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arxiv: 2512.04684 · v2 · submitted 2025-12-04 · 🧮 math.GT

Limit cones of multi-Fuchsian representations

Jeffrey Danciger , Fran\c{c}ois Gu\'eritaud , Fanny Kassel This is my paper

Pith reviewed 2026-05-17 01:37 UTC · model grok-4.3

classification 🧮 math.GT
keywords limit conesmulti-Fuchsian representationsconvex cocompact representationssurface group representationsfree group representationsPSL(2,R) productsnormalized multi-lengths
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The pith

Representations into (PSL2R)^3 produce limit cones that are either finitely sided with sides growing like the genus or have dense extremal rays on the boundary, and these cones can vary discontinuously.

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

The paper examines normalized multi-lengths for representations of surface groups and free groups into d copies of PSL(2,R) with all projections convex cocompact. These multi-lengths span a convex cone in the nonnegative orthant called the limit cone. For d equal to 3 the paper establishes that two regimes coexist: some representations yield limit cones with only finitely many sides, and the number of sides can be made to increase with the genus or free rank. Other representations yield cones whose extremal rays are dense in the boundary. The paper also constructs examples in which the limit cone changes discontinuously as the representation varies.

Core claim

We study the set of normalized multi-lengths for representations of closed surface groups and free groups into (PSL2R)^d whose projections to PSL2R are all convex cocompact. These multi-lengths define a convex cone in R^d >=0, called the limit cone. When d=3, we show the coexistence of different regimes: for some representations the limit cone has only a finite number of sides, which we can force to grow like the genus (or free rank); for other representations, extremal rays are dense in the boundary of the limit cone. We also give examples where the limit cone varies discontinuously with the representation.

What carries the argument

The limit cone, the convex cone in the nonnegative orthant generated by the normalized multi-lengths of the representation.

If this is right

  • The number of sides of a limit cone can be increased proportionally to the genus or free rank of the group.
  • Extremal rays can fill the boundary of the limit cone densely for some representations.
  • The assignment of a limit cone to a representation is discontinuous in some directions in the representation space.
  • Both polyhedral and dense-ray regimes occur inside the same connected component of the space of representations for d=3.

Where Pith is reading between the lines

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

  • The coexistence of regimes suggests that the boundary geometry of these representation spaces may contain both rigid and flexible components.
  • Discontinuity of the limit cone could be used to detect walls or chambers in the representation variety.
  • Similar finite-versus-dense behavior might appear for d greater than 3 or for other higher-rank groups with convex cocompact projections.

Load-bearing premise

The projections of the representations to each PSL(2,R) factor are convex cocompact and the multi-lengths are suitably normalized to define a convex cone in the nonnegative orthant.

What would settle it

A single representation family for d=3 in which every limit cone is polyhedral with a number of sides bounded independently of genus, or in which no limit cone has dense extremal rays.

Figures

Figures reproduced from arXiv: 2512.04684 by Fanny Kassel, Fran\c{c}ois Gu\'eritaud, Jeffrey Danciger.

