On Translation Lengths of Anosov Maps on Curve Graph of Torus

Changsub Kim; Hyungryul Baik; Hyunshik Shin; Sanghoon Kwak

REVIEW 3 major objections 4 minor 23 references

On Translation Lengths of Anosov Maps on Curve Graph of Torus

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Every Anosov map of the torus has a bi-infinite geodesic axis in the Farey graph, making its stable translation length a computable positive integer.

desk verdict The integer-translation-length theorem for Anosov maps on the Farey graph is plausible and likely true, but the paper's main construction rests on a genuine proof gap in Proposition 13 that must be filled before the theorem is fully secured. read the letter →

arxiv 1908.00472 v1 pith:4LTRO6KK submitted 2019-08-01 math.GT math.DS

Hyungryul Baik , Changsub Kim , Sanghoon Kwak , Hyunshik Shin This is my paper · ORCID

classification math.GTmath.DS

keywords anosovtranslationcurvegraphlengthmapslengthsnumber

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 works with the Farey graph, a network whose vertices are all rational numbers with infinity, and whose edges connect pairs of rationals whose difference is a unit fraction. This graph is the curve graph of a torus: each rational corresponds to an essential loop on the torus, and an edge means the two loops meet exactly once. Any area-preserving linear map of the torus with an expanding and a contracting direction is called Anosov; it acts on the Farey graph like a hyperbolic isometry of the hyperbolic plane. For any such map, the authors show there is a two-way infinite path in the Farey graph, a geodesic axis, along which the map acts by shifting. Because every edge has length 1, the distance the map shifts along an axis is an integer. The proof is constructive: the axis is found by recording the sequence of triangles of the Farey tessellation crossed by the map's hyperbolic axis, compressing that sequence into a periodic pattern, and reading off a shortest efficient path in one period. The paper supplies two algorithms, one to generate the periodic pattern and one to correct an off-by-one parity issue, and it demonstrates the computation on two examples. The authors then use the integer translation length to settle three further questions: which word in two Dehn twists minimizes translation length, how the curve-graph length compares with the Teichmüller translation length, and how many conjugacy classes share the same translation length.

Extended reading notes

Core claim

Theorem 18: For any Anosov map f, there exists a bi-infinite geodesic P in the Farey graph F on which f acts by translation. Consequently the stable translation length of an Anosov map on the curve graph of the torus is always a positive integer, and it can be computed exactly by the ladder-and-efficient-geodesic algorithm of Section 5.

Load-bearing premise

The proof of the axis theorem rests on Proposition 13, which states that in a periodic ladder one can find an efficient geodesic in a prime subladder whose bi-infinite concatenation is again an efficient geodesic. The proof is a case analysis of 'reluctant moves' at the ends of the subladder (Section 3.3, proof of Proposition 13, including Lemma 14); if the concatenation is not efficient at the period boundary, the invariant bi-infinite geodesic in the Farey graph would not be constructed. This is a load-bearing combinatorial step, distinct from the geometric claim itself.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper proves that every Anosov map on the torus curve graph has a bi-infinite geodesic axis, hence integral stable translation length, with a constructive algorithm. That is a clean result, and the worked examples check out. The ladder/cutting-sequence setup is clearly explained, and the applications (minimal word, ratio bound, length spectrum density) are reasonable. I think the main theorem is likely true.

What is actually new is the geodesic-axis statement in the Farey graph and the exact-integer consequence, plus the algorithmic computation. The machinery overlaps heavily with Series's cutting sequences and the Beardon–Hockman–Short ancestor paths, which the paper cites honestly. That does not make the result trivial, but the novelty is moderate and incremental.

Where the paper gets shaky is Proposition 13, the concatenation lemma for efficient geodesics in periodic ladders. Theorem 18 depends on it, and the proof is a quick case analysis with Figure 9 doing a lot of work. The stress-test note is right: Lemma 14 does not fully pin down every configuration where the two candidate geodesics meet, and the 'all coefficients are 1' contradiction is asserted rather than shown. This is the single load-bearing step, and it is the least rigorous part of the paper. I do not see a counterexample, and the examples work, so I would call it a proof gap rather than a false result. A referee should demand a complete proof.

