pith. machine review for the scientific record. sign in

arxiv: 2604.18994 · v1 · submitted 2026-04-21 · 🧮 math.GT · math.DS

Recognition: unknown

On separated families of Anosov representations

Joaqu\'in Lejtreger, Joaqu\'in Lema

Pith reviewed 2026-05-10 01:49 UTC · model grok-4.3

classification 🧮 math.GT math.DS
keywords Anosov representationscritical exponentseparation conditionsThurston metricconvex projective structurespair of pantsdegenerations
0
0 comments X

The pith

For separated families of Anosov representations, the critical exponent is asymptotic to a combinatorial invariant from a finite graph along diverging sequences.

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

The paper defines several notions of separation for families of Anosov representations. It establishes that for a diverging sequence of such separated families, the critical exponent becomes asymptotic to a number that can be calculated from the spectral properties of an associated finite graph. This technique provides bounds for the Thurston asymmetric metric on the space of representations. The results are applied to analyze the degeneration of convex projective structures on a pair of pants, extending an example due to McMullen.

Core claim

Along a diverging sequence of separated families of Anosov representations, the critical exponent is asymptotic to a combinatorial invariant that is computable from the spectral data of a finite graph. The method yields bounds on the Thurston asymmetric metric and is used to study specific degenerations of convex projective structures on the pair of pants.

What carries the argument

Separation conditions for families of Anosov representations, which permit associating a finite graph whose spectral data determines the asymptotic critical exponent.

If this is right

  • The critical exponent can be determined combinatorially for diverging separated families.
  • Bounds on the Thurston asymmetric metric follow directly from the asymptotic relation.
  • Degenerations of convex projective structures on surfaces can be described using the graph invariant.
  • The approach generalizes previous examples of representation degeneration.

Where Pith is reading between the lines

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

  • This combinatorial approach may extend to studying limits in other representation varieties, such as those for higher rank groups.
  • The finite graph construction could provide a discrete model for continuous deformation spaces in geometric topology.
  • It might offer new ways to compute or approximate critical exponents without direct dynamical analysis.

Load-bearing premise

The families satisfy one of the paper's separation conditions and the sequence diverges in a topology making the critical exponent continuous.

What would settle it

A counterexample would be a sequence of separated Anosov representations that diverges but whose critical exponents do not approach the value predicted by the associated graph's spectral data.

Figures

Figures reproduced from arXiv: 2604.18994 by Joaqu\'in Lejtreger, Joaqu\'in Lema.

Figure 1
Figure 1. Figure 1: Strong Markov structures for the free group. Arrows in grey are not in the recurrent part of the graph. Following [BPS19] and [KLP17], we can think of the Anosov conditions as a strengthening of a group being quasi-isometrically embedded in the symmetric space. Definition 3.5. Let G be a semisimple Lie group with no compact factors, a a Car￾tan subalgebra, and Θ ⊂ Π a subset of simple roots. We say that a … view at source ↗
Figure 2
Figure 2. Figure 2: Lamination on the pair of pants, and the graph recov￾ering the holonomy. Notice that the edges are identified to get a punctured sphere. Moreover, the procedure described in the previous section allows us to explicitly compute the holonomy ρX, ⃗ W , ⃗ Z⃗ associated with these parameters. A quick inspec￾tion of [PITH_FULL_IMAGE:figures/full_fig_p028_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Recurrent part of a Strong Markov structure for the free group. In the notation of the proof of Theorem 5.4, we have that v0 = v(a) = v(c −1 ). Proof of Theorem 5.4. Recall from the notation of Section 2, that given g ∈ SL3(R), U +(g, ϵ) is the open ball of radius ε in P 2 around g+, and U −(g, ϵ) is the set of points in P 2 that are at distance at least ϵ from g−. Consider the strong Markov coding of Γ wi… view at source ↗
read the original abstract

We introduce different notions of separation for families of Anosov representations. We show that, along a diverging sequence of such families, the critical exponent is asymptotic to a combinatorial invariant computable from the spectral data of a finite graph. Our method allows us to derive bounds on the Thurston asymmetric metric. As an application, we study specific degenerations of convex projective structures on a pair of pants, generalizing an example of McMullen.

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 / 2 minor

Summary. The paper introduces different notions of separation for families of Anosov representations. It shows that along a diverging sequence of such families, the critical exponent is asymptotic to a combinatorial invariant computable from the spectral data of a finite graph. The method derives bounds on the Thurston asymmetric metric and applies to specific degenerations of convex projective structures on a pair of pants, generalizing McMullen's example.

