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Oriented graphs on curve complex I: hyperbolic and extremal length

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper proves that the oriented graph induced by hyperbolic or extremal length on the curve complex uniquely determines the hyperbolic metric.

desk verdict Genuinely new rigidity result — comparing lengths only across disjoint pairs of curves determines the hyperbolic metric — but the main theorem is currently held up by an unproved configuration claim in the reduction to the graph-theoretic core. read the letter →

arxiv 2507.12728 v1 pith:KRRHNGJ5 submitted 2025-07-17 math.GT math.COmath.CV

Dong Tan , Wen Yang This is my paper
classification math.GTmath.COmath.CV MSC 20F6530F6057M1505C63
keywords curvecomplexorientedgraphhyperboliclengthextremalDehntwistTeichmüllerspacerigiditymappingclassgroup
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 establishes that an oriented graph on the curve complex—with a directed edge between two disjoint simple closed curves pointing from the shorter to the longer one—carries enough information to determine the hyperbolic metric. It introduces a class of functions, Dehn quasi-homothetic functions, whose growth under repeated Dehn twists matches that of hyperbolic length and the square root of extremal length. The main theorem shows that two such functions of the same type inducing the same oriented graph must be proportional. Consequently, knowing only which of any two disjoint simple closed curves is longer fixes the hyperbolic surface uniquely.

What carries the argument

The driving object is a Dehn quasi-homothetic function of type $(f,A)$: a function $L$ on curve classes such that $L(T_\alpha^n\beta)/f(n)\to A(\alpha,\beta)L(\alpha)$ whenever $\alpha$ and $\beta$ intersect, with $f(n)\to\infty$ and the ratios $f(k)/f(j)$ dense in $\mathbb{R}_+$. The paper's Theorem 22 isolates the combinatorial core: in an infinite graph equipped with a system of Dehn twists and a twist-invariant subgraph $W$ satisfying certain square and triangle configuration conditions, two non-negative functions of the same type inducing the same oriented graph must satisfy $L_1(\alpha)L_2(\beta)=L_2(\alpha)L_1(\beta)$ on every edge. The square pattern $\{\alpha,\beta,\xi,\eta\}$ lets the proof construct twisted curves whose relative order under $L_1$ and $L_2$ can be forced to disagree, so only proportional functions can survive.

What would settle it

One concrete check is to take a closed surface of genus 2, enumerate all pairs of disjoint separating curves, and search for non-separating curves forming the induced square or triangle configuration required by Theorem 22. An explicit pair for which no such non-separating curves exist would refute the reduction that produces Theorem 2.

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Extended reading notes

Core claim

The central result, Theorem 2, states that if $L_1$ and $L_2$ are non-negative Dehn quasi-homothetic functions of the same type on the curve complex of a closed surface of genus at least $2$, and they induce the same oriented graph, then $L_1(\alpha)L_2(\beta)=L_2(\alpha)L_1(\beta)$ for every pair of disjoint curves $\alpha,\beta$; when $L_2$ is positive, this yields $L_1=kL_2$ for some constant $k\ge 0$. Since hyperbolic length functions and square roots of extremal length functions are Dehn quasi-homothetic functions of the same type, Corollary 3 follows: if two points in Teichmüller space induce the same oriented graph through either hyperbolic or extremal length, the points are equal. The paper also proves Theorem 5: if a function has finite sublevel sets, every automorphism of its oriented graph is induced by a self-homeomorphism of the surface; for hyperbolic and extremal lengths, every such automorphism is induced by an isometry.

Load-bearing premise

The paper assumes, in a single unproved sentence, that for any two disjoint curves—even two that each cut the surface into two pieces—there are always non-separating curves arranged in the exact square or triangle pattern the proof needs. If that pattern is ever missing, the proportionality argument would not go through.

Editorial extensions

If this is right

  • Knowing only which of any two disjoint simple closed curves is longer—in hyperbolic or extremal length—uniquely determines the hyperbolic metric on the surface.
  • The oriented graphs induced by hyperbolic length and extremal length are separately injective and their images in the space of oriented graphs are disjoint.
  • For any function with finite sublevel sets, every automorphism of its oriented graph is induced by a self-homeomorphism of the surface, so the graph remembers topological structure.
  • For hyperbolic and extremal length functions, automorphisms of the oriented graph are exactly isometries of the surface.
  • The same type condition unifies hyperbolic length and square-root extremal length, so proportional growth behavior is enough for the rigidity conclusion.