Smaller issues: a few notational slips (the 277/227 matrix in Example 1), and Theorem 23's proof is only a few sentences and handwaves the key comparison. Those are fixable. The trace-count argument in Theorem 29 also glosses over the PSL vs SL conjugacy class count, though the conclusion is likely right.

This paper is for anyone working on curve graphs, Farey graph geodesics, or translation-length rigidity. It deserves a serious referee; with the Proposition 13 gap filled and the typos cleaned up, it would be a solid contribution. I would send it out rather than desk reject.

Referee Report

3 major / 4 minor

Summary. The paper studies the stable translation length of Anosov (hyperbolic) elements of PSL(2,Z) acting on the Farey graph of the torus. The main result (Theorem 18) asserts that every Anosov map has a bi-infinite geodesic axis in the Farey graph, so its stable translation length is a positive integer; the proof is constructive and the authors give an algorithm for computing it from the continued-fraction expansion of a fixed point. The framework is a combinatorial object called a ladder, together with 'efficient' geodesics satisfying a local move rule. Three applications are then developed: identifying a minimal word among products of two Dehn twists, bounding the ratio of Teichmuller to curve-graph translation lengths, and showing the length spectrum is evenly spread.

Significance. If the central theorem is established, this is a clean sporadic-case strengthening of Bowditch's rationality result: on the Farey graph the stable translation length is not merely rational but an integer. The constructive ladder-and-efficient-geodesic method is a useful feature, and the two worked examples give concrete integer lengths. The paper is self-contained against standard facts on cutting sequences and continued fractions, and the main construction has no fitted parameters. However, the central combinatorial concatenation lemma (Proposition 13) is not yet proved rigorously, and since Theorem 18 depends directly on it, the paper currently does not fully secure the advertised result.

major comments (3)

[§3.3, Proposition 13 and Lemma 14] Proposition 13 is load-bearing for Theorem 18, but its proof is incomplete. The proof asserts, without derivation, that the only possible obstruction to efficiency of the bi-infinite concatenation is a 'reluctant move' at the semi-final-to-final transition; that there are exactly two maximal efficient geodesics in a prime subladder with endpoints on the same side; and that if both candidates are problematic then the first and last coefficients of L' are 1. Figure 9 is used in place of a complete case analysis for the last assertion. Lemma 14, which is supposed to resolve the remaining case, only treats the situation in which the two geodesics do not intersect; it concludes that both are p...p and all coefficients are 1. It does not analyze intersections that occur after different earlier choices, and it does not establish the final claim in the proof of Proposition 13 that an all-1 ladder contradicts the presence of reluctant moves, since the proof does not rule out the possibility that an efficient path in an all-1 finite ladder contains a final t-move. A complete proof of this lemma is required before Theorem 18 can be accepted.
[§4, Theorem 17] The proof of Theorem 17 asserts that L' is a finite concatenation of copies of the prime subladder L''. This is needed to apply the 'more generally' clause of Proposition 13 in Theorem 18. The common-divisor argument states that minimality of the prime subladder forces the common divisor d to be odd, and then infers from the even length of a prime subladder that p and f(p) lie on different sides of L. Neither the minimality assertion nor the side-switching inference is proved. Please replace this passage with a formal argument, or prove directly that the f-translate of a rung in the periodic part is separated from the rung by an integral number of prime subladders.
[§5, algorithm after Proposition 21] The correctness of the computation algorithm is not established. After Proposition 21, the paper asserts that if the subladder ~L has even length then p is a pivot point and ~L is a finite concatenation of a prime subladder, and that if it has odd length then calibration (Algorithm 2) converts it into an even-length subladder whose efficient geodesic computes the translation length. No proof is supplied that the calibrated ladder is of the required form or that the length of its efficient geodesic equals l_C(f) rather than a multiple of it. Since the advertised contribution includes an algorithm that computes the exact translation length, this missing justification should be supplied.

minor comments (4)