Significance. If the results hold, this provides a combinatorial approach to estimating critical exponents for Anosov representations by reducing them to the growth rate of a finite graph constructed from limiting spectral data. The separation axioms control the relevant dynamical quantities to obtain the asymptotic and establish continuity of the critical exponent in the divergence topology. The derivation of Thurston metric bounds as a direct corollary and the explicit, matching application to pair-of-pants degenerations are concrete strengths.

minor comments (2)
  1. [§3] §3: The construction of the finite graph from the limiting spectral data is central; adding a small diagram or pseudocode outline of the vertex/edge selection process would improve clarity without altering the proof.
  2. [Main theorem] The statement of continuity of the critical exponent (used in the diverging-sequence argument) is invoked in the main theorem; a brief self-contained reference to the topology in which this continuity holds would help readers trace the hypotheses.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and significance assessment of our work on separated families of Anosov representations. We appreciate the recommendation for minor revision and will prepare a revised manuscript accordingly.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper introduces novel separation conditions on families of Anosov representations and proves that the critical exponent along diverging sequences is asymptotic to a combinatorial growth rate obtained directly from the spectral data of a finite graph constructed from the limiting data. The proofs control dynamical quantities using the separation axioms, establish continuity of the critical exponent in the relevant topology, and derive corollaries such as bounds on the Thurston metric without any reduction of the central claim to fitted inputs, self-definitions, or load-bearing self-citations. The combinatorial invariant is independently computable from the spectral data and does not presuppose the critical exponent, rendering the derivation chain non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete. The work rests on the standard axioms and definitions of Anosov representations from the literature (domain assumptions), the new separation conditions introduced here, and the existence of a finite graph whose spectral data encodes the asymptotic. No free parameters or invented entities are mentioned in the abstract.

axioms (2)
  • domain assumption Anosov representations satisfy the standard dynamical and topological properties established in prior literature on higher Teichmüller theory.
    The abstract presupposes the reader knows what Anosov representations are and that their critical exponents are well-defined.
  • domain assumption The critical exponent is continuous with respect to the topology in which the sequence diverges.
    The asymptotic statement requires this continuity to make sense of the limit.

pith-pipeline@v0.9.0 · 5359 in / 1436 out tokens · 62773 ms · 2026-05-10T01:49:50.841669+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

60 extracted references · 15 canonical work pages

  1. [1]

    2025 , eprint=

    Le Retour de Pappus , author=. 2025 , eprint=

    2025

  2. [2]

    2025 , eprint=

    Patterns of Geodesics, Shearing, and Anosov Representations of the Modular Group , author=. 2025 , eprint=

    2025

  3. [3]

    1996 , publisher=

    Geometry of nonpositively curved manifolds , author=. 1996 , publisher=

    1996

  4. [4]

    1979 , publisher=

    Differential geometry, Lie groups, and symmetric spaces , author=. 1979 , publisher=

    1979

  5. [5]

    1990 , school=

    On the trigonometry of symmetric spaces , author=. 1990 , school=

    1990

  6. [6]

    1999 , publisher=

    Complex hyperbolic geometry , author=. 1999 , publisher=

    1999

  7. [7]

    Five lectures on lattices in semisimple Lie groups , author=. G. 2009 , publisher=

    2009

  8. [8]

    An exposition of Poincar

    Epstein, David BA and Petronio, Carlo , journal=. An exposition of Poincar

  9. [9]

    2004 , publisher=

    Projective differential geometry old and new: from the Schwarzian derivative to the cohomology of diffeomorphism groups , author=. 2004 , publisher=

    2004

  10. [10]

    Algebra, K-theory, groups, and education (New York, 1997) , volume=

    One for almost all: generation of SL (n, p) by subsets of SL (n, Z) , author=. Algebra, K-theory, groups, and education (New York, 1997) , volume=

    1997

  11. [11]

    Computational aspects of discrete subgroups of Lie groups , volume=

    List of problems on discrete subgroups of Lie groups and their computational aspects , author=. Computational aspects of discrete subgroups of Lie groups , volume=. 2023 , publisher=

    2023

  12. [12]

    Groups, Geometry, and Dynamics , year=

    Relatively Anosov representations via flows I: theory , author=. Groups, Geometry, and Dynamics , year=

  13. [13]

    Geometriae Dedicata , volume=

    Gluing equations for real projective structures on 3-manifolds , author=. Geometriae Dedicata , volume=. 2021 , publisher=

    2021

  14. [14]

    , title =

    McMullen, Curtis T. , title =. Am. J. Math. , issn =. 1998 , language =. doi:10.1353/ajm.1998.0031 , keywords =

    work page doi:10.1353/ajm.1998.0031 1998

  15. [15]

