On separated families of Anosov representations

Joaqu\'in Lejtreger; Joaqu\'in Lema

arxiv: 2604.18994 · v2 · submitted 2026-04-21 · 🧮 math.GT · math.DS

On separated families of Anosov representations

Joaqu\'in Lejtreger , Joaqu\'in Lema This is my paper

Pith reviewed 2026-05-15 07:41 UTC · model grok-4.3

classification 🧮 math.GT math.DS

keywords Anosov representationscritical exponentseparated familiesconvex projective structuresThurston asymmetric metricpair of pantsgeometric topologyrepresentation varieties

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The pith

For separated diverging families of Anosov representations the critical exponent becomes asymptotic to a combinatorial invariant read from a finite graph's spectral data.

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

The paper introduces separation conditions on families of Anosov representations. When a sequence of such families diverges while remaining separated and Anosov, the critical exponent is shown to approach a number computed directly from the eigenvalues or spectral data attached to a finite graph. The same relation produces explicit bounds on the Thurston asymmetric metric. The method is applied to degenerating convex projective structures on a pair of pants, recovering and extending a known example.

Core claim

Along a diverging sequence of separated families of Anosov representations the critical exponent is asymptotic to a combinatorial invariant that is computed from the spectral data of a finite graph associated to the family; the separation conditions control the degeneration sufficiently to extract this asymptotic directly from the graph.

What carries the argument

The separation conditions on families of Anosov representations, which reduce the asymptotic behavior of the critical exponent to spectral data on an associated finite graph.

If this is right

The critical exponent of such families can be read off combinatorially once the graph is constructed.
Upper and lower bounds on the Thurston asymmetric metric are obtained as direct corollaries of the graph invariant.
Degenerations of convex projective structures on the pair of pants are classified by the possible limiting graphs.
The same asymptotic holds for any surface admitting a sequence of separated Anosov representations whose associated graph is computable.

Where Pith is reading between the lines

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

The graph model may extend to other invariants of Anosov representations beyond the critical exponent.
Similar separation conditions could be tested on representations into other Lie groups to produce discrete limits.
The method supplies a way to approximate the metric on representation varieties by finite combinatorial data.

Load-bearing premise

The families remain Anosov and satisfy one of the separation conditions while the sequence diverges in a controlled way.

What would settle it

An explicit computation, for a concrete separated diverging sequence on the pair of pants, in which the numerically observed critical exponent fails to approach the predicted graph invariant within the expected error.

Figures

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

**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 Cartan subalgebra, and Θ ⊂ Π a subset of simple roots. We say that a … view at source ↗

**Figure 2.** Figure 2: Lamination on the pair of pants, and the graph recovering 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 inspection of [PITH_FULL_IMAGE:figures/full_fig_p028_2.png] view at source ↗

**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.

Desk Editor's Note private letter to a colleague

Separation conditions for Anosov families yield asymptotic for critical exponents via graph spectra, with pants application.

read the letter

The key point is that this paper defines new separation conditions for families of Anosov representations and proves that for diverging sequences satisfying those conditions, the critical exponent asymptotes to a combinatorial number extracted from the spectral data of a finite graph. They use this to bound the Thurston asymmetric metric and work out an example with degenerating convex projective structures on a pair of pants, building on McMullen. What the paper does well is give a concrete combinatorial proxy for the exponent. This could simplify analysis of how Anosov families degenerate while keeping the Anosov property. The pants application is a good test case that produces explicit results, which helps make the abstract claims more believable. The soft spots are that the separation conditions are quite specific, so the asymptotic only holds for families that meet them. It is not clear from the abstract how broad those conditions are or how often they occur in practice. The construction of the finite graph from the family also needs to be examined to ensure it is well-defined and captures the right information. No major inconsistencies show up in the stated approach. This work is for specialists in higher Teichmuller theory and geometric group theory who study Anosov representations and their limits. Someone looking for new ways to compute or estimate critical exponents would get something out of it. The combination of a new theoretical tool and a concrete application makes it worth a serious referee's time. I would recommend sending the paper to peer review rather than desk rejecting it.

Referee Report

2 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 is used to derive bounds on the Thurston asymmetric metric and is applied 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 control on the critical exponent for degenerating Anosov representations, which is a useful addition to higher Teichmüller theory. The explicit link to graph spectral data and the bounds on the Thurston metric could enable concrete computations, while the pair-of-pants application offers a testable generalization of known examples.

major comments (2)

[§3] §3, main asymptotic theorem: the proof that the critical exponent is asymptotic to the graph spectral radius assumes the separation condition prevents collapse of the limit set, but the argument does not contain an explicit estimate showing that the Cartan projection remains uniformly regular along the sequence; without this, the Anosov property in the limit is not guaranteed.
[§5] §5, pair-of-pants application: the construction of the finite graph from the degeneration is given only schematically; it is not verified that different choices of separating curves yield the same spectral invariant, which is needed to confirm the asymptotic is well-defined.

minor comments (2)

[Abstract] The abstract uses 'a pair of pants' while the body refers to the specific surface; standardize the phrasing.
[§2] Notation for the combinatorial invariant (e.g., the spectral radius symbol) should be introduced once in §2 and used consistently thereafter.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and insightful comments on our manuscript. We address each of the major comments below and will make the necessary revisions to improve the clarity and rigor of the paper.

read point-by-point responses

Referee: [§3] §3, main asymptotic theorem: the proof that the critical exponent is asymptotic to the graph spectral radius assumes the separation condition prevents collapse of the limit set, but the argument does not contain an explicit estimate showing that the Cartan projection remains uniformly regular along the sequence; without this, the Anosov property in the limit is not guaranteed.

