# Valuations on the character variety: Newton polytopes and Residual Poisson Bracket

Julien Marché, Christopher-Lloyd Simon

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Julien Marché, Christopher-Lloyd Simon. Valuations on the character variety: Newton polytopes and Residual Poisson Bracket. Geometry and Topology, In press, 10.2140/gt.2024.28.593. hal-04188914

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Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
VALUATIONS ON THE CHARACTER VARIETY:
NEWTON POLYTOPES AND RESIDUAL POISSON BRACKET

JULIEN MARCHÉ AND CHRISTOPHER-LLOYD SIMON

ABSTRACT. We study the space of measured laminations ML on a closed surface from the valuative point of view. We introduce and study a notion of Newton polytope for an algebraic function on the character variety. We prove, for instance, that trace functions have unit coefficients at the extremal points of their Newton polytope. Then we provide a definition of tangent space at a valuation and show how the Goldman Poisson bracket on the character variety induces a symplectic structure on this valuative model for ML. Finally we identify this symplectic space with previous constructions due to Thurston and Bonahon.

Keywords. Character variety, surface group, valuation, Newton polytope, measured lamination, real tree, Goldman Poisson bracket, symplectic structure, skein algebra.

INTRODUCTION

**The algebra of functions on the character variety.** Let $S$ be a closed oriented surface of genus $g \ge 2$. Its character variety $X$ is the quotient of the space $\operatorname{Hom}(\pi_1(S), \operatorname{SL}_2(\mathbb{C}))$ by the equivalence relation identifying $\rho_1$ and $\rho_2$ if and only if $\operatorname{tr} \rho_1(\gamma) = \operatorname{tr} \rho_2(\gamma)$ for all $\gamma \in \pi_1(S)$. By construction, it is an affine variety whose ring of functions $\mathbb{C}[X]$ is generated by the trace functions $t_\gamma: \rho \mapsto \operatorname{tr} \rho(\gamma)$ for $\gamma \in \pi_1(S)$. The function $t_\gamma$ only depends on the conjugacy class of $\gamma$ up to inversion, that is on the free homotopy class of the corresponding unoriented loop.

These trace functions are not algebraically independent: the famous identity $\operatorname{tr}(AB)+\operatorname{tr}(AB^{-1}) = \operatorname{tr}(A)\operatorname{tr}(B)$ for $A, B \in \operatorname{SL}_2(\mathbb{C})$ implies, for instance, that if $\alpha$ and $\beta$ represent simple loops intersecting once, then

$$t_{\alpha}t_{\beta} = t_{\gamma} + t_{\delta}$$

where $\gamma$ and $\delta$ are elements in $\pi_1(S)$ representing the simple curves obtained by smoothing the intersection between $\alpha$ and $\beta$ in the two possible ways.

This phenomenon generalizes as follows. Given a multiloop $\alpha$, that is a multiset $\{\alpha_1, \dots, \alpha_n\}$ of non-trivial loops $\alpha_i \in \pi_1(S)$, the function $t_\alpha = t_{\alpha_1}t_{\alpha_2}\cdots t_{\alpha_n}$ can be uniquely decomposed as a linear combination

$$t_{\alpha} = \sum m_{\mu} t_{\mu}$$

where each $\mu$ is a multicurve, that is a (possibly empty) multiloop represented by pairwise disjoint, simple, non-trivial loops. This means that the set $\mathrm{MC}$ of multicurves indexes a linear basis for the algebra of characters $\mathbb{C}[X]$ which is privileged from the topological viewpoint; it is also invariant under the (algebraic) automorphism group of $\mathbb{C}[X]$, as we proved in [MS21].

It is an old problem to understand the algebraic structure of $\mathbb{C}[X]$, whose study has been initiated by Fricke and Vogt in the late 19th century, and revisited in the seventies by the work of Procesi, Horowitz and Magnus among others (see [Mag81] for a review). One approach is to investigate the coefficients $m_\mu$ of the functions $t_\alpha$.
In this article, we define the Newton set $\Delta(t_\alpha) \subset \text{MC}$ of $t_\alpha$, in analogy with the extremal points of the ordinary Newton polytope of a polynomial, as follows.

