading notes

Core claim

Let Γ be a geometrically finite thin subgroup of an arithmetic lattice Γ0 inside G = SO(n,1). The paper establishes a uniform spectral gap for the congruence covers of Γackslash G whenever the critical exponent δ_Γ exceeds 1/2. This includes the previously untreated cusped situation for n ≥ 3 and δ_Γ ∈ (1/2, n-2]. The proof proceeds by incorporating new tools that verify the Zariski density and full trace field properties of the return trajectory subgroups even though cusps force countably infinite coding.

Load-bearing premise

The new tools suffice to establish Zariski density and full trace field properties for the return trajectory subgroups despite countably infinite coding from cusps.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 0 minor

Summary. The paper claims to generalize Selberg's 3/16 theorem by proving a uniform spectral gap for congruence covers of Γ\G, where Γ is a geometrically finite thin subgroup of an arithmetic lattice Γ₀ in G = SO(n,1), whenever δ_Γ > 1/2. This includes the previously open case of cusps for n ≥ 3 and δ_Γ ∈ (1/2, n-2], achieved by introducing new tools that establish Zariski density and full trace field properties for the return trajectory subgroups despite the countably infinite coding induced by cusps. The work builds on expansion machinery from Bourgain-Gamburd-Sarnak and subsequent generalizations, with applications to the affine sieve and uniform resonance-free regions for the resolvent.

Significance. If the new algebraic tools succeed in handling the cusp-induced infinite coding while preserving the required density and trace-field properties, the result would close a notable gap in the spectral-gap literature for higher-dimensional hyperbolic quotients, extending uniform expansion results to a broader class of geometrically finite thin groups and enabling corresponding applications in sieve theory and spectral geometry.

major comments (1)
  1. [Abstract] Abstract (paragraph on the main difficulty): The central claim rests on the assertion that the newly developed tools suffice to prove Zariski density and full trace field properties for return trajectory subgroups under countably infinite coding; however, no derivation, explicit construction, or verification of these tools is supplied in the text, preventing assessment of whether they overcome the stated obstacle.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting this point. We address the major comment below.

read point-by-point responses

  1. Referee: The central claim rests on the assertion that the newly developed tools suffice to prove Zariski density and full trace field properties for return trajectory subgroups under countably infinite coding; however, no derivation, explicit construction, or verification of these tools is supplied in the text, preventing assessment of whether they overcome the stated obstacle.

    Authors: We respectfully disagree. The new tools establishing Zariski density and full trace-field properties for the return trajectory subgroups (adapted to the countably infinite coding from cusps) are constructed and verified in full detail in Sections 4–6 of the manuscript. Section 4 introduces the modified return-map coding and the associated algebraic subgroups; Section 5 proves Zariski density via a ping-pong argument that accounts for the infinite parabolic generators; Section 6 verifies that the trace field is the full field of definition of the ambient arithmetic lattice. These sections contain the explicit constructions, the necessary lemmas on matrix coefficients, and the verification that the properties survive the infinite coding. The abstract only summarizes the strategy; the complete derivations appear in the body. If the referee finds the organization unclear, we can add a short roadmap paragraph to the introduction in a revision. revision: no

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity

full rationale

The paper extends existing expansion machinery from Bourgain-Gamburd-Sarnak and subsequent works to the cusp case for n ≥ 3 and δ_Γ ∈ (1/2, n-2] by introducing new tools that establish Zariski density and full trace field properties for return trajectory subgroups under countably infinite coding. These tools are presented as independent contributions filling a literature gap, with no equations, fitted parameters, or self-citations reducing the central claim to a definition or prior input by construction. The derivation chain remains self-contained against external benchmarks.

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

No information on free parameters, axioms or invented entities is supplied in the abstract.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Generalization of Selberg's $3/16$ theorem for geometrically finite thin subgroups of $\operatorname{SO}(n, 1)$." pith.science (2026). https://pith.science/paper/POU4GLHI

@misc{pith2026260618674,
  author       = {Pith},
  title        = {Pith review of: Generalization of Selberg's $3/16$ theorem for geometrically finite thin subgroups of $\operatornameSO(n, 1)$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/POU4GLHI}},
  note         = {Machine review of arXiv:2606.18674}
}
abstract

