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arxiv: 2606.08842 · v1 · pith:VK77C6EPnew · submitted 2026-06-07 · 🧮 math.NT · math.DS

Transcendence of simple geodesics on finite modular covers

Christopher-Lloyd Simon This is my paper

Pith reviewed 2026-06-27 17:45 UTC · model grok-4.3

classification 🧮 math.NT math.DS
keywords simple geodesicsmodular coversgeodesic laminationstranscendencefinite index subgroupshyperbolic geometrymodular groupPSL(2,Z)
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The pith

If a simple geodesic on a finite modular cover belongs to a minimal lamination, its forward endpoint is rational, quadratic, or transcendental.

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

The paper conjectures that the forward endpoint of a simple geodesic on a finite cover of the modular orbifold is rational, quadratic, or transcendental. It proves the statement when the geodesic is a leaf of a minimal geodesic lamination. This matters because it connects the geometric property of simplicity for geodesics in the hyperbolic plane to the algebraic degree or transcendence of their boundary points. The conjecture already holds completely for the modular torus cover associated to the derived subgroup of PSL(2,Z). A reader would care as the result suggests an arithmetic filter on which numbers can serve as endpoints for simple geodesics.

Core claim

We conjecture that if ξ′ is simple, then ξ+ is either rational or quadratic or transcendental. We prove this conjecture for leaves of minimal geodesic laminations. We explain why the conjecture is known for all simple geodesics in the modular torus cover associated to the derived subgroup Γ′=[Γ,Γ]. The geodesics are pairs of points on the projective line, and the finite-index subgroup Γ′ acts on the hyperbolic plane to produce the cover.

What carries the argument

Projection of geodesics from the upper half-plane model to a finite cover of the modular orbifold, restricted to leaves of minimal geodesic laminations.

Load-bearing premise

That the simple geodesics considered are leaves of minimal geodesic laminations on the cover.

What would settle it

A simple geodesic on one of the covers whose forward endpoint is an algebraic irrational of degree three or higher would disprove the conjecture.

Figures

Figures reproduced from arXiv: 2606.08842 by Christopher-Lloyd Simon.

Figure 1
Figure 1. Figure 1: Projective model of HP, with its ideal triangulation △ and dual tree T . Fundamental domain under the action of Γ = PSL2(Z). The orbifold M = Γ\HP. The modular orbifold M = Γ\HP has genus zero, a cusp associated to the fixed point ∞ ∈ Q P 1 ⊂ ∂HP of R, as well as two conical singularities of order two and three associated to the fixed points i, j ∈ HP of S and T. Thus Γ = π1(M) is the free amalgam of its s… view at source ↗
Figure 2
Figure 2. Figure 2: The geodesic (0, ξ) mod PSL2(Z) penetrates B(h) ⊂ M each time n ∈ N satisfies 1 2 (⌊ 0, xn−1, . . . , x0 ⌋ + ⌊ xn, xn+1, . . . ⌋) > h. For a geodesic (ξ −, ξ+) ∈ G(M), its Markov constant C(ξ −, ξ+) is the supremum of 2h ∈ R+ such that it is disjoint from B(h). If (ξ −, ξ+) ∈ [−1, 0) × [1, ∞) with −1/ξ− = ⌊ x−1; x−2, . . . ⌋ and ξ + = ⌊ x0, x1, x2, . . . ⌋, the geodesic (ξ −, ξ+) ⊂ HP intersects the ideal … view at source ↗
Figure 3
Figure 3. Figure 3: The embedded graphs T ′ ⊂ M′ when Γ ′ = Γ(N) for N ∈ {2, 3, 4, 5}. Simple geodesics yield bounded continued fraction expansions The (complete oriented) geodesics of HP identify with the space G(HP) = ∂HP×∂HP\diagonal, and those of M′ with its quotient G(M′ ) = Γ′\G(HP) by the diagonal Γ ′ -action. The geodesics ξ ∈ G(HP) with a given projection ξ ′ ∈ G(M′ ) form the Γ ′ orbit of ξ ∈ G(HP) (note that its Γ-… view at source ↗
Figure 4
Figure 4. Figure 4: A geodesic (lamination) carried by T (3) ⊂ Γ(3) yields a train-track structure. A geodesic lamination Ξ ′ ⊂ M′ is minimal when every one of its leaves ξ ′ has both its past and future ξ ± mod Γ′ that are dense in Ξ ′ . By [Bon01, Proposition 3]: every geodesic lamination is a disjoint union of finitely many minimal sublaminations, and finitely many isolated leaves whose ends spiral along the minimal sublam… view at source ↗
Figure 5
Figure 5. Figure 5: The free group PSL2(Z) ′ acts on HP with quotient a cusped torus M′ . The Galois group PSL2(Z)/ PSL2(Z) ′ = Z/6 acts on M′ with quotient M. The homotopy classes of loops in M′ correspond to the conjugacy classes in π1(M′ ), hence to the reduced cyclic words on {A, A−1 , B, B−1}. For n ∈ Z ∗ , the n-th power of the commutator [A, B] corresponds to the loop winding n times around the cusp; every other non-tr… view at source ↗
Figure 6
Figure 6. Figure 6: Action of Γ/Γ ′′ on the honeycomb graph H with base arc [i ′ , j′ ⟩, and cusps Γ ′′\Γ/⟨R⟩. Paths encoded by RLRLL and (RLRLL) 2 in the honeycomb graph H. Let us describe the Galois action of Γ/Γ ′′ on the metabelian cover M′′ → M combining the action of Γ ′/Γ ′′ on M′′ → M′ with the action of Γ/Γ ′ on M′ → M. (Recall that Galois actions are given by left multiplication whereas monodromy actions are given b… view at source ↗
Figure 7
Figure 7. Figure 7: Simple closed geodesics in M′ lift to lines of rational slope in H1(M′ ; R) through i ′ . The corresponding combinatorial hex-paths are unique only up to cyclic permutation. We now recall the notion of Sturmian sequences from [Fog02, Chapter 6] to associate them pair of real numbers (which are transcendental as we will recall 3.18). Definition 3.8 (Sturmian sequences and numbers). Define the set of Sturmia… view at source ↗
read the original abstract