[Abstract and Introduction] The abstract says f acts 'transitively' on the invariant geodesic; the precise statement, as in Theorem 18, is that f acts by translation. Please correct this wording.
[Example 1] The displayed matrix in the text is (277 60; 337 73), but the figure caption and the stated fixed points (77±sqrt(26149))/337 correspond to the matrix (227 60; 337 73). Please correct this inconsistency.
[§3.3, Proposition 11] The proof of Proposition 11 uses the undefined term 'locus' and dismisses endpoint cases as 'obvious'. Please define the term and expand the exceptional cases, since the local-to-global geodesic argument is not fully formal.
[§2, Lemma 3] Lemma 3 is cited with the one-line proof 'Refer to [Ser15]'. Since the lemma is used in the proof of Theorem 17, a precise reference to the statement in [Ser15] or a short self-contained explanation would be helpful.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central theorem is built on classical continued-fraction, cutting-sequence, and Farey-graph facts cited from the literature; the paper's own constructs (ladder, efficient moves, calibration) are explicitly defined and proved, so they are not postulated inputs. No free parameters or invented entities appear.

assumptions (5)

standard math Lagrange's theorem: a continued fraction is periodic if and only if it represents a quadratic irrational (cited as [Ste92]).
Used in Theorem 17 to get periodicity of the cutting sequence from the quadratic irrational fixed points of f.
standard math Cutting-sequence exponents equal the coefficients of the positive continued fraction (Proposition 1, taken from [Ser15]).
Basis for identifying the ladder type with the cutting sequence in Proposition 5, hence for the periodicity of the ladder.
standard math Two oriented geodesics with the same endpoint eventually have the same cutting sequence (Lemma 3, taken from [Ser15]).
Used to pass from the eventually periodic cutting sequence of a fixed point to a periodic subladder in Theorem 17.
standard math Every geodesic path between vertices x,y of the Farey graph has each internal vertex incident to an edge separating x,y (Fact 7, taken from [Minsky 1996]).
Used in Corollary 8 to prove ladders are geodesically convex, which underpins Proposition 10 and the whole geodesic-in-ladder strategy.
standard math The number H(t) of conjugacy classes in SL(2,Z) with trace t satisfies H(t) > |t|^{1-theta} for every theta>0 (Lemma 30, taken from [CCC80]).
Used in Theorem 29 to bound N(R) from below by a super-polynomial function of log R.

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Pith. "Pith review of On Translation Lengths of Anosov Maps on Curve Graph of Torus." pith.science (2026). https://pith.science/paper/4LTRO6KK

@misc{pith2026190800472,
  author       = {Pith},
  title        = {Pith review of: On Translation Lengths of Anosov Maps on Curve Graph of Torus},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4LTRO6KK}},
  note         = {Machine review of arXiv:1908.00472}
}

read the original abstract

We show that an Anosov map has a geodesic axis on the curve graph of a torus. The direct corollary of our result is the stable translation length of an Anosov map on the curve graph is always a positive integer. As the proof is constructive, we also provide an algorithm to calculate the exact translation length for any given Anosov map. The application of our result is threefold: (a) to determine which word realizes the minimal translation length on the curve graph within a specific class of words, (b) to establish the effective bound on the ratio of translation lengths of an Anosov map on the curve graph to that on Teichm\"uller space, and (c) to estimate the overall growth of the number of Anosov maps which have a sufficient number of Anosov maps with the same translation length.

Figures

Figures reproduced from arXiv: 1908.00472 by the authors.

**Figure 1.** Components of Ladder Definition. An endpoint of a ladder is a degree 2 vertex of a ladder. Connect two endpoints with an oriented geodesic g. While recording the cutting sequence of g, call every L or R-labeled vertex as a pivot point. Include two endpoints as pivot points as well. An edge in a ladder is called a rung if its interior and g intersect. The spine K of a ladder L is the path in L with the following prop… view at source ↗

**Figure 2.** Spine for (1, 1)-type ladder. Remark. In fact, if a ladder is other than (1, 1)-type, then its spine is uniquely determined. This is because except for the (1, 1)-type ladder, we can connect each pivot point to another pivot point via the unique rung, except two endpoints. After joining them via rungs, we get a path P 0 in the ladder except for two endpoints. Now we can attach two endpoints to P in a unique way. Se… view at source ↗

**Figure 3.** How spine can be formed uniquely: Connect each pivot point to [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (11 more)

**Figure 4.** Figure 4: Type of Ladder Definition. We say a ladder is of type (a1, · · · , an) if the ladder has a1, · · · , an consecutive numbers of Farey triangles with the same pivot points read off in the orientation given to the geodesic. As an example, see [PITH_FULL_IMAGE:figures/ful…

