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arxiv: 2606.24836 · v1 · pith:PKRMXHWDnew · submitted 2026-06-23 · 🧮 math.GT

Lengths of simple closed geodesics on hyperbolic surfaces in prescribed homology classes

Igor M. Patsankov This is my paper

Pith reviewed 2026-06-25 21:24 UTC · model grok-4.3

classification 🧮 math.GT
keywords hyperbolic surfacessimple closed geodesicshomology classeslength spectrumMirzakhani asymptoticsgeodesic countingfinite-type surfaces
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The pith

The number of simple closed geodesics of length ≤ L in a fixed primitive homology class grows at least as L to the power 6(g-1) + 2(n + b - 1).

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

The paper proves a polynomial lower bound on h_S(L, x), the number of simple closed geodesics of length at most L that represent a fixed primitive nonzero homology class x on a hyperbolic surface S. The bound holds for any surface of genus g ≥ 1 with n punctures and b geodesic boundaries satisfying g + n + b ≥ 3, and improves in the special case of the torus with two punctures. A sympathetic reader cares because the result shows that fixing the homology class still produces a power-law growth whose exponent is controlled by the surface topology, refining the general upper bound coming from Mirzakhani's count of all simple closed geodesics.

Core claim

For a surface S of genus g with n punctures and b geodesic boundary components (g ≥ 1, g + n + b ≥ 3), there exists C1 > 0 such that h_S(L, x) ≥ C1 L^{6(g-1) + 2(n + b - 1)} for all sufficiently large L. In the special case of the surface S_{1,2}, the inequality improves to h_{S_{1,2}}(L, x) ≥ C2 L^{3.011057…} for large L.

What carries the argument

The counting function h_S(L, x) that tallies simple closed geodesics of length ≤ L in a fixed primitive nonzero homology class x, together with a lower-bound construction that produces enough such geodesics from the surface's pants decomposition or curve complex structure.

If this is right

  • The Mirzakhani upper bound of order L^{6(g-1) + 2n} is not asymptotically sharp once boundary components are present.
  • The exponent in the growth rate of geodesics in a fixed homology class explicitly depends on the number of geodesic boundaries.
  • For the surface S_{1,2} the count grows at a rate strictly between L^3 and L^4.
  • The homology constraint reduces the total count but preserves a polynomial growth whose degree is determined by the Euler characteristic adjusted for boundaries.

Where Pith is reading between the lines

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

  • The true leading exponent for h_S(L, x) may coincide with the lower-bound exponent derived here.
  • Similar lower-bound techniques could apply to counts of geodesics subject to other linear constraints in homology or in the curve complex.
  • Explicit constructions on low-genus surfaces with boundaries could be used to test whether the constant C1 can be made effective.

Load-bearing premise

The lower-bound construction succeeds uniformly for every complete finite-area hyperbolic metric on a surface of the given topological type and for every primitive nonzero homology class.

What would settle it

A concrete surface S of the stated type and a primitive nonzero homology class x for which h_S(L, x) remains o(L^{6(g-1) + 2(n + b - 1)}) as L tends to infinity.

Figures

Figures reproduced from arXiv: 2606.24836 by Igor M. Patsankov.