The real projective line $\mathbb{R}\mathbf{P}^1$ is the boundary of $\mathbf{HP}=\{z\in \mathbb{C}\colon \Im(z)>0\}$, a model of the hyperbolic plane whose space of geodesics identifies with $\mathcal{G}(\mathbf{HP})=\mathbb{R}\mathbf{P}^1 \times \mathbb{R}\mathbf{P}^1 \setminus \mathrm{diagonal}$. The modular group $\Gamma=\operatorname{PSL}_2(\mathbb{Z})$ acts on $\mathbf{HP}$ with quotient the modular orbifold $\mathbf{M}=\Gamma\backslash \mathbf{HP}$. Consider a finite-index subgroup of the modular group $\Gamma^\prime \subset \Gamma = \operatorname{PSL}_2(\mathbb{Z})$ corresponding to a finite cover $\mathbf{M} \to \mathbf{M}^\prime$. A geodesic $(\xi^-,\xi^+)\in \mathcal{G}(\mathbf{HP})$ projects $\bmod{\Gamma^\prime}$ to a geodesic $\xi^\prime \subset \mathbf{M}^\prime$. We conjecture that if $\xi^\prime$ is simple, then $\xi^+$ is either rational or quadratic or transcendental. We prove this conjecture for leaves of minimal geodesic laminations. We explain why the conjecture is known for all simple geodesics in the modular torus cover associated to the derived subgroup $\Gamma^\prime = [\Gamma, \Gamma]$.

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 manuscript states a conjecture that if ξ′ is a simple geodesic on a finite modular cover M' of the modular orbifold, then the endpoint ξ+ is rational, quadratic, or transcendental. It proves the conjecture when ξ′ is a leaf of a minimal geodesic lamination and separately recalls that the full statement is already known for all simple geodesics on the modular torus cover arising from the commutator subgroup Γ' = [Γ, Γ].

Significance. The partial result for minimal laminations supplies a concrete, non-vacuous case of the conjectured transcendence dichotomy and connects the dynamics of the PSL(2,ℤ) action on the geodesic space to classical questions in transcendence theory. The explicit delimitation of the proved subclass and the reference to the known torus case are strengths; the argument relies on standard facts about group actions and hyperbolic geometry rather than ad-hoc parameters or self-referential definitions.

minor comments (2)
  1. [Abstract] Abstract, paragraph on the conjecture: the phrasing 'we prove this conjecture for leaves of minimal geodesic laminations' is clear, but the introduction should include a brief sentence indicating how large this subclass is among all simple geodesics on finite covers.
  2. [Abstract] Notation: the symbols HP, G(HP), and M' are introduced with boldface; verify that the same conventions are used uniformly in all subsequent sections and that the diagonal exclusion in G(HP) is recalled when needed.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive assessment of the manuscript. We are gratified that the concrete nature of the result for minimal laminations, its connection to transcendence questions, and the explicit reference to the known torus case were viewed as strengths. The recommendation to accept is appreciated.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper presents a conjecture that the positive endpoint ξ+ of a simple geodesic ξ′ on a finite modular cover is rational, quadratic, or transcendental, and proves the statement only for the subclass of leaves of minimal geodesic laminations. It separately notes that the full statement is already known for the modular torus arising from the commutator subgroup. The provided text contains no equations, parameter fits, or self-referential reductions; the argument relies on standard facts about the action of PSL₂(ℤ) and its finite-index subgroups on the hyperbolic plane and its boundary. No load-bearing step reduces by construction to an input, self-citation chain, or renamed empirical pattern. The derivation is therefore self-contained against external benchmarks in hyperbolic geometry.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract: the work rests on the standard axioms of hyperbolic geometry (upper half-plane model, geodesic identification), the action of PSL(2,Z), and the definition of finite-index subgroups and simple geodesics. No free parameters, invented entities, or ad-hoc axioms are introduced in the provided text.

axioms (2)
  • standard math The upper half-plane with the PSL(2,Z) action yields the modular orbifold whose geodesics are identified with pairs of distinct points on RP^1.
    Invoked in the opening paragraph of the abstract to set up the space of geodesics.
  • domain assumption Finite-index subgroups correspond to finite covers on which projected geodesics can be simple or not.
    Used to formulate the conjecture for M'.

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