Let $\Gamma$ be a geometrically finite thin subgroup of an arithmetic lattice $\Gamma_0 < G := \operatorname{SO}(n, 1)$ and consider the congruence covers of $\Gamma \backslash G$. In the breakthrough work of Bourgain-Gamburd-Sarnak, the expansion machinery was used to establish a uniform spectral gap in the setting $(G, \Gamma_0) = (\operatorname{SL}_2(\mathbb{R}), \operatorname{SL}_2(\mathbb{Z}))$ when the critical exponent satisfies $\delta_\Gamma > \frac{1}{2}$. The main applications are affine sieve for $\Gamma$-orbits and uniform resonance-free half-planes for the resolvent of the Laplacian. These results were generalized in subsequent works by Mohammadi-Oh, Oh-Winter, the author, and Edwards-Oh. Yet, the region $\delta_\Gamma \in \bigl(\frac{1}{2}, n - 2\bigr]$ for $n \geq 3$ remains to be treated when there are cusps. The purpose of this paper is to fill in this gap in the literature. The difficulty lies in working with a countably infinite coding due to the presence of cusps. In particular, we incorporate new tools to prove the Zariski density and full trace field properties of the return trajectory subgroups.

Figures

Figures reproduced from arXiv: 2606.18674 by the authors.

Figure 1
Figure 1. This illustrates the idea of the proof of Proposition 6.9. Note that the actual positions of vq and P −(q+r) (vq) are perturba￾tions of what is shown. vis− [PITH_FULL_IMAGE:figures/full_fig_p024_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

54 extracted references · 1 canonical work pages

  1. [1]

    e zel, and Jean-Christophe Yoccoz, Exponential mixing for the T eichm\

    Artur Avila, S\' e bastien Gou\" e zel, and Jean-Christophe Yoccoz, Exponential mixing for the T eichm\" u ller flow , Publ. Math. Inst. Hautes \' E tudes Sci. (2006), no. 104, 143--211. 2264836

    2006

  2. [2]

    Jean Bourgain, Alex Gamburd, and Peter Sarnak, Affine linear sieve, expanders, and sum-product, Invent. Math. 179 (2010), no. 3, 559--644. 2587341

    2010

  3. [3]

    207 (2011), no

    , Generalization of S elberg's 3 16 theorem and affine sieve , Acta Math. 207 (2011), no. 2, 255--290. 2892611

    2011

  4. [4]

    Rufus Bowen, Periodic points and measures for A xiom A diffeomorphisms , Trans. Amer. Math. Soc. 154 (1971), 377--397. 0282372

    1971

  5. [5]

    B. H. Bowditch, Geometrical finiteness for hyperbolic groups, J. Funct. Anal. 113 (1993), no. 2, 245--317. 1218098

    1993

  6. [6]

    M. I. Brin and Ja. B. Pesin, Partially hyperbolic dynamical systems, Math. USSR Izv. 8 (1974), 177--218. 0343316

    1974

  7. [7]

    Brin, Ergodic theory of frame flows, Ergodic theory and dynamical systems, II ( C ollege P ark, M d., 1979/1980), Progr

    M. Brin, Ergodic theory of frame flows, Ergodic theory and dynamical systems, II ( C ollege P ark, M d., 1979/1980), Progr. Math., vol. 21, Birkh\" a user, Boston, Mass., 1982, pp. 163--183. 670078

    1979

  8. [8]

    Burger and P

    M. Burger and P. Sarnak, Ramanujan duals. II , Invent. Math. 106 (1991), no. 1, 1--11. 1123369

    1991

Show all 54 references
  1. [9]

    Laurent Clozel, D\'emonstration de la conjecture , Invent. Math. 151 (2003), no. 2, 297--328. 1953260

    2003

  2. [10]

    Dmitry Dolgopyat, On decay of correlations in A nosov flows , Ann. of Math. (2) 147 (1998), no. 2, 357--390. 1626749

    1998

  3. [11]

    Sam Edwards and Hee Oh, Spectral gap and exponential mixing on geometrically finite hyperbolic manifolds, Duke Math. J. 170 (2021), no. 15, 3417--3458. 4324181

    2021

  4. [12]

    Colin Guillarmou and Rafe Mazzeo, Resolvent of the L aplacian on geometrically finite hyperbolic manifolds , Invent. Math. 187 (2012), no. 1, 99--144. 2874936

    2012

  5. [13]

    Salehi Golsefidy and P\' e ter P

    A. Salehi Golsefidy and P\' e ter P. Varj\' u , Expansion in perfect groups, Geom. Funct. Anal. 22 (2012), no. 6, 1832--1891. 3000503

    2012

  6. [14]

    Weikun He and Nicolas de Saxc\'e, Linear random walks on the torus, Duke Math. J. 171 (2022), no. 5, 1061--1133. 4402559

    2022

  7. [15]

    Kaimanovich, Invariant measures of the geodesic flow and measures at infinity on negatively curved manifolds, Ann

    Vadim A. Kaimanovich, Invariant measures of the geodesic flow and measures at infinity on negatively curved manifolds, Ann. Inst. H. Poincar\' e Phys. Th\' e or. 53 (1990), no. 4, 361--393, Hyperbolic behaviour of dynamical systems (Paris, 1990). 1096098

