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arxiv: 2607.00957 · v1 · pith:BEZ3IUBVnew · submitted 2026-07-01 · 🧮 math.GT

Gromov boundary of the Grand Arc graph

Carolyn Abbott , Assaf Bar-Natan , Arya Vadnere This is my paper

Pith reviewed 2026-07-02 02:59 UTC · model grok-4.3

classification 🧮 math.GT
keywords grand arc graphGromov boundarygeodesic laminationsinfinite-type surfacecurve complexbounded geodesic image theoremnon-compact boundary
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The pith

Geodesic laminations form a dense subset of the Gromov boundary of the grand arc graph on an infinite-type surface.

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

The paper shows that the Gromov boundary of the grand arc graph admits a dense subset that can be identified with the space of geodesic laminations on the surface. This identification follows the same pattern used for the curve complex and depends on first establishing a bounded geodesic image theorem for the grand arc graph. The work further demonstrates that the resulting boundary fails to be compact. A reader would care because the description supplies a concrete topological object that controls the large-scale geometry of the graph. The result therefore opens a route to study infinite-type surfaces through their laminations rather than through the graph alone.

Core claim

We describe a dense subset of the Gromov boundary of the grand arc graph of an infinite-type surface as a space of geodesic laminations, analogous to Klarreich's description of the Gromov boundary of the curve complex. After showing that the grand arc graph satisfies a bounded geodesic image theorem, we also prove that the boundary is not compact.

What carries the argument

The grand arc graph on an infinite-type surface, with its Gromov boundary partially identified with geodesic laminations via the bounded geodesic image theorem.

If this is right

  • The Gromov boundary of the grand arc graph is not compact.
  • The bounded geodesic image theorem holds for the grand arc graph.
  • Geodesic laminations serve as a dense set of points in the boundary.
  • Results known for the curve complex boundary have direct analogues for the grand arc graph.

Where Pith is reading between the lines

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

  • The same lamination description may apply to other arc or curve graphs on infinite-type surfaces once a bounded geodesic image theorem is verified.
  • Non-compactness of the boundary could constrain the dynamics of the mapping class group action on the graph.
  • The identification supplies a candidate for studying quasi-isometric rigidity questions for these graphs.

Load-bearing premise

The grand arc graph satisfies a bounded geodesic image theorem.

What would settle it

A sequence of grand arcs whose geodesics in the graph fail to remain within bounded distance of a fixed geodesic lamination while approaching a boundary point would falsify the dense-subset identification.

Figures

Figures reproduced from arXiv: 2607.00957 by Arya Vadnere, Assaf Bar-Natan, Carolyn Abbott.