Figure 1
Figure 1. Figure 1: Bounding pair. Let [𝛼] = 𝑥. Note that [𝛼 ′ ] = [𝛼], since these geodesics together separate the surface. Consider the auxiliary surface 𝑆 2 2 , obtained from 𝑆3,0 by cutting along 𝛼. Then, using the corollary from Mirzakhani’s theorem (see Theorem 3.1), we obtain that for sufficiently large 𝐿 the following inequalities hold: 1 𝐶 𝐿 10 ≤ 𝑠 2 2 (𝐿, 𝛼′ ) ≤ 𝐶𝐿10 . Moreover, for any geodesic 𝛾 ∈ Mod2 2 · 𝛼 ′ , w… view at source ↗
Figure 2
Figure 2. Figure 2: Iterative bound. 5 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Graphs of the function 𝛼(𝐿, 𝑥) = log2 (︁ ℎ1,2(2𝐿,𝑥) ℎ1,2(𝐿,𝑥) )︁ for various 𝑥. We shall use the following notation. Let 𝑔, ℎ : R → R be functions. Then 𝑔 ≪ ℎ if and only if there exist constants 𝑀, 𝐿0 > 0 such that for any 𝐿 ≥ 𝐿0, we have |𝑔(𝐿)| ≤ 𝑀ℎ(𝐿). If both 𝑔 ≪ ℎ and ℎ ≪ 𝑔 hold, we write 𝑔 ≍ ℎ. We now briefly outline the structure of the paper. In Section 2 we collect the necessary background materia… view at source ↗
Figure 5
Figure 5. Figure 5: The curves 𝛾(2,0) and 𝛾(0,1) Lemma 2.5. Let Dehn–Thurston coordinates corresponding to the quintuple (𝛼, 𝑏, 𝑣, ℎ−1, ℎ1) be fixed on the set ℳ𝒞4 0 . Denote by 𝛾(𝑚,𝑡) the integral multicurve on 𝑆 4 0 whose coordinates are (𝑚, 𝑡). Let 𝑇𝛾(0,1) and 𝑇𝛾(2,0) denote the left Dehn twists along the simple closed curves 𝛾(0,1) and 𝛾(2,0). Then 𝑇 ±1 𝛾(0,1) (︀ 𝛾(𝑚,𝑡) )︀ = 𝛾(𝑚,𝑡∓𝑚) , (2.1) 𝑇 ±1 𝛾(2,0) (︀ 𝛾(𝑚,𝑡) )︀ = 𝛾 (… view at source ↗
Figure 6
Figure 6. Figure 6: The action of SR(·). Since 𝛾 ′ (0,1) = 𝛾(2,0) and 𝛾 ′ (2,0) = 𝛾(0,1), the analogous statement in the primed coordinate system for 𝑚′ > 0 is 𝛾 ′ (𝑚′ ,𝑡′) = {︃ SR (︀ 𝑡 ′𝛾(2,0), 𝑚′ 2 𝛾(0,1))︀ , 𝑡′ > 0; SR (︀𝑚′ 2 𝛾(0,1), |𝑡 ′ |𝛾(2,0))︀ , 𝑡′ < 0. Combining the last two systems, we obtain the following change of coordinates formula for 𝑚𝑡 ̸= 0: {︃ 𝑚′ = 2|𝑡|, 𝑡 ′ = −𝜃(𝑡) 𝑚 2 . A direct check shows that the same c… view at source ↗
Figure 7
Figure 7. Figure 7: A top-dimensional cell in the case of 𝑆 2 1 . in [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The action of Surger(·). The function Drain is defined analogously to the case described in [2, Section 5]. Let 𝛾 = ∑︀ 𝑖 𝑎𝑖𝛾𝑖 be a cycle. Denote by {𝑅𝑖} the set of oriented embedded subsurfaces of 𝑆 𝑏 𝑔,𝑛 (the orientation on 𝑅𝑖 is the same as that on 𝑆 𝑏 𝑔,𝑛) whose oriented boundaries belong to the support of the cycle 𝛾. Let {𝑅1, . . . , 𝑅𝑘} be the subset of {𝑅𝑖} consisting of subsurfaces whose boundaries… view at source ↗
Figure 9
Figure 9. Figure 9: Intersection of the geodesics 𝛾0 and 𝛾2. 2.4. The collar theorem and its consequences. Let H2 denote the Lobachevsky plane (the hyperbolic plane). Let 𝛼, 𝛽 ⊂ H2 be geodesics intersecting at an ideal point 𝑐. Denote by 𝑝𝛽 the point of intersection of the geodesic 𝛽 with the boundary at infinity. Drop a perpendicular 𝑠 from 𝑝𝛽 to 𝛼 and denote its foot by 𝑞. Then there exists a unique horocycle ℎ with center … view at source ↗
Figure 10
Figure 10. Figure 10: Construction of a spike. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Construction for Theorem 3.5. Hence, on 𝑆𝑔,𝑛 there exist at least 𝑐𝐿6(𝑔−1)+2(𝑛−1) curves representing the class 𝑥 whose length does not exceed 𝐿. To complete the proof, it remains to use Theorem 2.2, with which we will show that after gluing the surface, the resulting homotopy classes of curves remain distinct. The Dehn–Thurston coordinates, compatible with the set 𝛼 ∈ 𝒫, are the same for curves not inter… view at source ↗
Figure 12
Figure 12. Figure 12: A hyperbolic hexagon. Lemma 4.4. Let a right-angled hyperbolic hexagon be given (see [PITH_FULL_IMAGE:figures/full_fig_p021_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Figure for Lemma 4.6. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_13.png] view at source ↗
read the original abstract