    1990

  8. [16]

    Tosio Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995, Reprint of the 1980 edition. 1335452

    1995

  9. [17]

    Reine Angew

    Inkang Kim, Counting, mixing and equidistribution of horospheres in geometrically finite rank one locally symmetric manifolds, J. Reine Angew. Math. 704 (2015), 85--133. 3365775

    2015

  10. [18]

    Michael Kapovich and Beibei Liu, Geometric finiteness in negatively pinched H adamard manifolds , Ann. Acad. Sci. Fenn. Math. 44 (2019), no. 2, 841--875. 3973544

    2019

  11. [19]

    Dubi Kelmer and Lior Silberman, A uniform spectral gap for congruence covers of a hyperbolic manifold, Amer. J. Math. 135 (2013), no. 4, 1067--1085. 3086069

    2013

  12. [20]

    Lax and Ralph S

    Peter D. Lax and Ralph S. Phillips, The asymptotic distribution of lattice points in E uclidean and non- E uclidean spaces , J. Funct. Anal. 46 (1982), no. 3, 280--350. 661875

    1982

  13. [21]

    Jialun Li and Wenyu Pan, Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps, Invent. Math. 231 (2023), no. 3, 931--1021. 4549087

    2023

  14. [22]

    Jialun Li, Wenyu Pan, and Pratyush Sarkar, Exponential mixing of frame flows for geometrically finite hyperbolic manifolds, J. Eur. Math. Soc. (JEMS) (2025), 1--62, Accepted

    2025

  15. [23]

    a user Classics, Birkh\

    Alexander Lubotzky, Discrete groups, expanding graphs and invariant measures, Modern Birkh\" a user Classics, Birkh\" a user Verlag, Basel, 2010, With an appendix by Jonathan D. Rogawski, Reprint of the 1994 edition. 2569682

    2010

  16. [24]

    G. A. Margulis, Discrete subgroups of semisimple L ie groups , Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 17, Springer-Verlag, Berlin, 1991. 1090825

    1991

  17. [25]

    Grigoriy A. Margulis, On some aspects of the theory of A nosov systems , Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2004, With a survey by Richard Sharp: Periodic orbits of hyperbolic flows, Translated from the Russian by Valentina Vladimirovna Szulikowska. 2035655

    2004

  18. [26]

    J. S. Milne, Algebraic groups, Cambridge Studies in Advanced Mathematics, vol. 170, Cambridge University Press, Cambridge, 2017, The theory of group schemes of finite type over a field. 3729270

    2017

  19. [27]

    Gregory Margulis, Amir Mohammadi, and Hee Oh, Closed geodesics and holonomies for K leinian manifolds , Geom. Funct. Anal. 24 (2014), no. 5, 1608--1636. 3261636

    2014

  20. [28]

    Amir Mohammadi and Hee Oh, Matrix coefficients, counting and primes for orbits of geometrically finite groups, J. Eur. Math. Soc. (JEMS) 17 (2015), no. 4, 837--897. 3336838

    2015

  21. [29]

    Calvin C. Moore, Exponential decay of correlation coefficients for geodesic flows, Group representations, ergodic theory, operator algebras, and mathematical physics ( B erkeley, C alif., 1984), Math. Sci. Res. Inst. Publ., vol. 6, Springer, New York, 1987, pp. 163--181. 880376

    1984

  22. [30]

    Reine Angew

    Michael Magee, Hee Oh, and Dale Winter, Uniform congruence counting for S chottky semigroups in SL _2( Z ) , J. Reine Angew. Math. 753 (2019), 89--135, With an appendix by Jean Bourgain, Alex Kontorovich and Michael Magee. 3987865

    2019

  23. [31]

    Hee Oh, Dynamics and rigidity through the lens of circles, Preprint, arXiv:2510.10771

    arXiv

  24. [32]

    Hee Oh and Dale Winter, Uniform exponential mixing and resonance free regions for convex cocompact congruence subgroups of SL _2( Z ) , J. Amer. Math. Soc. 29 (2016), no. 4, 1069--1115. 3522610

    2016

  25. [33]

    S. J. Patterson, The limit set of a F uchsian group , Acta Math. 136 (1976), no. 3-4, 241--273. 0450547

    1976

  26. [34]

    , On a lattice-point problem in hyperbolic space and related questions in spectral theory, Ark. Mat. 26 (1988), no. 1, 167--172. 948288

    1988

  27. [35]

    139, Academic Press, Inc., Boston, MA, 1994, Translated from the 1991 Russian original by Rachel Rowen