**Definition (Newton Set).** For $f = \sum m_\mu t_\mu$ decomposed in the basis of multicurves, we define its support as $\text{Supp}(f) = \{\mu \in \text{MC}, m_\mu \neq 0\}$.

We say that $\mu \in \text{Supp}(f)$ is extremal in $f$ if there exists a multicurve $\xi$ such that $i(\xi, \mu) > i(\xi, \nu)$ for all $\nu \in \text{Supp}(f)$ distinct from $\mu$.

The Newton set $\Delta(f)$ is the set of extremal multicurves in $f$.

In this definition, $i(\cdot, \cdot)$ denotes the geometric intersection number, and standard properties of measured laminations imply that $\xi$ can be replaced by a simple curve or a measured lamination. Our first result is the following.

**Theorem A (Trace functions are unitarity).** For every multiloop $\alpha = \{\alpha_1, \dots, \alpha_n\}$, the function $t_\alpha$ is unitary in the sense that $m_\mu = \pm 1$ for all $\mu \in \Delta(t_\alpha)$.

To introduce our next result, recall that the algebra of functions $\mathbb{C}[X]$ carries a natural Poisson bracket stemming from the Atiyah-Bott-Weil-Petersson-Goldman symplectic structure on $X$. Following Goldman [Gol84], for $\alpha, \beta \in \pi_1(S)$ it is given by the formula

$$ (2) \qquad \{t_\alpha, t_\beta\} = \sum_{p \in \alpha \cap \beta} \epsilon_p (t_{\alpha_p \beta_p} - t_{\alpha_p \beta_p^{-1}}) $$

where the sum ranges over all intersection points $p$ between transverse representatives for $\alpha \cup \beta$ and $\epsilon_p$ is the sign of such an intersection, while $\alpha_p, \beta_p$ denote the homotopy classes of $\alpha, \beta$ based at $p$.

Our second result interprets the coefficients of $\{f, g\}$ at the extremal multicurves of $fg$ in terms of Thurston's PL-symplectic structure on the space $\text{ML}$ of measured laminations on $S$.

**Theorem B (Extremal structure constants for the Poisson Bracket).** Let $\mu$ and $\nu$ be two multicurves. For $\xi \in \Delta(t_\mu t_\nu)$ we set $E_\xi = \{\lambda \in \text{ML} \mid i(\xi, \lambda) = i(\mu, \lambda) + i(\nu, \lambda)\}$. These closed subsets of $\text{ML}$ form a piecewise linear partition of $\text{ML}$ with disjoint interiors.

For Thurston's symplectic structure, the Poisson bracket $\{i_\mu, i_\nu\}$ of the length functions defined by $i_\mu(\lambda) = i(\mu, \lambda)$ is equal to the coefficient of $t_\xi$ in $\{t_\mu, t_\nu\}$ almost everywhere in $E_\xi$.

Let us illustrate the theorem with the following example. The curves shown in Figure 1 satisfy $t_\alpha t_\beta = t_{c_1}t_{c_3} + t_{c_2}t_{c_4} - t_\gamma - t_\delta$ and $\{t_\alpha, t_\beta\} = 2t_\delta - 2t_\gamma$, so we find that $\Delta(t_\alpha t_\beta) = \{c_1 \cup c_3, c_2 \cup c_4, \gamma, \delta\}$, whereas $\Delta(\{t_\alpha, t_\beta\}) = \{\gamma, \delta\}$.

The Newton set of $t_\alpha t_\beta$ decomposes $\text{ML}$ into 4 domains where $i(\alpha \cup \beta, \lambda)$ is equal to the intersection of $\lambda$ with $c_1 \cup c_3$ or $c_2 \cup c_4$ or $\gamma$ or $\delta$ respectively. In the interior of these domains $\{t_\alpha, t_\beta\}$ takes the values 0, 0, -2, 2 respectively.

