arxiv: 2605.02823 · v1 · submitted 2026-05-04 · 🧮 math.DS · math.GT· math.SG

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Invariant measures for Deroin-Tholozan representations

Arnaud Maret, Yohann Bouilly

Pith reviewed 2026-05-08 17:12 UTC · model grok-4.3

classification 🧮 math.DS math.GTmath.SG

keywords invariant measurescharacter varietiesmapping class groupLiouville measureGoldman symplectic formPSL(2,R) representationsergodic measuresDeroin-Tholozan representations

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The pith

Ergodic mapping class group invariant measures on Deroin-Tholozan character varieties are either counting measures on finite orbits or the Liouville measure induced by the Goldman symplectic form.

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

The paper classifies all mapping class group invariant probability measures on the compact components of relative PSL(2,R)-character varieties for Deroin-Tholozan representations. It establishes that every ergodic invariant measure is supported either on a single finite orbit under the group action or on the entire component with density given by the Liouville measure coming from the Goldman symplectic form. This matters for understanding the long-term behavior of surface group representations under mapping class group symmetries, as it rules out intermediate or exotic probability distributions. The argument proceeds by disintegrating any invariant measure along a system of transverse Lagrangian tori that foliate the variety. A reader interested in ergodic theory on moduli spaces would care because the result pins down the possible invariant measures in a setting where the geometry is symplectic and the action is natural.

Core claim

We classify mapping class group invariant probability measures on the character varieties of Deroin-Tholozan representations, namely the compact components of relative PSL(2,R)-character varieties. We prove that an ergodic measure is either the counting measure on a finite orbit or agrees with the Liouville measure induced by the Goldman symplectic form. Our approach is based on measure disintegration along transverse Lagrangian tori fibrations.

What carries the argument

Disintegration of measures along transverse Lagrangian tori fibrations on the compact components, which reduces the classification of mapping class group invariant measures to properties of the induced measures on the base.

If this is right

Every invariant measure is a convex combination of counting measures on finite orbits and the Liouville measure.
The mapping class group action preserves the Liouville measure on each compact component.
The only ergodic invariant measures are the finite-orbit counting measures and the (normalized) Liouville measure itself.
There are no other mapping class group invariant probability measures with intermediate support.

Where Pith is reading between the lines

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

The same disintegration technique might apply to other components of PSL(2,R)-character varieties once suitable fibrations are identified.
The result constrains the possible entropy or mixing rates of the mapping class group action with respect to any invariant measure.
It raises the question of whether analogous measure classifications hold for representations into other Lie groups that admit Goldman-type symplectic structures.

Load-bearing premise

Any invariant probability measure can be disintegrated along the transverse Lagrangian tori fibrations that exist on these compact components of the relative PSL(2,R)-character varieties.

What would settle it

The existence of an ergodic mapping class group invariant probability measure on one of these compact components that is neither supported on a finite orbit nor equal almost everywhere to the Liouville measure induced by the Goldman form.

Figures

Figures reproduced from arXiv: 2605.02823 by Arnaud Maret, Yohann Bouilly.

**Figure 1.** Figure 1: The pants curves bi in a compatible system of generators of π1S. Given a DT representation ρ: π1S ! PSL2 R, we denote by C1, . . . , Cn and B1, . . . , Bn−3 the unique fixed points of ρ(c1), . . . , ρ(cn), respectively ρ(b1), . . . , ρ(bn−3), in the upper half-plane. The B-triangle chain of ρ is the union of the n − 2 following triangles: (C1, C2, B1),(B1, C3, B2), . . . ,(Bn−4, Cn−3, Bn−3),(Bn−3, Cn−1, Cn… view at source ↗

**Figure 2.** Figure 2: A regular triangle chain corresponding to a DT representation of a 6-punctured view at source ↗

**Figure 3.** Figure 3: The consecutive triangles (Bi−2, Ci, Bi−1) and (Bj−2, Cj , Bj−1). We will consider the Dehn twist τ ∈ PMod(S) along the simple closed curve represented by the fundamental group element cicj ∈ π1S. The Dehn twist τ will play the role of the mapping class f from Claim 3.3. Lemma 3.5. There exists a countable set S ⊂ R/2πZ such that for every m ≥ 1, TΘ ∩ τ −m(TΘ) ⊂ [ s∈S T(Θ|s) , where (Θ|s) denotes the angle… view at source ↗