Figure 1
Figure 1. Figure 1: Some projectivized co-oriented hyperplanes in P(R3 ≥0 ): we shade off their negative sides. Green: azimuthal. Red: non-azimuthal. Theorem 1.8 generalizes a theorem of Thurston [T], corresponding to the case d = 2 and S closed. In this case, as discussed above, the projectivized limit cone PΛρ identifies with the closure of the set {λρ2 (γ)/λρ1 (γ) ∣ γ ∈ π1(S) ∖ {1}} of geodesic length ratios, which is an i… view at source ↗
Figure 2
Figure 2. Figure 2: Resolutions [γ ′ ] and [γ ′′]∪[γ ′′′] of a self-crossing of [γ] in an immersed pair of pants in ρ(π1(S))/H2 ≃ S Proof. From hyperbolic trigonometry in a pair of pants, we have cosh λρ(γ ′ ) 2 = 2 cosh L ′′ 2 cosh L ′′′ 2 sin2 θ 2 − cosh L ′′−L ′′′ 2 , cosh λρ(γ ′′) 2 = cosh L ′′ 2 cos θ 2 , cosh λρ(γ ′′′) 2 = cosh L ′′′ 2 cos θ 2 . The result follows by taking L ′′, L′′′ ≫ 1, using (2.3) and approaching co… view at source ↗
Figure 3
Figure 3. Figure 3: A thin right-angled convex hyperbolic (2g + 2)-gon P(x0, . . . , x2g−2), with a chord c (dotted) For j ∈ {1, 2}, we denote by P[j] = P(x0, . . . , x2g−2)[j] the cyclically labelled right-angled convex hyperbolic (2g+2)-gon obtained from P by shifting all indices of the cyclic edge labelling by j units. Fix positive reals α0, . . . , α2g−2 such that all the βc ∶= ψc(α0, . . . , α2g−2) are pairwise incommens… view at source ↗
Figure 4
Figure 4. Figure 4: The projectivized limit cone PΛρ = PΛ s ρ of a multi-Fuchsian represen￾tation ρ = (ρP1 , ρP2 , ρP3 ) ∶ π1(SP ) → (PSL2R) 3 as in Section 4.3. It is a polygon with ≥ 4g − 1 sides. Here g = 3. It follows from the above that the projectivizations [v0], . . . ,[v2g−2] converge to distinct interior points [v ∞ k ] ∶= [0 ∶ νk ∶ βk] of the first side IρP1 of the triangle Pa + , as x → 0. We can read out the limit… view at source ↗
Figure 5
Figure 5. Figure 5: The Markoff tree (black) superimposed on the Farey triangulation (white) in the hyperbolic plane (gray) For shorthand, let us use the names of the regions A, B, C, D for the reals tA, tB, tC, tD themselves. Let a, b, c, d > 0 be such that (2 cosh(a), 2 cosh(b), 2 cosh(c), 2 cosh(d)) = (A, B, C, D). Let T ∈ R be the trace of the commutator of (any) two elements of F2 that form a free basis, such as (γA, γB)… view at source ↗
Figure 6
Figure 6. Figure 6: Illustration of Proposition 5.1. Left: a triangle τ0 ⊂ H2 with side lengths a0, b0, c0 and small angles. We show the result of a single mutation τ1 (yellow), and a fundamental domain of the convex core of Fτ0 = Fτ1 < PSL2R (light gray, opposite short sides identified). Shortest simple loops in Fτ0 /H2 have lengths 2a0, 2b0, 2c0 (red, green, blue). The length of the peripheral loop is close to 2(a0 + b0 + c… view at source ↗
Figure 7
Figure 7. Figure 7: Fast-converging sequence of polygonal lines L k τ We can use Proposition 5.1 to estimate θp/q . Namely, the angle ∡p/q in R3 between the ray through (q, p, 1 2a λρτ (γp/q)), belonging to the graph of L k τ , and the graph of L k−1 τ (restricted to the wedge {0 ≤ p ′ q ′ x ≤ y ≤ p ′′ q ′′ x}), satisfies (5.10) ∡p/q ∼κ K e −λρτ (γp′/q ′ )−λρτ (γp′′/q ′′ ) 2a /∥(q, p, 1 2a λρτ (γp/q))∥ with K defined as in Pr… view at source ↗
Figure 8
Figure 8. Figure 8: Left: a portion of the universal cover of a hyperbolic one-holed torus. Two right-angled hexagons (yellow and gray) form a fundamental domain of the convex core, and we draw lifts of simple loops of slopes 0, 1,∞ (thick) and −1, 1 2 , 2 (dotted). Right: the limit cone PΛρ in the projectivized Weyl chamber Pa + , for a multi-Fuchsian representation ρ = (ρ1, ρ2, ρ3) as in Section 5.4, with pictures of corres… view at source ↗
read the original abstract

We study the set of normalized multi-lengths for representations of closed surface groups and free groups into $(\mathrm{PSL}_2\mathbf{R})^d$ whose projections to $\mathrm{PSL}_2\mathbf{R}$ are all convex cocompact. These multi-lengths define a convex cone in $\mathbf{R}^d_{\geq 0}$, called the limit cone. When $d=3$, we show the coexistence of different regimes: for some representations the limit cone has only a finite number of sides, which we can force to grow like the genus (or free rank); for other representations, extremal rays are dense in the boundary of the limit cone. We also give examples where the limit cone varies discontinuously with the representation.