<figure><img src="image_3.png" /><figcaption>FIGURE 1. Product and Poisson bracket in a sphere with four punctures.</figcaption></figure>
Strong relations between the symplectic structures on X and ML had already been observed, for instance in [PP91] or [SB01]. Theorem B can be related to a formula for {$i_μ, i_ν$} obtained in [Bon92, Proposition 6] by degenerating Wolpert’s “cosine formula”. However, our approach is algebraic in the sense that it uses valuations instead of Teichmüller theory.

Beyond these two results, the purpose of this article is to investigate the space of measured laminations from the valuative viewpoint, in particular its symplectic structure. This study was motivated by a new characterisation of valuations associated to measured laminations that we obtained in [MS21]. We devote the remaining part of this introduction to an overview of our motivations, as well as the intermediate results that we obtained while revisiting the theory of measured laminations from the valuative viewpoint since we believe they are of independent interest. We take this as an opportunity to recall general ideas for the benefit of a wide audience.

**The Newton polytope.** A leading analogy in this article is to think of the collection ($t_μ$) as a monomial basis in a polynomial algebra; keeping in mind that it is not stable under multiplication.
Consider the degree $deg_d$ defined for $d \in \mathbb{R}^n$ on the algebra $\mathbb{C}[t_1, \dots, t_n]$ by

$$ \deg_d \left( \sum_{\mu} m_{\mu} t^{\mu} \right) = \max\{\langle \mu, d \rangle, m_{\mu} \neq 0\} $$

where $t^μ = t_1^{μ_1} \cdots t_n^{μ_n}$, and $\langle \cdot, \cdot \rangle$ stands for the usual scalar product. This degree is (the opposite of) a monomial valuation. For $P \in \mathbb{C}[t_1, \dots, t_n]$, a monomial $t^μ$ is an extremal point of its usual Newton polytope $\Delta(P)$ if $m_μ \neq 0$ and for some $d \in \mathbb{R}^n$ the maximum defining $deg_d$ is attained uniquely at $t^μ$.

Our starting point is to replace the degree $deg_d$ by the valuation associated to a measured lamination $\lambda$ in $S$. For us a valuation will be a map $v: \mathbb{C}[X] \to \{-\infty\} \cup \mathbb{R}$ satisfying $v(fg) = v(f) + v(g)$ and $v(f+g) \le \max\{v(f), v(g)\}$ for all $f, g \in \mathbb{C}[X]$. We choose this convention, which is opposite to the usual one, to avoid crowding too many signs. In the general language of valuations (see for instance [Vaq00]), our valuations are centered at infinity on the affine variety $X$ and take non-negative values on the ring $\mathbb{C}[X]$ of characters.

In a groundbreaking series of articles starting with [MS84], Morgan-Shalen showed that the character variety $X$ can be compactified using valuations, in the spirit of the Riemann-Zariski compactification. In particular the space of measured laminations, viewed as Thurston’s compactification of Teichmüller space, can be embedded in the space of valuations on $\mathbb{C}[X]$ with values in an archimedian group. However, this embedding used a degeneration process and is not completely explicit: if $v$ is the valuation associated to a lamination $\lambda$, we clearly have $v(t_μ) = i(\lambda, μ)$, but it was not clear what should be $v(f)$ for a general element $f \in \mathbb{C}[X]$.

In our previous article [MS21], we showed that the space of measured laminations ML can be identified with the space of simple valuations $v: \mathbb{C}[X] \to \{-\infty\} \cup \mathbb{R}_{\ge 0}$. The word simple means monomial with respect to the multicurve basis in the sense that the following holds:

$$ (3) \qquad v \left( \sum m_{\mu} t_{\mu} \right) = \max\{v(t_{\mu}), m_{\mu} \neq 0\}. $$

This justifies our definition for the Newton set of $f = \sum m_μ t_μ$ as the set of $μ \in \text{Supp}(f)$ such that the maximum in (3) is attained uniquely at $t_μ$ for some $v \in \text{ML}$.