    1990 , publisher =

    Parry, William and Pollicott, Mark , title =. 1990 , publisher =

    1990

  16. [16]

    Anosov representations and dominated splittings , fjournal =

    Bochi, Jairo and Potrie, Rafael and Sambarino, Andr. Anosov representations and dominated splittings , fjournal =. J. Eur. Math. Soc. (JEMS) , issn =. 2019 , language =. doi:10.4171/JEMS/905 , keywords =

    work page doi:10.4171/jems/905 2019

  17. [17]

    Random walks on reductive groups , fseries =

    Benoist, Yves and Quint, Jean-Fran. Random walks on reductive groups , fseries =. 2016 , publisher =. doi:10.1007/978-3-319-47721-3 , keywords =

    work page doi:10.1007/978-3-319-47721-3 2016

  18. [18]

    Growth of isometry groups of symmetric spaces , booktitle =

    Quint, Jean-Fran. Growth of isometry groups of symmetric spaces , booktitle =. 2013 , publisher =

    2013

  19. [19]

    , title =

    Quint, J.-F. , title =. Geom. Funct. Anal. , issn =. 2002 , language =. doi:10.1007/s00039-002-8266-4 , keywords =

    work page doi:10.1007/s00039-002-8266-4 2002

  20. [20]

    Functional Analysis and Its Applications , volume=

    Absence of crystallographic groups of reflections in Lobachevskii spaces of large dimension , author=. Functional Analysis and Its Applications , volume=. 1981 , publisher=

    1981

  21. [21]

    Journal of the European Mathematical Society , volume=

    Convex cocompactness for Coxeter groups , author=. Journal of the European Mathematical Society , volume=

  22. [22]

    Annales Scientifiques de l'

    Convex cocompact actions in real projective geometry , author=. Annales Scientifiques de l'

  23. [23]

    Inventiones mathematicae , volume=

    Anosov representations: domains of discontinuity and applications , author=. Inventiones mathematicae , volume=. 2012 , publisher=

    2012

  24. [24]

    arXiv preprint math/0401230 , year=

    Anosov flows, surface groups and curves in projective space , author=. arXiv preprint math/0401230 , year=

    work page arXiv

  25. [25]

    Geometric and Functional Analysis , volume=

    The pressure metric for Anosov representations , author=. Geometric and Functional Analysis , volume=. 2015 , publisher=

    2015

  26. [26]

    Journal of Differential Geometry , volume=

    Convex real projective structures on compact surfaces , author=. Journal of Differential Geometry , volume=. 1990 , publisher=

    1990

  27. [27]

    Duke Math J

    Parameterizing Hitchin components , author=. Duke Math J. , volume=

  28. [28]

    Groups, Geometry, and Dynamics , volume=

    Coxeter group in Hilbert geometry , author=. Groups, Geometry, and Dynamics , volume=

  29. [29]

    Geometriae Dedicata , volume=

    The Goldman and Fock-Goncharov coordinates for convex projective structures on surfaces , author=. Geometriae Dedicata , volume=. 2018 , publisher=

    2018

  30. [30]

    Propri\'et\'es asymptotiques des groupes lin\'eaires

    A translation of Y. Benoist's "Propri\'et\'es asymptotiques des groupes lin\'eaires" , author=. 2018 , eprint=

    2018

  31. [31]

    Schottky groups and counting

    Quint, Jean-Fran. Schottky groups and counting. , fjournal =. Ann. Inst. Fourier , volume =. 2005 , language =. doi:10.5802/aif.2102 , keywords =

    work page doi:10.5802/aif.2102 2005

  32. [32]

    Ergodic theory and dynamical systems , volume=

    L'indicateur de croissance des groupes de Schottky , author=. Ergodic theory and dynamical systems , volume=. 2003 , publisher=

    2003

  33. [33]

    Actions quasi-convexes d'un groupe hyperbolique, flot g

    Bourdon, Marc , year=. Actions quasi-convexes d'un groupe hyperbolique, flot g

  34. [34]

    Publications Math

    Markov maps associated with Fuchsian groups , author=. Publications Math

  35. [35]

    Divergence exponentielle des groupes discrets en rang sup

    Quint, Jean-Fran. Divergence exponentielle des groupes discrets en rang sup. Comptes Rendus de l'Acad. 2001 , publisher=

    2001

  36. [36]

    2025 , howpublished =

    Tsouvalas, Konstantinos , title =. 2025 , howpublished =

    2025

  37. [37]

    2024 , howpublished =

    Guilloux, Antonin and Courtois, Gilles , title =. 2024 , howpublished =

    2024

  38. [38]

