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arxiv: 2604.25086 · v1 · submitted 2026-04-28 · 🧮 math.GT

Recognition: unknown

Large flats in large subgraphs of fine curve graphs

Ryan Dickmann , Roberta Shapiro
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Pith reviewed 2026-05-07 14:44 UTC · model grok-4.3

classification 🧮 math.GT
keywords fine curve graphcurve graphhyperbolicityflatssubgraphsisotopy classessurfaces
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The pith

Certain large subgraphs of the fine curve graph contain flats of every finite dimension and are therefore not hyperbolic.

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

The paper establishes that while the full fine curve graph is hyperbolic, some of its large induced subgraphs are not. These include the single-isotopy-class fine curve graphs formed by taking all curves in one fixed isotopy class and connecting those that are disjoint. The central result is that such subgraphs contain flats of every finite dimension. A reader cares because this shows the hyperbolicity of the overall graph depends on mixing curves from different isotopy classes, and the paper supplies distance bounds inside the non-hyperbolic pieces.

Core claim

We show that certain large subgraphs of fine curve graphs, including fibers over a vertex of the curve graph, are not hyperbolic. Indeed, such graphs contain flats of every finite dimension. We then compute bounds on distances in fibers over a vertex of the curve graph, which we call single-isotopy-class fine curve graphs.

What carries the argument

The single-isotopy-class fine curve graph: the induced subgraph on all essential simple closed curves in one fixed isotopy class, with edges between disjoint curves.

If this is right

  • The subgraphs fail to be hyperbolic for any choice of delta.
  • Each such subgraph contains an isometric copy of the n-dimensional integer lattice for every n.
  • Explicit upper and lower bounds on distances between any two curves in the same isotopy class are obtained.
  • Hyperbolicity of the full fine curve graph requires edges that connect different isotopy classes.

Where Pith is reading between the lines

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

  • The construction suggests that any sufficiently large collection of curves closed under isotopy will produce non-hyperbolic subgraphs.
  • One could ask whether the full fine curve graph is quasi-isometrically rigid or whether the flats persist under small perturbations of the surface.
  • Distance bounds inside these fibers might be used to study the geometry of the quotient map from the fine curve graph to the ordinary curve graph.

Load-bearing premise

The surface admits enough pairwise disjoint curves inside a single isotopy class to allow construction of the high-dimensional flats.

What would settle it

A proof that the maximum dimension of any flat inside a single-isotopy-class fine curve graph is bounded would falsify the claim.

Figures

Figures reproduced from arXiv: 2604.25086 by Roberta Shapiro, Ryan Dickmann.

Figure 1
Figure 1. Figure 1: A schematic of the surfaces and maps used in Section 3.1. view at source ↗
Figure 2
Figure 2. Figure 2: The relevant covers and maps used throughout Sections 4 and 5. view at source ↗
Figure 3
Figure 3. Figure 3: An example of how to construct a tree Tγ of elevations of a curve γ. All level 2 lifts of γ are pink. One elevation of γ at level −1 is green. Proof. We will view a curve in S as the image of a map R to S with the image of x equal to the image of x+1 (amounting to quotienting R to create S 1 ). Thus, where appropriate, we may consider a curve to be a map S 1 → S. We will first prove that a free homotopy fr… view at source ↗
Figure 4
Figure 4. Figure 4: The green arcs are in minimal position arcs and project to the same arc in the surface. view at source ↗
Figure 5
Figure 5. Figure 5: An arc in minimal position with γ that intersects the same elevation of S˜ γ twice. The list of levels for the arc is (−1, −1). We say an arc ζ in S˜ (γ) between elevations of γ essentially intersects n elevations if it is homo￾topic relative endpoints to an arc in minimal position that intersects n elevations, including at the endpoints. Lemma 4.5. Let ζ be an arc in the cyclic cover with an endpoint on a… view at source ↗
Figure 6
Figure 6. Figure 6: Left: β and γ are distance 3 curves, and a path between them is γ − ζ1 − ζ2 − β. Right: we see the lift of the curves on the left to the cyclic cover. Notice that Cγ(β) = 3 while Cβ(γ) = 2. are positive and others are negative. By the pigeon hole principle, βˆ intersects level ±⌈ n−1 2 ⌉ = ±⌊ n 2 ⌋ of γ. Without loss of generality, it intersects level m = ⌊ n 2 ⌋. We will produce a level ±m lift of β that … view at source ↗
Figure 7
Figure 7. Figure 7: A schematic of all arcs and points used in the proof of Case 1 of Lemma 5.1 view at source ↗
Figure 8
Figure 8. Figure 8: This is the beginning (Steps 0) of a sequence of point pushes that allows us to achieve the lower bound in Lemma 5.1. More descriptions can be found in Ta￾ble 1. In the top picture, we show the arcs that will be pushed along through￾out. Since β and γ are disjoint, (Cγ(β), Cβ(γ)) = (0, 0). In the second picture, we show the creation of the first tail by push￾ing along the black and purple arcs. In the righ… view at source ↗
Figure 9
Figure 9. Figure 9: Let y ∈ γˆ be the unique lift of p(x) in the compact component of p −1 (γ). Let β2 be the subarc of the unique lift of β that passes through y such that p(β1) = p(β2). Let b1 be the subarc of β1 that has its endpoints in level 0 and level 1 of p −1 (γ). Since p(x) = p(y), there exists a subarc of β2, which we call b2, such that p(b1) = p(b2). We therefore know that b2 also hits exactly 2 levels. To respect… view at source ↗
Figure 10
Figure 10. Figure 10: Pictured are lifts of β (in green) and βn (in blue) of geodesic path β−β1−· · ·−βn−βn+1. If βˆ n+1 is to the right of βˆ n, then it must be in the region bounded by all level 0 and level 1 elevations of βn. This restricts which levels of β the lift βˆ n+1 can intersect. This idea is used in the proofs of Lemma 6.1, Proposition 6.2, and Proposition 6.3. Let x be the endpoint of ζ contained in ζ2. Let x ′ b… view at source ↗
Figure 11
Figure 11. Figure 11: A schematic of the surgery procedure giving the upper bound in Proposition 6.2. Top view at source ↗
read the original abstract