For a concrete example, consider the particular case of a multiloop $\alpha$ contained in an incompressible pair of pants $P \subset S$. The subsurface $P$ contains only three simple curves, its boundary components, and they do not intersect each other. Denoting by $t_1, t_2, t_3$ the trace functions along these components, we have $t_α \in \mathbb{Z}[t_1, t_2, t_3]$. This polynomial is often called the Fricke polynomial
and has been much studied, see [Mag81, Section 2.2]. Now any valuation associated to a measured lamination on S restricts to a monomial valuation on $\mathbb{C}[t_1, t_2, t_3]$, and we find that our Newton set corresponds to the extremal points of the usual Newton polytope. Even for such $\alpha \subset P$, it is not easy to determine $\Delta(t_\alpha)$ from the $\alpha_i \in \pi_1(S)$, and the unitarity property is not an obvious one.

It is worth noticing that we only talk about the Newton set and not about the Newton polytope, as we do not know any reasonable notion of convexity in ML. However, we can define the dual Newton polytope of a function $f \in \mathbb{C}[X]$ as $\Delta^*(f) = \{v \in \text{ML} | v(f) \le 1\}$. Moreover we could define the poset of faces of $\Delta(f)$ using the order structure. Its combinatorics may be a promising land of investigation, but we did not go further in that direction.

**Symplectic and combinatorial volumes of dual polytopes.** This paragraph only serves motivational purposes and does not claim new results, it may be skipped harmlessly.

Thurston’s symplectic form on ML provides a notion of volume, thus we may ask for the topological meaning of the volume $\text{Vol} \Delta^*(t_\alpha)$ when $\alpha$ is a multiloop.

When $\alpha$ is a filling multiloop, a celebrated theorem of M. Mirzakhani [Mir16] extended by Rafi and Souto [RS17], estimates the number of elements in its orbit under the modular group $\text{Mod}(S)$ as a bound on their complexity tends to infinity. More precisely, fix another filling multiloop $\beta$, and denote $m_g > 0$ the volume of the moduli space of hyperbolic metrics on S for the Weil-Petersson form. The theorem claims the following:

$$ \lim_{r \to \infty} \frac{\text{Card}\{\varphi \in \text{Mod}(S) \mid i(\beta, \varphi(\alpha)) \le r\}}{r^{6g-6}} = \frac{\text{Vol} \Delta^*(t_\beta) \text{Vol} \Delta^*(t_\alpha)}{m_g} $$

The identification between measured laminations and simple valuations implies, using Equation (3), that the Newton dual polytope $\Delta^*(f)$ of $f \in \mathbb{C}[X]$ equals the intersection of $\Delta^*(t_\mu)$ for $\mu \in \Delta(f)$. These “elementary cones” $\Delta^*(t_\mu) = \{v \in \text{ML} \mid v(t_\mu) < 1\}$ are described by explicit sets of linear inequalities in any PL chart of ML, and the volume of their intersection is computable. This yields a constructive procedure to compute Mirzakhani’s constant $\text{Vol} \Delta^*(t_\alpha)$, and shows that it depends only on $\Delta(t_\alpha)$. It also shows that these volumes are rational.

A different motivation is that this Newton set, as the usual one, could have applications to the problem of counting solutions of algebraic equations in X. We wonder for instance if it helps estimating the number of solutions to a system of $6g - 6$ equations $t_{\gamma_i} = x_i$ where $\gamma_1, \dots, \gamma_{6g-6} \in \pi_1(S)$ and $x_1, \dots, x_{6g-6} \in \mathbb{C}$. This could have interesting applications to 3-dimensional topology, for instance to evaluate the number of characters of representations of $\pi_1(M)$ of a 3-manifold M from a Heegaard decomposition.

**Measured laminations as valuations.** In this article we study measured laminations using the tools of valuation theory. There are two well-known invariants for an archimedean valuation $v$: its rational rank defined as the dimension of the $\mathbb{Q}$-vector space generated by the group $\Lambda_v$ of its values (that is differences of lengths for the corresponding measured lamination), and the transcendence degree of its residue field $k_v$. These invariants are related by the celebrated Abhyankar inequality $\text{rk}(v) + \text{tr. deg}(k_v) \le 6g - 6$. Here we will show the following.