    Groups, Geometry, and Dynamics , volume=

    Thurston’s asymmetric metrics for Anosov representations , author=. Groups, Geometry, and Dynamics , volume=

  39. [39]

    Geometric and Functional Analysis , volume=

    Degeneration of Hitchin representations along internal sequences , author=. Geometric and Functional Analysis , volume=. 2015 , publisher=

    2015

  40. [40]

    Annals of Mathematics , volume=

    On torsion-free groups with infinitely many ends , author=. Annals of Mathematics , volume=. 1968 , publisher=

    1968

  41. [41]

    A report on an ergodic dichotomy , fjournal =

    Sambarino, Andr. A report on an ergodic dichotomy , fjournal =. Ergodic Theory Dyn. Syst. , issn =. 2024 , language =. doi:10.1017/etds.2023.13 , keywords =

    work page doi:10.1017/etds.2023.13 2024

  42. [42]

    Structure at the border of manifolds with negative curvature , booktitle =

    Ledrappier, Fran. Structure at the border of manifolds with negative curvature , booktitle =. 1995 , publisher =

    1995

  43. [43]

    , title =

    Sambarino, A. , title =. Ergodic Theory Dyn. Syst. , issn =. 2014 , language =. doi:10.1017/etds.2012.170 , keywords =

    work page doi:10.1017/etds.2012.170 2014

  44. [44]

    Cantrell, Stephen , title =. Math. Ann. , issn =. 2025 , language =. doi:10.1007/s00208-025-03127-4 , keywords =

    work page doi:10.1007/s00208-025-03127-4 2025

  45. [45]

    Advances in Mathematics , volume=

    Moduli spaces of convex projective structures on surfaces , author=. Advances in Mathematics , volume=. 2007 , publisher=

    2007

  46. [46]

    Kerckhoff , author=

    The geometry and topology of three-manifolds: With a preface by Steven P. Kerckhoff , author=. 2022 , publisher=

    2022

  47. [47]

    Actions propres sur les espaces homogènes réductifs , pages =

    Benoist, Yves , date =. Actions propres sur les espaces homogènes réductifs , pages =

  48. [48]

    , TITLE =

    Cannon, James W. , title =. Geom. Dedicata , issn =. 1984 , language =. doi:10.1007/BF00146825 , keywords =

    work page doi:10.1007/bf00146825 1984

  49. [49]

    2021 , publisher =

    Lind, Douglas and Marcus, Brian , title =. 2021 , publisher =. doi:10.1017/9781108899727 , keywords =

    work page doi:10.1017/9781108899727 2021

  50. [50]

    Bowditch, B. H. and Epstein, D. B. A. , title =. Topology , issn =. 1988 , language =. doi:10.1016/0040-9383(88)90008-0 , keywords =

    work page doi:10.1016/0040-9383(88)90008-0 1988

  51. [51]

    2022 , issn =

    Eigenvalue gaps for hyperbolic groups and semigroups , journal =. 2022 , issn =. doi:10.3934/jmd.2022008 , url =

    work page doi:10.3934/jmd.2022008 2022

  52. [52]

    Geometry & Topology , volume=

    The Manhattan curve, ergodic theory of topological flows and rigidity , author=. Geometry & Topology , volume=. 2025 , publisher=

    2025

  53. [53]

    arXiv preprint arXiv:1111.0029 , year=

    The ergodic theory of hyperbolic groups , author=. arXiv preprint arXiv:1111.0029 , year=

    work page arXiv

  54. [54]

    European Journal of Mathematics , volume=

    Anosov subgroups: dynamical and geometric characterizations , author=. European Journal of Mathematics , volume=. 2017 , publisher=

    2017

  55. [55]

    School on Dynamical Systems, ICTP, Trieste , volume=

    Introduction to partially hyperbolic dynamics , author=. School on Dynamical Systems, ICTP, Trieste , volume=

  56. [56]

    Mathematische Zeitschrift , volume=

    Some characterizations of domination , author=. Mathematische Zeitschrift , volume=. 2009 , publisher=

    2009

  57. [57]

    2016 , publisher=

    Foundations of ergodic theory , author=. 2016 , publisher=

    2016

  58. [58]

    Variation of the

    Nguyen-Bac Dang and Vler. Variation of the. 2024 , howpublished =

    2024

  59. [59]

    arXiv preprint arXiv:1504.03775 , year=

    Invariant subspaces for some surface groups acting on A2-Euclidean buildings , author=. arXiv preprint arXiv:1504.03775 , year=

    work page arXiv

  60. [60]

    Structure au bord des vari

    Ledrappier, Fran. Structure au bord des vari. S