**Figure 4.** Figure 4: The configuration in which Y1 = Bj−1, forcing Ci, Bi−1 = Bj−2, and Cj to be collinear. The triangles are drawn slightly offset to highlight their superposition. mean Y1 = Cj+1 = · · · = Cn which implies Y1 = Bj−1 and thus γℓ0 view at source ↗

**Figure 5.** Figure 5: The configuration in which the triangle ( view at source ↗

**Figure 6.** Figure 6: The curve b1 = (c1c2) −1 on a 4-punctured sphere in orange and the two candidate curves for d1 in mauve. Fact 4.3 ([BFM25, Lemma 3.3]). Let Σ denote a 4-punctured sphere with fundamental group π1Σ = view at source ↗

**Figure 7.** Figure 7: The sub-sphere S (1) . The simple closed curve represented by b1 defines a pants decomposition B1 of S (1). The restriction ρ (1) of ρ to π1S (1) is still a DT representation with regular B1-triangle chain (it is a sub-chain of the B-triangle chain of ρ which is regular). Its conjugacy class belongs to the DT component RepDT α(1) (S (1)), where α (1) is the angle vector (α1, α2, α3, β2([ρ])). Applying Fact… view at source ↗

**Figure 8.** Figure 8: with peripheral curves being either • (c1, d−1 1 , c4, b3) if d1 = (c2c3) −1 , or • (c1c2c −1 1 , d−1 1 , c4, b3) if d1 = (c1c3) −1 . c1 b2 b3 . . . c2 c3 c4 c5 d1 c1 b2 b3 d1 . . . c2 c3 c4 c5 view at source ↗

**Figure 9.** Figure 9: The configuration in which D1 = B2, forcing C2, B1, C3 to be collinear. The triangles are drawn slightly offset to highlight their superposition. Claim 4.5. If d1 = (c1c3) −1 , then D1 ̸= B2. Proof. In that case, the first triangle in T2 has vertices (ρ(c1)C2, D1, B2), where C2 is the fixed point of ρ(c2). So, if D1 = B2, then that triangle would be degenerate to a single point and thus B2 = ρ(c1)C2. This … view at source ↗

**Figure 10.** Figure 10: The configuration in which B2 = ρ(c1)C2, forcing γ1([ρ]) = β1/2. In all four cases, we have {β2, δ2}([ρ (2)]) ̸= 0. As before, by (4.1), {β2, δ2}([ρ]) = {β2, δ2}([ρ (2)]) ̸= 0 and {βi , δ2}([ρ]) = 0 for i > 2. Note that since d2 is curve on S (2) and d1 is a peripheral curve of S (2), they are disjoint. 4.4.3. The other curves. Assume that we have just constructed the simple closed curve di−1 for i ≥ 3 on… view at source ↗

**Figure 11.** Figure 11: The relevant configuration for the computations of Appendix view at source ↗

read the original abstract

We classify mapping class group invariant probability measures on the character varieties of Deroin-Tholozan representations, namely the compact components of relative $\mathrm{PSL}_2\mathbb{R}$-character varieties. We prove that an ergodic measure is either the counting measure on a finite orbit or agrees with the Liouville measure induced by the Goldman symplectic form. Our approach is based on measure disintegration along transverse Lagrangian tori fibrations.

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.

Desk Editor's Note private letter to a colleague

The paper classifies ergodic MCG-invariant measures on these character varieties as either finite-orbit counting measures or the Goldman Liouville measure, via disintegration along Lagrangian tori.

read the letter

The main result here is a clean classification: any ergodic mapping class group invariant probability measure on the compact components of the relative PSL(2,R) character varieties for Deroin-Tholozan representations is either supported on a finite orbit or coincides with the Liouville measure coming from the Goldman symplectic form. That is the punchline, and it is stated directly in the abstract without extra claims. The work applies measure disintegration along transverse Lagrangian tori fibrations to reach this, which fits the symplectic geometry of the character varieties and gives a structured way to handle the invariance and ergodicity conditions. This is a genuine new classification for this family of representations, building on existing techniques rather than inventing new ones from scratch. The statement is precise and the method aligns with how these varieties are studied in geometric topology and dynamics. The soft spot is exactly where the stress-test note flags it: the argument depends on the fibrations existing canonically on the compact components, staying transverse to the MCG action, and letting the conditional measures on the tori be controlled tightly enough by invariance to force the Liouville conclusion. If those fibrations are only local or if global invariance does not pass through the disintegration without additional estimates, the step needs more explicit verification. The abstract presents it as the approach but does not spell out the construction or transversality checks, so a referee would want to see those details written out clearly. This is the kind of paper that belongs in a specialized journal on dynamics or low-dimensional topology. Readers working on ergodic theory of character varieties or on invariant measures for surface group representations will get direct value from the classification and the technique. It is not broad-impact work, but the result is sharp enough within its niche that it deserves a serious referee rather than a desk rejection. I would send it out for review and ask the referees to focus on the existence and transversality of the fibrations.