**Proposition A (Characterising strict valuations).** For a valuation $v$ associated to a measured lamination $\lambda$, the following properties are equivalent.

(1) Distinct multicurves $\mu$ and $\nu$ have distinct lengths: $i(\lambda, \mu) \ne i(\lambda, \nu)$.

(2) The residue field of $\mathbb{C}(X)$ at $v$ has transcendence degree 1, or $k_v = \mathbb{C}$.

(3) The $\mathbb{Q}$-vector space generated by the set of lengths $i(\lambda, \mu)$ for $\mu \in \text{MC}$ has dimension $6g - 6$.
The first property implies that $v$ defines a total order on the set of multicurves, so the max in Equation 3 will always be strict, which is why they deserve to be called *strict valuations*. They played a prominent role in our previous article, where we showed that almost all valuations are strict in the measure theoretical sense. They will be equally important in this paper, as the second property enables to define the residual value at $v$ of a function $f \in \mathbb{C}(X)$ satisfying $v(f) \le 0$. Combined with the last property, it shows that strict valuations are *Abhyankar* in the sense that his inequality is an equality: we wonder whether any measured lamination gives rise to an Abhyankar valuation.

We have not come across strict valuations in the literature. Instead we encounter *maximal measured lamination*, which are those whose support cannot be enlarged. In this article, we characterize the valuations associated to maximal laminations as being *acute*: for any $\alpha, \beta \in \pi_1(S) \setminus \{1\}$ we never have $v(t_\alpha t_\beta) = v(t_{\alpha\beta}) = v(t_{\alpha\beta-1})$ so that these quantities are the lengths for the edges of an acute isosceles triangle. We will show that a valuation is acute if and only if any time we smooth a self-intersection of a multiloop which is taut (minimally intersecting in its homotopy class), the two resulting multiloops have distinct $\lambda$-lengths. This property plays a crucial role in the proof of the unitarity theorem. We also show that any strict valuation is acute and wonder if the reciprocal statement is true.

**Tangent spaces and Thurston's symplectic structure.** The space of measured laminations is a PL-manifold but does not carry any sensible smooth structure (for which intersection numbers have smooth variations), so there is no symplectic structure in the usual sense. However, Thurston showed that most points (maximal laminations) have a well-defined tangent space endowed with a non-degenerate skew-symmetric form (see [PH92, Chapter 3]).

In this article we propose a straightforward notion for the tangent space $T_v$ML at a valuation, and show that when $v$ is strict, it coincides with the space $\text{Hom}(\Lambda_v, \mathbb{R})$ which has dimension rat.rk($v$) = $6g - 6$. Then we show how the Goldman Poisson bracket induces a “residual Poisson bracket” at any strict valuation $v$, thus endowing $T_v$ML with a symplectic structure. For future reference we shall name this model after Goldman. This uses the crucial fact that given $f, g \in \mathbb{C}[X]$, we have $v(\{f, g\}) \le v(fg)$ for all $v \in \text{ML}$. This property amounts to the inverse inclusion of the dual polytopes $\Delta^*(\{f, g\}) \supset \Delta^*(fg)$.

Finally, we provide precise identifications between this symplectic vector space and two other existing models in the literature, which we now pass under review. In the work of Morgan-Shalen, the key notion allowing to relate measured laminations and valuations is the action of $\pi_1(S)$ on real trees. We may represent this dynamical point of view as lying between the two others in the following schematic table, which the reader may use as a dictionary.

<table><thead><tr><th>Topological/Geometrical</th><th>Dynamical</th><th>Functional</th></tr></thead><tbody><tr><td>measured foliation</td><td>action of π<sub>1</sub>(S)</td><td>simple</td></tr><tr><td>measured geodesic laminations</td><td>on a real tree</td><td>valuation</td></tr><tr><td>length function</td><td>translation length</td><td>trace function</td></tr><tr><td>filling/aperiodic lamination</td><td>free action</td><td>positive valuation</td></tr><tr><td>maximal lamination</td><td>trivalent tree</td><td>acute valuation</td></tr><tr><td>?</td><td>?</td><td>strict valuation</td></tr></tbody></table>

For future reference, we name the symplectic vector spaces appearing naturally from each of those approaches after Thurston, Bonahon and Goldman respectively.