Reading between the lines

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

  • The paper leaves open whether two Dehn quasi-homothetic functions with genuinely different growth types can induce the same oriented graph; if such examples exist, they would mark the boundary of this rigidity phenomenon.
  • Because the argument relies only on the abstract graph-theoretic Theorem 22, the same approach may apply to other complexes with systems of Dehn twists, yielding rigidity for other natural families of length-like functions.
  • The lemmas relating graph distance to intersection number suggest that the oriented graph may encode more than order—possibly intersection data up to a monotone transform—which would make it an even richer discrete invariant.
  • A natural testable extension is whether a finite subgraph of the oriented graph, rather than the whole graph, already determines the hyperbolic metric; the paper does not address this quantitative question.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper introduces Dehn quasi-homothetic functions on the curve complex of a closed surface and studies the oriented graph they induce on disjoint simple closed curves. The main rigidity theorem (Theorem 2) states that two non-negative Dehn quasi-homothetic functions of the same type that induce the same oriented graph must be proportional; an immediate consequence (Corollary 3) is that the oriented graph induced by hyperbolic length or by extremal length determines the hyperbolic metric on the surface. The paper also proves (Theorem 5) that every automorphism of the oriented graph induced by a function with finite sublevel sets is induced by a self-homeomorphism of the surface, and that for hyperbolic or extremal length functions the automorphisms are induced by isometries (Corollary 6). The general graph-theoretic framework is developed in Theorems 20 and 22, and the applications to length functions are given in Section 4.

Significance. If fully established, the main result is a striking and novel rigidity statement: relative comparisons of lengths of disjoint simple closed curves, a very coarse piece of data, determine the hyperbolic metric and, separately, the extremal length structure. The paper also contributes a clean abstract framework—Dehn quasi-homothetic functions on arbitrary infinite graphs—that may be useful beyond the curve complex. The algebraic core, especially Theorems 20 and 22, is carefully written and appears correct. However, the specialization to the curve complex rests on an unproved geometric configuration assertion, and the proof of Theorem 5 relies on several 'see Figure' existence claims. These gaps are load-bearing for the paper's central claims, so the current version is not yet complete.

major comments (3)
  1. [§3.1, proof of Theorem 2] The proof of Theorem 2 consists of the single sentence 'Setting G=C(S) and choosing W to be the set of non-separating curves, we see that Theorem 22 specializes to Theorem 2.' No verification is given that the two configuration hypotheses of Theorem 22 are satisfied for C(S) with W the non-separating curves. These hypotheses require, for every edge {α,β} in C(S), the existence of non-separating curves arranged in the specified square or triangle pattern, including the cases where one or both of α and β are separating. This is a genuine geometric statement, not a purely formal reduction, and the proportionality conclusion of Theorem 2 and hence Corollary 3 depend on it for all edges of the curve complex. The authors should supply a proof or a precise lemma establishing these configurations.
  2. [§3.2, proof of Theorem 5, pages 20–25] The proof of Theorem 5 relies on several curve configurations that are only justified by reference to figures: (i) the existence of τ disjoint from α, α2, ..., α3g−4 with i(β,τ)>0 in Figures 4 and 5; (ii) in the first case, the existence of α'_m and ξ' with the five-vertex pattern of Figure 6; (iii) in the second case, the existence of distinct ξ, η with the four-vertex pattern of Figure 7. These existence statements are used essentially in the contradiction arguments, and no construction or proof is provided. The paper should include explicit constructions or a lemma showing that such curves exist for an arbitrary system of decomposing curves and arbitrary automorphism images, rather than appealing to the figures.
  3. [§3.2, Lemma 26, page 18] In the proof of the 'only if' direction, it is asserted that for a non-separating α and distinct β1, β2 with i(α,β1)=i(α,β2)=0, there exists a curve γ0 with i(α,γ0)=0 and i(β1,γ0)>0, i(β2,γ0)>0. This existence is not proved and is not immediately obvious, since the relative positions of β1 and β2 in the connected surface S∖α can vary. The construction of the infinite sequence {γ_k} depends on this assertion, and the lemma is used later to show that automorphisms preserve separating versus non-separating curves (Corollary 27), which is itself used in the proof of Theorem 5. A proof or reference for this geometric fact is needed.
minor comments (4)
  1. [§5, line 1] The word 'donote' should be 'denote' in the phrase 'Let DQL(S) donote the space'.
  2. [Definition 1 and Definition 19] The diagram notation such as 'G[α,β,ξ,η] = α β / η ξ' is used without a verbal explanation of which pairs are adjacent; a short sentence defining the square and triangle patterns would improve readability.
  3. [§4.1, first case] The derivation that k=1 from dTh(X,Y)≥0 and dTh(Y,X)≥0 is correct but could be spelled out more explicitly, since the paper writes only 'k=1 since dTh≥0'.
  4. [§5, last paragraph] The statement that DQL(S) equipped with the Thurston metric embeds Teichmüller space isometryrically is not proved or cited in this paper; as written it is an announcement, not a result.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main rigidity theorem is derived from external asymptotic and automorphism theorems; the sole weakness is an unproved combinatorial specialization, not a circular step.