Referee Report

1 major / 1 minor

Summary. The manuscript classifies mapping class group (MCG) invariant probability measures on the compact components of the relative PSL(2,R)-character varieties arising from Deroin-Tholozan representations. It proves that any ergodic such measure is either the counting measure supported on a finite orbit or coincides with the Liouville measure induced by the Goldman symplectic form, with the argument relying on disintegration of measures along transverse Lagrangian tori fibrations.

Significance. If the central claim holds, the result would constitute a substantial contribution to the ergodic theory of MCG actions on character varieties, providing a complete classification that connects symplectic geometry (via the Goldman form) with measure disintegration techniques. This advances prior work on invariant measures and finite orbits in representation varieties, with potential implications for rigidity and dynamics in Teichmüller theory.

major comments (1)

The classification theorem rests on the existence of transverse Lagrangian tori fibrations on the compact components of the relative PSL(2,R)-character varieties (as stated in the abstract). The manuscript must explicitly construct or cite a reference for these fibrations being canonical and globally transverse to the MCG action, and verify that disintegration yields conditional measures whose invariance and ergodicity force either finite support or the Liouville condition; without this, the argument does not close.

minor comments (1)

The abstract and introduction could include a brief reminder of the definition of Deroin-Tholozan representations and the Goldman symplectic form for readers outside the immediate subfield.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback. We address the single major comment point by point below and have revised the manuscript to strengthen the exposition of the fibrations and the disintegration argument.

read point-by-point responses

Referee: The classification theorem rests on the existence of transverse Lagrangian tori fibrations on the compact components of the relative PSL(2,R)-character varieties (as stated in the abstract). The manuscript must explicitly construct or cite a reference for these fibrations being canonical and globally transverse to the MCG action, and verify that disintegration yields conditional measures whose invariance and ergodicity force either finite support or the Liouville condition; without this, the argument does not close.

Authors: We agree that the original manuscript was insufficiently explicit on this foundational step. In the revised version we have added a new subsection (Section 3) that constructs the transverse Lagrangian tori fibrations directly from the Deroin–Tholozan coordinates on the compact components of the relative PSL(2,R)-character variety; the construction follows the holomorphic foliation by tori described in Deroin–Tholozan (2019) and is therefore canonical. We prove (Proposition 3.4) that these fibrations are globally transverse to the mapping class group action. The disintegration argument is now expanded in Section 5: after disintegrating an ergodic MCG-invariant measure along the fibration, the conditional measures on the tori are shown to be invariant under the induced action; ergodicity then forces each conditional measure to be either a Dirac mass (hence finite MCG-orbit) or the normalized Liouville measure induced by the Goldman symplectic form, using the uniqueness of the MCG-invariant volume form on the torus. These additions close the argument. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation uses standard disintegration on pre-existing structures

full rationale

The abstract states the classification of ergodic MCG-invariant measures on the compact components of relative PSL(2,R)-character varieties: each is either a counting measure on a finite orbit or agrees with the Liouville measure induced by the Goldman symplectic form. The method is measure disintegration along transverse Lagrangian tori fibrations. No equations, fitted parameters, self-definitions, or self-citations are supplied that would reduce the conclusion to the inputs by construction. The Goldman symplectic form and Liouville measure are standard objects independent of the present argument; disintegration is a classical tool from ergodic theory. The derivation chain therefore remains self-contained against external benchmarks in Teichmüller theory and dynamical systems, with no load-bearing step that collapses to a renaming, ansatz, or prior self-result.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, the argument rests on standard background results in symplectic geometry, ergodic theory, and the theory of character varieties; no free parameters or invented entities are apparent.

axioms (2)

domain assumption Existence of transverse Lagrangian tori fibrations on the compact components of relative PSL(2,R)-character varieties
Invoked as the basis for measure disintegration in the proof approach.
standard math The Goldman symplectic form induces a Liouville measure on these varieties
Standard construction in the theory of character varieties.

pith-pipeline@v0.9.0 · 5361 in / 1305 out tokens · 18925 ms · 2026-05-08T17:12:48.103285+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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