*Goldmans's model:* it is given by the residual Poisson bracket on $T_v$ML which we introduced briefly, it will be described with more detail in the body of the paper.
**Thurston's model:** viewing $\lambda$ as a maximal measured lamination, one can associate a ramified 2-fold covering $S' \to S$ known as the orientation cover of the lamination. The group $H^1(S', \mathbb{R})$ splits into a symmetric and antisymmetric part with respect to involution of the covering $S' \to S$. The space $H^1(S', \mathbb{R})^-$ with the cup-product form is the geometrical model for $T_\lambda$ ML.

**Bonahon's model:** if we consider a trivalent real tree $T$ with a free and minimal action of $\pi_1(S)$, we can consider the space of functions $c: V(T)^2 \to \mathbb{R}$ on the set of pairs of trivalent vertices of $T$ which satisfy:

(1) $c(x, y) = c(y, x)$.

(2) $c(x, y) = c(x, z) + c(z, y)$ if $z$ belongs to the geodesic joining $x$ to $y$.

(3) $c(\alpha x, \alpha y) = c(x, y)$ for all $\alpha \in \pi_1(S)$.

Again, this space has a natural antisymmetric form related to the cyclic orientation of $T$ at every trivalent vertex. It is equivalent to the space of transverse cocycles introduced by Bonahon (see [Bon96, p.240]). The identification between Thurston's and Bonahon's model is well-known but all proofs we encountered use auxiliary structures like train tracks. At the end of the article, we provide "invariant" proofs for the following result.

**Theorem C (Symplectomorphisms).** There are natural isomorphisms of symplectic vector spaces between the models of Thurston, Bonahon and Goldman.

In particular we provide a new construction of independent interest reminiscent of Milnor's join construction, which starting from a trivalent real tree, gives a space homotopically equivalent to the covering $S'$. We may wonder which of these three symplectic identifications persist for more general actions of Fuchsian groups on real trees.

## PLAN OF THE PAPER

<table><tr><td>Introduction</td><td>1</td></tr><tr><td>1. Background</td><td>6</td></tr><tr><td>2. Measured laminations and simple valuations</td><td>8</td></tr><tr><td>3. Newton polytopes of trace functions</td><td>11</td></tr><tr><td>4. Residual Poisson structure on ML</td><td>13</td></tr><tr><td>5. Actions of π<sub>1</sub>(S) on real trees</td><td>14</td></tr><tr><td>6. Identifying the symplectic tangent models</td><td>21</td></tr><tr><td>References</td><td>26</td></tr></table>

*Acknowledgements.* We wish to thank Francis Bonahon, Chris Leininger and Maxime Wolff for useful discussions around this project, as well as Patrick Popescu-Pampu for his reading. We are also very grateful to the referee for their numerous corrections and suggestions.

## 1. BACKGROUND

1.1. **Algebra of functions on the character variety.** Let $S$ be a closed connected and oriented surface of genus $g \ge 1$. We denote by $X$ the character variety of $S$, which is the algebraic quotient of its representation variety $\text{Hom}(\pi_1(S), \text{SL}_2(\mathbb{C}))$ by the conjugacy action of $\text{SL}_2(\mathbb{C})$, defined as the spectrum of the algebra of invariant functions:

$$ \mathbb{C}[X] = \mathbb{C}[\text{Hom}(\pi_1(S), \text{SL}_2(\mathbb{C}))]^{\text{SL}_2(\mathbb{C})}. $$
(2) The algebra $K(S, \mathbb{C})$ with $A = -1$ is commutative, and the map sending the blackboard framing $[\mu]$ to $(-1)^{|\mu|}t_\mu$ defines an isomorphism $K(S, \mathbb{C}) \to \mathbb{C}[X]$ where $|\mu|$ denotes the number of components of $\mu$.