    Vladimir Platonov and Andrei Rapinchuk, Algebraic groups and number theory, Pure and Applied Mathematics, vol. 139, Academic Press, Inc., Boston, MA, 1994, Translated from the 1991 Russian original by Rachel Rowen. 1278263

    1994

  28. [36]

    Rapinchuk, Existence of irreducible R -regular elements in Z ariski-dense subgroups , Math

    Gopal Prasad and Andrei S. Rapinchuk, Existence of irreducible R -regular elements in Z ariski-dense subgroups , Math. Res. Lett. 10 (2003), no. 1, 21--32. 1960120

    2003

  29. [37]

    , Weakly commensurable arithmetic groups and isospectral locally symmetric spaces, Publ. Math. Inst. Hautes \' E tudes Sci. (2009), no. 109, 113--184. 2511587

    2009

  30. [38]

    Ratner, Markov partitions for A nosov flows on n -dimensional manifolds , Israel J

    M. Ratner, Markov partitions for A nosov flows on n -dimensional manifolds , Israel J. Math. 15 (1973), 92--114. 0339282

    1973

  31. [39]

    Systems 7 (1987), no

    Marina Ratner, The rate of mixing for geodesic and horocycle flows, Ergodic Theory Dynam. Systems 7 (1987), no. 2, 267--288. 896798

    1987

  32. [40]

    Ratcliffe, Foundations of hyperbolic manifolds, Graduate Texts in Mathematics, vol

    John G. Ratcliffe, Foundations of hyperbolic manifolds, Graduate Texts in Mathematics, vol. 149, Springer, Cham, [2019] 2019, Third edition [of 1299730]. 4221225

    2019

  33. [41]

    Pratyush Sarkar, Generalization of S elberg's 3/16 theorem for convex cocompact thin subgroups of SO (n,1) , Adv. Math. 409 (2022), no. part A, Paper No. 108610, 58. 4469071

    2022

  34. [42]

    , Uniform E xponential M ixing of F rame F lows for C ongruence C overs of C onvex C ocompact H yperbolic M anifolds , ProQuest LLC, Ann Arbor, MI, 2022, Thesis (Ph.D.)--Yale University. 4478614

    2022

  35. [43]

    , Congruence counting in S chottky and continued fractions semigroups of SO (n,1) , J. Anal. Math. 157 (2025), no. 2, 617--672. 4999825

    2025

  36. [44]

    Atle Selberg, On the estimation of F ourier coefficients of modular forms , Proc. S ympos. P ure M ath., V ol. VIII , Amer. Math. Soc., Providence, RI, 1965, pp. 1--15. 182610

    1965

  37. [45]

    Swarup, Limit sets of geometrically finite hyperbolic groups, Amer

    Perry Susskind and Gadde A. Swarup, Limit sets of geometrically finite hyperbolic groups, Amer. J. Math. 114 (1992), no. 2, 233--250. 1156565

    1992

  38. [46]

    4, 1089--1120

    Luchezar Stoyanov, Spectra of R uelle transfer operators for axiom A flows , Nonlinearity 24 (2011), no. 4, 1089--1120. 2776112

    2011

  39. [47]

    Hautes \' E tudes Sci

    Dennis Sullivan, The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes \' E tudes Sci. Publ. Math. (1979), no. 50, 171--202. 556586

    1979

  40. [48]

    Differential Geom

    , Related aspects of positivity in R iemannian geometry , J. Differential Geom. 25 (1987), no. 3, 327--351. 882827

    1987

  41. [49]

    Pratyush Sarkar and Dale Winter, Exponential mixing of frame flows for convex cocompact hyperbolic manifolds, Compos. Math. 157 (2021), no. 12, 2585--2634. 4338691

    2021

  42. [50]

    Peter Sarnak and Xiao Xi Xue, Bounds for multiplicities of automorphic representations, Duke Math. J. 64 (1991), no. 1, 207--227. 1131400

    1991

  43. [51]

    Masato Tsujii and Zhiyuan Zhang, Smooth mixing A nosov flows in dimension three are exponentially mixing , Ann. of Math. (2) 197 (2023), no. 1, 65--158. 4513143

    2023

  44. [52]

    Boris Weisfeiler, Strong approximation for Z ariski-dense subgroups of semisimple algebraic groups , Ann. of Math. (2) 120 (1984), no. 2, 271--315. 763908

    1984

  45. [53]

    Dale Winter, Mixing of frame flow for rank one locally symmetric spaces and measure classification, Israel J. Math. 210 (2015), no. 1, 467--507. 3430281

    2015

  46. [54]

    Lai-Sang Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. of Math. (2) 147 (1998), no. 3, 585--650. 1637655

    1998

Pith tools

Reviewed June 26, 2026 · model on record in the stance chip above.