full rationale

The paper does not exhibit a circular derivation. Theorem 2 is proved from the abstract graph statements Theorem 20 and Theorem 22, whose assumptions (Dehn quasi-homothetic of the same type and equal oriented graph) do not contain the proportionality conclusion; the conclusion is derived, not assumed. The asymptotic type (f, A) is verified independently for hyperbolic length by Slegers' Lemma 14 and for the square root of extremal length by Kerckhoff's Proposition 15, so no parameter is fitted from the oriented-graph data and then relabeled as a prediction. The citations to Ivanov, Kerckhoff, Slegers, Kahn-Pilgrim-Thurston, and Parlier-Vo-Xu are external theorems used as premises, and the reference list contains no self-citations by the present authors; hence no self-citation chain is load-bearing. The only noticeable weakness is the one-line reduction in the proof of Theorem 2: 'Setting G=C(S) and choosing W to be the set of non-separating curves, we see that Theorem 22 specializes to Theorem 2' leaves the square/triangle configuration hypotheses for W unverified. That is a possible completeness or correctness gap in a geometric configuration lemma, not circularity, because the hypotheses of Theorem 22 do not assume the proportionality statement they are used to prove. Theorem 5 similarly reduces to Ivanov's automorphism theorem and standard facts about surfaces. Accordingly, the derivation chain is not circular, and the honest finding is a non-finding for circularity.

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

The paper introduces no fitted constants and no new postulated physical or mathematical entities in the sense of added structure; the Dehn quasi-homothetic condition is a new definition, not an entity. The central proof relies on standard external theorems and on an unproved geometric configuration assertion about the curve complex.

assumptions (7)
  • standard math Ivanov's theorem: every automorphism of the curve complex C(S) is induced by an element of the extended mapping class group Mod±(S).
    Used as the target for Theorem 5: to show an automorphism of G_L is induced by a homeomorphism, it suffices to show it preserves disjointness. Cited as [Iva97].
  • standard math Kerckhoff's theorem: the Teichmüller distance between X and Y is 1/2 ln sup_α Ext_Y(α)/Ext_X(α).
    Used in the proof of Corollary 3 (second case) and Corollary 6 to turn proportionality of extremal lengths into equality of points in Teichmüller space. Cited as [Ker80, Theorem 4].
  • standard math The 9g-9 theorem: there exist 9g-9 curves whose hyperbolic lengths determine the point in Teichmüller space.
    Used in Corollary 6 to conclude that if a homeomorphism preserves all hyperbolic lengths, it is an isometry. Cited as Proposition 7.11 of [FLP12].
  • standard math Slegers' estimate: for a non-positively curved metric, L_ρ(T_α^k β) = k i(α,β) L_ρ(α) + O(1).
    Used in Subsection 2.6 to show hyperbolic length functions are Dehn quasi-homothetic of type (k↦k, (α,β)↦i(α,β)). Cited as [Sle22, Lemma 2.2].
  • standard math Kerckhoff's continuity of extremal length on MF: Ext_X extends continuously to measured foliations and is homogeneous of degree 2.
    Used to show sqrt(Ext_X) is Dehn quasi-homothetic of the same type. Cited as [Ker80, Proposition 3].
  • standard math The curve complex C(S) is connected and its automorphism group is Mod±(S) (Ivanov).
    Connectivity is used for the 'if G is connected' part of Theorem 2 and Theorem 22; the automorphism theorem is used in Theorem 5.
  • ad hoc to paper Geometric fact: for any edge in the curve complex, there exist non-separating curves forming the square and triangle configurations required by Theorem 22 conditions (1) and (2).
    This is the paper's unproved geometric input. It is asserted in the proof of Theorem 2 ('Setting G=C(S) and choosing W to be the set of non-separating curves, we see that Theorem 22 specializes to Theorem 2') without a proof or reference. This is the weakest assumption.

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Pith. "Pith review of Oriented graphs on curve complex I: hyperbolic and extremal length." pith.science (2026). https://pith.science/paper/KRRHNGJ5

@misc{pith2026250712728,
  author       = {Pith},
  title        = {Pith review of: Oriented graphs on curve complex I: hyperbolic and extremal length},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KRRHNGJ5}},
  note         = {Machine review of arXiv:2507.12728}
}
abstract

We investigate oriented graphs based on the curve complex $C(S)$ of a closed surface $S$ and induced by functions on the vertex set of $C(S)$. In particular, we introduce the Dehn quasi-homothetic functions, which behave similarly to homotheties under repeated Dehn twists. We prove that any two positive such functions of the same type induce different oriented graphs unless they are proportional. This leads to a new rigidity result for closed hyperbolic surfaces -- distinct from the $9g-9$ theorem and length spectrum rigidity -- knowing only for any two disjoint simple closed curves which one is longer (in terms of hyperbolic or extremal length) suffices to determine the hyperbolic metric on the surface. We also prove that each automorphism of the oriented graph induced by a function with sublevel sets finite is induced by a self-homeomorphism of $S$.

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