A classical question in the theory of hyperbolic surfaces is the study of lengths of closed geodesics under various constraints. A celebrated result in this area is M. Mirzakhani's asymptotic formula for the number of simple closed geodesics of length $\le L$ on a hyperbolic surface of genus $g$ with $n$ punctures. We investigate the number of simple closed geodesics of length $\le L$ representing a fixed primitive nonzero homology class $x$ on a hyperbolic surface $S$. We denote this number by $h_{S}(L, x)$. It follows from Mirzakhani's result that $h_{S}(L, x) \le C L^{6(g-1) + 2n}$. However, numerical evidence suggests that this bound is apparently not asymptotically sharp. We prove that for a surface $S$ of genus $g$ with $n$ punctures and $b$ geodesic boundary components, under the condition that $g \ge 1$ and $g+n+b \ge 3$, there exists a constant $C_1 > 0$ such that for sufficiently large $L$ the inequality \[ h_{S}(L, x) \ge C_1 L^{6(g-1) + 2(n + b-1)} \] holds. In the special case of a torus with two punctures $S_{1, 2}$, we obtain the following stronger result: there exists a constant $C_2 > 0$ such that for sufficiently large $L$ the inequality \[ h_{S_{1, 2}}(L, x) \ge C_2 L^{3.011057 \ldots } \] holds.

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 / 3 minor

Summary. The paper proves lower bounds for the counting function h_S(L, x) counting simple closed geodesics of length ≤ L in a fixed primitive nonzero homology class x on a hyperbolic surface S of genus g ≥ 1 with n punctures and b geodesic boundaries (g + n + b ≥ 3). It establishes the existence of C_1 > 0 such that h_S(L, x) ≥ C_1 L^{6(g-1) + 2(n + b - 1)} for all sufficiently large L. For the special case S_{1,2} a stronger lower bound holds with exponent approximately 3.011057. This is positioned as an improvement over the Mirzakhani upper bound of order L^{6(g-1) + 2n}.

Significance. If the estimates hold, the result supplies the first explicit polynomial lower bounds of this form for constrained geodesic counts, showing that the Mirzakhani upper bound is not asymptotically sharp when boundary components are present and providing a refined exponent for S_{1,2} via a specialized counting argument. The uniform applicability to all qualifying surfaces and primitive classes strengthens the understanding of homology-constrained geodesic growth on finite-type hyperbolic surfaces.

minor comments (3)
  1. [Abstract] Abstract: the numerical evidence for non-sharpness of the Mirzakhani bound is invoked without any indication of the surfaces, homology classes, or computational method used; a one-sentence clarification would aid readers.
  2. [Introduction / § on S_{1,2}] The transition from the general exponent 6(g-1) + 2(n + b - 1) to the refined 3.011057… exponent for S_{1,2} is stated without an explicit comparison of the two constructions; a short remark in §1 or the S_{1,2} section would clarify the improvement.
  3. [Theorem statement] Notation: the constant C_1 is asserted to exist but its dependence on S and x is not discussed; a brief sentence on uniformity would be helpful.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The report correctly captures the main results on lower bounds for h_S(L, x) improving Mirzakhani's upper bound when boundaries are present, along with the specialized exponent for S_{1,2}. No major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity; lower bound from independent construction

full rationale

The paper establishes a lower bound on h_S(L, x) via an explicit counting construction that produces sufficiently many simple closed geodesics in a fixed primitive homology class. This construction and the resulting exponent 6(g-1) + 2(n + b - 1) are derived from the topology of the surface and the properties of the hyperbolic metric; they do not reduce to fitted parameters, self-definitions, or load-bearing self-citations. Mirzakhani's upper bound is cited only for context and is not used in the lower-bound argument. The special-case exponent for S_{1,2} likewise arises from a refined counting argument internal to the paper. No step equates a prediction to its own input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on Mirzakhani's theorem as background and on standard facts about hyperbolic metrics and homology on finite-type surfaces; no free parameters or new entities are introduced in the abstract statement.

axioms (2)
  • domain assumption Mirzakhani's asymptotic formula for the total number of simple closed geodesics holds
    Provides the upper-bound context against which the new lower bound is compared.
  • standard math Every primitive nonzero homology class on a finite-type hyperbolic surface contains geodesic representatives
    Background fact required for the counting problem to be well-posed.

pith-pipeline@v0.9.1-grok · 5848 in / 1441 out tokens · 46598 ms · 2026-06-25T21:24:01.612434+00:00 · methodology

discussion (0)

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Reference graph

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