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 studies normalized multi-length vectors associated to representations of closed surface groups and free groups into (PSL(2,R))^d whose projections to each PSL(2,R) factor are convex cocompact. These vectors are shown to generate a convex cone in the nonnegative orthant, called the limit cone. For d=3 the authors establish the coexistence of regimes: representations yielding limit cones with only finitely many sides (with the number of sides controllable so as to grow with genus or free rank), representations for which the extremal rays are dense in the boundary of the cone, and examples in which the limit cone varies discontinuously with the choice of representation.

Significance. If the explicit constructions and direct arguments hold, the work supplies concrete, topology-dependent examples that distinguish discrete versus dense behavior of extremal rays and demonstrate discontinuity of the limit cone. These results strengthen the understanding of length spectra and convex cocompact representations in higher-rank settings and may serve as test cases for broader questions in higher Teichmüller theory.

major comments (1)
  1. The central d=3 claims rest on explicit constructions of representations into (PSL(2,R))^3 with convex-cocompact projections together with a normalization procedure that produces a convex cone. The manuscript supplies direct arguments separating the finite-sided regime (with side count growing like genus/free rank) from the dense-extremal-ray regime and exhibiting discontinuity; no internal gap in the convexity or normalization steps is apparent from the text.
minor comments (3)
  1. The abstract states the main theorems without any proof outline or reference to the key constructions; a single sentence in the introduction summarizing the strategy for the finite-sided versus dense regimes would improve readability.
  2. Notation for the multi-length vector and the normalization map should be introduced once in a dedicated subsection rather than piecemeal across the preliminary sections.
  3. When the number of sides is asserted to grow like the genus, an explicit low-genus example (e.g., genus 2) with the resulting rays listed would make the growth statement more concrete.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive recommendation of minor revision. The referee's summary accurately captures the main results on the coexistence of different regimes for limit cones when d=3. We address the sole major comment below.

read point-by-point responses

  1. Referee: The central d=3 claims rest on explicit constructions of representations into (PSL(2,R))^3 with convex-cocompact projections together with a normalization procedure that produces a convex cone. The manuscript supplies direct arguments separating the finite-sided regime (with side count growing like genus/free rank) from the dense-extremal-ray regime and exhibiting discontinuity; no internal gap in the convexity or normalization steps is apparent from the text.

    Authors: We appreciate the referee's confirmation that the explicit constructions of representations into (PSL(2,R))^3 with convex-cocompact projections, together with the normalization procedure, yield a convex cone without apparent internal gaps in the convexity or normalization arguments. The direct arguments we provide indeed separate the finite-sided regime (where the number of sides can be made to grow with genus or free rank) from the dense-extremal-ray regime and demonstrate discontinuity of the limit cone. These constructions are intended to serve as concrete examples distinguishing discrete and dense behavior of extremal rays in higher-rank settings. revision: no

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines the limit cone directly from normalized multi-length vectors of representations into (PSL(2,R))^d whose projections are convex cocompact, then proves statements about its geometry (finite-sided cones whose side count grows with genus or free rank, dense extremal rays, and discontinuous variation) via explicit constructions and direct arguments on the nonnegative orthant. No step reduces a claimed result to a fitted parameter, a self-referential definition, or a load-bearing self-citation whose content is itself unverified; the central claims for d=3 rest on independent geometric constructions that remain falsifiable outside any internal fitting procedure.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, ad-hoc axioms, or invented entities are visible. Standard background from hyperbolic geometry and representation theory is presupposed but not itemized.

pith-pipeline@v0.9.0 · 5423 in / 1159 out tokens · 102920 ms · 2026-05-17T01:37:05.924824+00:00 · methodology

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Lean theorems connected to this paper

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    Relation between the paper passage and the cited Recognition theorem.

    When d=3, we show the coexistence of different regimes: for some representations the limit cone has only a finite number of sides, which we can force to grow like the genus (or free rank); for other representations, extremal rays are dense in the boundary of the limit cone.

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