Figure 1
Figure 1. Figure 1: Every bicorn arc between the grand arcs a and b intersects b infinitely many times. There are several reasons that bicorn arcs are better suited for this paper than unicorn arcs. First, in the context of the grand arc graph, when using unicorn arcs, one must be careful about the orientations on the two arcs so that the resulting unicorn arc converges to distinct ends. On the other hand, all bicorn arcs sta… view at source ↗
Figure 2
Figure 2. Figure 2: Constructing a bicorn arc disjoint from a given unicorn arc. [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Constructing a unicorn arc disjoint from a given bicorn arc. [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Constructing a bicorn arc q ∈ B (a ′ , b) intersecting p in at most one point. a b a ′ p q Q P [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Constructing a unicorn arc in U (a ′ , b) disjoint from p. which intersects p in at most one point; see [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Left: Y is a partial witness for A(S, P). Right: W and W ′ are disjoint subsurfaces of S, both of which are witnesses, while Y is a partial witness for S. Example 4.5. Suppose S is an infinite-type surface with |S(Σ)| ≥ 4 and G = G (Σ) is the grand arc graph. If γ is a separating simple closed curve that separates two of the elements of the grand splitting from the rest, then S \ γ consists of two infinite… view at source ↗
Figure 7
Figure 7. Figure 7: Left: γ separates the surface into two partial witnesses. Right: cutting the surface along the collection of curves in red separates it into two partial witnesses, both limiting to all three maximal ends. We will primarily consider witnesses and partial witnesses for G(Σ) or A(S, Γ) for some S ⊆ Σ. When this is the case and no confusion is possible, we abuse terminology and do not specify “for G(Σ)” or “fo… view at source ↗
Figure 8
Figure 8. Figure 8: The bicorn arc x and ∂Y bound a disk, and so x may have essential intersections with arcs of b ∩ Y . a uniform quality quasi-geodesic sequence connecting a boundary curve α of Y with any curve β intersecting Y , such that dY (x, β) is uniformly bounded for each curve x in this quasi-geodesic except the two closest to α. These quasi-geodesics provide shortcuts for the full geodesic in C (Y ). Webb construct… view at source ↗
Figure 9
Figure 9. Figure 9: If l ′ is sufficienlty close to l, then bicorns between a and l are also bicorns between a and l ′ . Definition 6.6. Suppose a ∈ G(Σ), L is a minimal lamination on Σ, and l ∈ L. A sequence (bn)n∈N ⊂ B(a, l) is an ending bicorn sequence between a and L if there exists R ∈ N such that: (1) b0 = a, (2) bn geod −−−→ l, (3) (bn)n∈N is an R–quasi-path, i.e., dG (bn, bn+1) < R for all n ∈ N, and (4) the l-part of… view at source ↗
Figure 10
Figure 10. Figure 10: Constructing the next bicorn bi+1 in Case 1 (left) and Case 2 (right). It remains to show that this sequence (bi)i∈N geodesically converges to l. Pick a lift ˜l of l in the universal cover H2 . Each point of a ∩ l corresponds to a lift of a intersecting ˜l in H2 . The segments li defined above correspond to a nested sequence of segments ˜li along ˜l, and the bicorn arc bi is obtained by joining ˜li to the… view at source ↗
Figure 11
Figure 11. Figure 11: A sequence of bicorn arcs defined by lifts [PITH_FULL_IMAGE:figures/full_fig_p036_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: θ θ − κ a l l P0 [PITH_FULL_IMAGE:figures/full_fig_p037_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Since the length of every pants seam is 1, we can find [PITH_FULL_IMAGE:figures/full_fig_p042_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Since ln[−t, t] and l[−t, t] are close for n ≫ 0, the angle of intersection between a and ln is close to the angle of intersection between a and l. Therefore, the angles of transverse intersections of a and ln[−t, t] are bounded below by θ := θ0 − β1 for all t ≥ 1, and so (2) holds. 42 [PITH_FULL_IMAGE:figures/full_fig_p042_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: The lifts ˜aj ’s of a shrink to the endpoint of ˜ℓ. The curve τ is the concatenation of γ and ˜aj at their unique intersection x. Its straightening ¯τ is the lift of the geodesic representative of (∞x)γ ∪ (x∞)a, and it lands in Ij ⊂ V . Proof of Proposition A.1. Suppose ℓ is not high-filling, or equivalently by the characteriza￾tion in [BW18, Lemma 3.5.4], the infinite unicorn path P(a, ℓ) is bounded in R… view at source ↗
read the original abstract

We describe a dense subset of the Gromov boundary of the grand arc graph of an infinite-type surface as a space of geodesic laminations, analogous to Klarreich's description of the Gromov boundary of the curve complex. After showing that the grand arc graph satisfies a bounded geodesic image theorem, we also prove that the boundary is not compact.

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 / 1 minor

Summary. The paper claims to describe a dense subset of the Gromov boundary of the grand arc graph of an infinite-type surface as a space of geodesic laminations (analogous to Klarreich's theorem for the curve complex), after establishing that the grand arc graph satisfies a bounded geodesic image theorem; it further proves that this boundary is not compact.

Significance. If the results hold, the work extends boundary descriptions from finite-type to infinite-type surfaces via standard hyperbolic geometry techniques for arc graphs, and the non-compactness result distinguishes the infinite-type case. The strategy of proving a bounded geodesic image theorem to identify the boundary with laminations follows established methods in the field.

minor comments (1)
  1. [Abstract] The abstract states the main theorems clearly but supplies no proof outlines or verification steps; the full manuscript should include explicit references to the sections containing the bounded geodesic image theorem proof and the boundary identification argument to facilitate checking.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our work and for recommending minor revision. The referee's summary accurately captures the main results of the paper. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation relies on independent proof of BGIT and external analogy

full rationale

The paper explicitly states it proves the bounded geodesic image theorem for the grand arc graph before using it to identify a dense subset of the Gromov boundary with geodesic laminations, mirroring Klarreich's external result on the curve complex. No self-citations are load-bearing, no parameters are fitted and renamed as predictions, and no ansatz or uniqueness theorem is smuggled in via prior author work. The central claim is derived from a stated internal proof plus standard techniques, making the chain self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review based solely on the abstract; no specific free parameters, axioms, or invented entities can be identified from the given text.

pith-pipeline@v0.9.1-grok · 5569 in / 943 out tokens · 53530 ms · 2026-07-02T02:59:53.705235+00:00 · methodology

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Reference graph

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