**Figure 5.** Figure 5: An orientation of a ladder induces an order of pivot points. The point [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 6.** Figure 6: The pivot point a has two options to move toward adjacent pivot points: b, c. We say a path P in a ladder L satisfies the efficient moving condition if for each move between adjacent pivot points in P, • Move t whenever the move p passes through more than one edge in L…

**Figure 7.** Figure 7: Six paths in a (2, 3, 4, 1, 1, 1)-ladder. Two blue points for each figure indicate the beginning points and the final points of paths. Green segments stand for efficient moves, and red ones for non-efficient moves. Only (a) and (e) satisfy the efficient moving conditio…

**Figure 8.** Figure 8: Illustration of Reluctant move. Within a ladder, there are two maxi [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]

**Figure 9.** Figure 9: Both efficient geodesics have problematic reluctant moves(dotted [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]

**Figure 10.** Figure 10: 4pqr, 4f(p)f(q)f(r): Two triangles with different orientations To show f stabilizes each side of L, it suffices to show f sends a pivot point of L to another pivot point on the same side of L. Assume there is some 15 [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]

**Figure 11.** Figure 11: Periodic part of bi-infinite ladder L. Let L 00 be its prime subladder, with orientation inherited from Af , and p, q be its first two pivot points. As f(p), f(q) are also pivot points of L, Now define L 0 to be the finite subladder of L bounded by two edges pq and f(…

**Figure 12.** Figure 12: Invariant geodesic for f represented by 227 60 337 73 Now we provide a non-standard ladder example which forces to use Proposition 21 to find a rung of the associated ladder. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_12.png]

**Figure 13.** Figure 13: Invariant geodesic for f represented by 65 −56 101 −87 6 Applications 6.1 Minimal Word Shin and Strenner [SS16] showed that there is a non-Penner type pseudo-Anosov mapping class Mod(S) when S is a non-sporadic surface. They also showed this is not the case when S …

**Figure 14.** Figure 14: Description of Process (ii). Note the efficient geodesic(Blue lines) does not change under the process. In both cases, the efficient geodesic never visits red vertices, but always does green ones. Recall that the efficient geodesic in a ladder solely depends on the ty…

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Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Aougab and S

T. Aougab and S. J. Taylor. Pseudo- A nosovs optimizing the ratio of T eichm \" u ller to curve graph translation length. In the tradition of Ahlfors-Bers, VII, Contemporary Mathematics , 696:17--28, 2017

work page 2017

[2] [2]

An extremal problem for quasiconformal mappings and a theorem by T hurston

Lipman Bers. An extremal problem for quasiconformal mappings and a theorem by T hurston. Acta Mathematica , 141(1):73--98, 1978

work page 1978

[3] [3]

A. F. Beardon, M. Hockman, and I. Short. Geodesic continued fractions. Michigan Mathematical Journal , 61(1):133--150, 2012

work page 2012

[4] [4]

Bowditch

Brian H. Bowditch. Tight geodesics in the curve complex. Inventiones mathematicae , 171(2):281--300, 2008

2008

[5] [5]

Is a typical bi- P erron algebraic unit a pseudo- A nosov dilatation? Ergodic Theory and Dynamical Systems , pages 1--6, 2017

Hyungryul Baik, Ahmad Rafiqi, and Chenxi Wu. Is a typical bi- P erron algebraic unit a pseudo- A nosov dilatation? Ergodic Theory and Dynamical Systems , pages 1--6, 2017

work page 2017

[6] [6]

Minimal asymptotic translation lengths of T orelli groups and pure braid groups on the curve graph

Hyungryul Baik and Hyunshik Shin. Minimal asymptotic translation lengths of T orelli groups and pure braid groups on the curve graph. Math. Res. Not. IMRN. , 2018. Electronically published on Dec 4, 2018. (To appear in print)

work page 2018

[7] [7]

Polynomial-time algorithms for the curve graph

Mark C Bell and Richard CH Webb. Polynomial-time algorithms for the curve graph. arXiv preprint arXiv:1609.09392 , 2016

work page Pith review arXiv 2016

[8] [8]

Chowla, J

S. Chowla, J. Cowles, and M. Cowles. On the number of conjugacy classes in SL (2, Z ) . Journal of Number Theory , 12(3):372--377, 1980

work page 1980