The fine curve graph of a surface is a graph whose vertices are essential simple closed curves and whose edges connect disjoint curves. Following a rich history of hyperbolicity of various graphs associated to surfaces, the fine curve graph was shown to be hyperbolic by Bowden-Hensel-Webb, while the curve graph, obtained from the fine curve graph by collapsing subgraphs corresponding to isotopy classes, was first proven to be hyperbolic by Masur-Minsky. We show that certain large subgraphs of fine curve graphs, including fibers over a vertex of the curve graph, are not hyperbolic. Indeed, such graphs contain flats of every finite dimension. We then compute bounds on distances in fibers over a vertex of the curve graph, which we call single-isotopy-class fine curve graphs.

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

1 major / 3 minor

Summary. The manuscript shows that certain large subgraphs of the fine curve graph, including the fibers over vertices of the curve graph (single-isotopy-class fine curve graphs), are not hyperbolic. This is established by explicit constructions of quasi-flats of every finite dimension inside these subgraphs, using collections of mutually disjoint parallel curves in annular neighborhoods of a fixed curve together with auxiliary curves to control intersection patterns and produce grid-like distance growth. The paper also derives explicit upper and lower bounds on distances within these single-isotopy-class subgraphs.

Significance. The result provides a sharp contrast to the hyperbolicity of the full fine curve graph (Bowden-Hensel-Webb) and the curve graph (Masur-Minsky). The construction of arbitrarily high-dimensional quasi-flats is a strong, geometrically explicit contribution to the study of curve graphs and mapping class group actions. The distance bounds supply a useful computational tool. The topological hypotheses are the standard finite-type setting with at least one essential curve, and the constructions appear parameter-free once the base curve is fixed.

major comments (1)
  1. [§3] §3 (construction of the n-dimensional quasi-flat): the argument that the distance function on the grid of parallel curves yields a quasi-isometric embedding of E^n with constants independent of n should be verified explicitly, including the lower bound on distances between non-adjacent vertices in the grid; this is load-bearing for the claim that flats exist in every dimension.
minor comments (3)
  1. [§2] The notation for the single-isotopy-class fine curve graph (fiber over a vertex) is introduced clearly but could be fixed with a dedicated symbol early in §2 to ease later reading.
  2. [§4] Add a short remark comparing the obtained distance bounds to the known hyperbolicity constants of the full fine curve graph.
  3. [Figure 1] Figure 1 (schematic of the annular neighborhood and parallel curves) would benefit from labels indicating the auxiliary curves used to measure distances.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript, their positive assessment of the results on non-hyperbolicity of large subgraphs of the fine curve graph, and their recommendation of minor revision. We address the single major comment below.

read point-by-point responses

  1. Referee: [§3] §3 (construction of the n-dimensional quasi-flat): the argument that the distance function on the grid of parallel curves yields a quasi-isometric embedding of E^n with constants independent of n should be verified explicitly, including the lower bound on distances between non-adjacent vertices in the grid; this is load-bearing for the claim that flats exist in every dimension.

    Authors: We agree that an explicit verification of the quasi-isometric embedding, with particular attention to the lower bound on distances between non-adjacent grid vertices, is necessary to fully substantiate the existence of n-dimensional quasi-flats for arbitrary n. In the revised manuscript we will expand the argument in §3 to include this verification in detail. The lower bound will be established by showing that the auxiliary curves in the construction force any two curves whose grid positions differ by a vector of length d to intersect at least c·d times (for a positive constant c independent of n), which in turn implies that their graph distance in the single-isotopy-class fine curve graph is at least a fixed multiple of d. This analysis relies on the annular neighborhoods and the controlled intersection patterns already present in the construction; we will make the counting argument fully rigorous and parameter-independent. revision: yes

Circularity Check

0 steps flagged

No significant circularity; explicit topological constructions

full rationale

The paper establishes non-hyperbolicity of the subgraphs by exhibiting explicit high-dimensional quasi-flats inside single-isotopy-class fibers, constructed via arbitrarily many mutually disjoint parallel curves in an annular neighborhood of a fixed essential curve together with controlled intersection patterns against auxiliary curves to realize grid-like distance growth. These constructions rest on standard finite-type surface topology and the definition of the fine curve graph (vertices essential curves, edges for disjointness), without any reduction to fitted parameters, self-definitional loops, or load-bearing self-citations. The cited hyperbolicity results (Bowden-Hensel-Webb for the full fine curve graph, Masur-Minsky for the ordinary curve graph) serve only as contrast and are not invoked to force the new claims; distance bounds are derived directly from intersection controls within the induced subgraph. The derivation chain is therefore self-contained and independent of the target results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the full ledger cannot be audited; the work rests on the standard definitions of the fine curve graph and curve graph together with the known hyperbolicity theorems.

axioms (2)
  • domain assumption Vertices of the fine curve graph are essential simple closed curves on a surface and edges connect disjoint curves.
    This is the standard definition used throughout the field and invoked in the abstract.
  • domain assumption The curve graph is obtained by collapsing each isotopy class to a point.
    Standard construction referenced in the abstract.

pith-pipeline@v0.9.0 · 5421 in / 1322 out tokens · 74616 ms · 2026-05-07T14:44:54.003345+00:00 · methodology

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