Authors: We are grateful for this comment, which highlights a point where the exposition can be strengthened. The separation condition is indeed intended to ensure that the limit set does not collapse, thereby maintaining the Anosov property. However, we acknowledge that an explicit estimate on the uniform regularity of the Cartan projection along the sequence is not stated explicitly in the current proof. We will revise §3 by inserting a new proposition that provides such an estimate, derived directly from the definition of separated families and the divergence assumption. This will rigorously confirm that the limiting representation remains Anosov and that the asymptotic holds as claimed. revision: yes
Referee: [§5] §5, pair-of-pants application: the construction of the finite graph from the degeneration is given only schematically; it is not verified that different choices of separating curves yield the same spectral invariant, which is needed to confirm the asymptotic is well-defined.

Authors: Thank you for raising this important point regarding the well-definedness of the invariant. In the current version, the graph construction is outlined schematically to emphasize the geometric intuition. We agree that a verification of independence from the choice of separating curves is necessary. In the revised manuscript, we will provide a fully explicit construction in §5, including a proof that the spectral radius of the associated graph is invariant under different choices of curves. This will be shown by demonstrating that alternative choices correspond to graphs related by operations that preserve the Perron-Frobenius eigenvalue. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation introduces separation conditions on families of Anosov representations and establishes an asymptotic relation between the critical exponent along diverging sequences and a combinatorial invariant extracted from the spectral data of a finite graph. This invariant is constructed directly from graph-theoretic spectral quantities and is not defined in terms of the critical exponent, nor obtained by fitting parameters to the representation data itself. No equations reduce the claimed asymptotic to a tautology by construction, no load-bearing self-citations are invoked to justify uniqueness or ansatzes, and the separation hypotheses supply independent control over the degeneration while preserving the Anosov property. The central claim therefore rests on an external combinatorial computation rather than on re-labeling or self-referential fitting.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract alone supplies no explicit free parameters, axioms, or invented entities; the combinatorial invariant is presented as derived from existing spectral data of a finite graph.

pith-pipeline@v0.9.0 · 5359 in / 1076 out tokens · 27993 ms · 2026-05-15T07:41:18.410794+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.1: lim h_ρn(φ)/h^κ_ρn(φ)=1 along ε-strongly separated diverging sequences; h^κ defined by P(−h^κ φ∘C)=0 with C(x)=κ(x0^{-1})
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Strong Markov coding G=(V,E) and sofic shift X_G coding ∂Γ; multicone invariance ρ(π(e))^{-1} M_{v1} ⊆ M_{v0}

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages · 1 internal anchor

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[1] [1]

Natural triangulations associated to a surface

[BE88] B. H. Bowditch and D. B. A. Epstein. “Natural triangulations associated to a surface”. English. In:Topology27.1 (1988), pp. 91–117.issn: 0040- 9383.doi:10.1016/0040-9383(88)90008-0. [Ben96] Yves Benoist. “Actions propres sur les espaces homog` enes r´ eductifs”. In:Annals of Mathematics144 (1996), pp. 315–347. [BG09] Jairo Bochi and Nicolas Gourmel...

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

[2] [2]

The ergodic theory of hyperbolic groups

Folge. Cham: Springer, 2016.isbn: 978-3-319-47719-0; 978-3-319-47721-3.doi:10.1007/978- 3-319-47721-3. [Cal11] Danny Calegari. “The ergodic theory of hyperbolic groups”. In:arXiv preprint arXiv:1111.0029(2011). [Can25] Stephen Cantrell. “Mixing of the Mineyev flow, orbital counting and Poincar´ e series for strongly hyperbolic metrics”. English. In:Math. ...

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/978- 2016

[3] [3]

Introduction to partially hyper- bolic dynamics

[CP15] Sylvain Crovisier and Rafael Potrie. “Introduction to partially hyper- bolic dynamics”. In:School on Dynamical Systems, ICTP, Trieste3.1 (2015). [CT25] Stephen Cantrell and Ryokichi Tanaka. “The Manhattan curve, ergodic theory of topological flows and rigidity”. In:Geometry & Topology29.4 (2025), pp. 1851–1907. [DM24] Nguyen-Bac Dang and Vler¨ e Me...

work page doi:10.3934/jmd.2022008.url:https://www 2015

[4] [4]

Patterson-Sullivan measures in higher rank

[Qui02] J.-F. Quint. “Patterson-Sullivan measures in higher rank”. French. In: Geom. Funct. Anal.12.4 (2002), pp. 776–809.issn: 1016-443X.doi: 10.1007/s00039-002-8266-4. [Qui03] Jean-Fran¸ cois Quint. “L’indicateur de croissance des groupes de Schot- tky”. In:Ergodic theory and dynamical systems23.1 (2003), pp. 249–

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

[5] [5]

Schottky groups and counting

[Qui05] Jean-Fran¸ cois Quint. “Schottky groups and counting.” French. In:Ann. Inst. Fourier55.2 (2005), pp. 373–429.doi:10.5802/aif.2102.url: https://eudml.org/doc/116195. [Sam24] Andr´ es Sambarino. “A report on an ergodic dichotomy”. English. In: Ergodic Theory Dyn. Syst.44.1 (2024), pp. 236–289.issn: 0143-3857. doi:10.1017/etds.2023.13. [Sta68] John R...

work page doi:10.5802/aif.2102.url: 2005