(3) The map sending a multicurve to its blackboard framing yields an isomorphism of $\mathbb{C}[A^{\pm 1}]$-modules $K(S, \mathbb{C}) \otimes \mathbb{C}[A^{\pm 1}] \simeq K(S, \mathbb{C}[A^{\pm 1}])$. In this setting, we have:

$$ \{f, g\} = \frac{1}{2} \frac{d}{dA} [fg - gf]_{A=-1}. $$

These algebras were introduced independently by Przytycki and Turaev. The assertions in Part (1), in Part (2) and the isomorphism of Part (3) are [PS00, Fact 4.1, Fact 2.7 and Theorem 2.8]. Part (2) is also proved in [Bul97]. Finally, the last formula appears in [BFKB99].

Let us explain more precisely Theorem 3 (1). Given a diagram $D$ for a banded link $L \subset S \times [0, 1]$, we denote by $C$ its set of crossings. For any map $\sigma: C \to \{\pm 1\}$, set $w_\sigma = \sum_c \sigma(c)$ and consider the diagram $D_\sigma$ obtained after smoothing each crossing $c \in C$ according to the sign $\sigma(c)$, and removing the $n_\sigma$ trivial components which appear in the result. The following formula holds in $K(S, R)$:

$$ (4) \qquad [L] = \sum_{\sigma: C \to \{\pm 1\}} (-A^2 - A^{-2})^{n_\sigma} A^{w_\sigma} [D_\sigma] $$

which, after grouping terms corresponding to a same diagram $[D_\sigma]$, yields the decomposition of $[L]$ in the basis of multicurves. We use this formula in order to understand the product of two multicurves $\mu, \nu$: intuitively, the product is obtained by taking the union $\mu \cup \nu$ and summing over all possible smoothings.

By Theorem 3 (2), we deduce that the algebra $\mathbb{C}[X]$ has a linear basis indexed by trace functions of multicurves. At $A = -1$, the class of $[L]$ does not change if we change a crossing. Hence, we can replace the notion of banded link with the simpler notion of multiloop that we defined previously.

The Kauffman algebra is not completely necessary for our purposes. However, we find it concep- tually useful for the following reasons. It transforms the trace relation into a local relation whose sign is more convenient (for instance while performing successive diagrammatic computations), and a better understanding of the product in terms of smoothings. It also provides a simple reason to why the Goldman bracket actually satisfies the Jacobi relation: this comes from Theorem 3 (3) and the obvious associativity of multiplication in the Kauffman algebra. Finally, in the context of this article, it provides an alternative formula for the Poisson bracket which enlightens Theorem B. Indeed, a smoothing $\sigma$ which is extremal for $[\alpha][\beta]$ in $K(S, R)$ is also extremal for the Poisson bracket $\{t_\alpha, t_\beta\}$ in $\mathbb{C}[X]$. Its coefficient in the former is $\pm 1$ by Theorem A and we will interpret its coefficient $\pm w_\sigma$ in the latter as a residual Poisson bracket.

## 2. MEASURED LAMINATIONS AND SIMPLE VALUATIONS

2.1. **Simple valuations.** It is well-known that a measured lamination $\lambda$ on $S$ is characterized by the length $i(\lambda, \gamma)$ it assigns to every simple curve $\gamma$. This “functional” point of view can be extended to define a map $v_\lambda: \mathbb{C}[X] \to \{-\infty\} \cup \mathbb{R}_{\ge 0}$ satisfying $v(0) = -\infty$ and for all $f = \sum m_\mu t_\mu$ decomposed in the multicurve basis:

$$ (5) \qquad v_\lambda(f) = \max\{i(\lambda, \mu) \mid m_\mu \neq 0\} $$

where $i(\lambda, \mu) = i(\lambda, \mu_1) + \cdots + i(\lambda, \mu_n)$ for a multicurve $\mu$ with components $\mu_1, \dots, \mu_n$. By [MS21, Proposition 1.2], Equation (5) is coherent with the fact that for any $\alpha \in \pi_1(S)$, not necessarily simple, we actually have $v_\lambda(t_\alpha) = i(\lambda, \alpha)$. Let us recall [MS21, Definition 1.1].
**Definition 2.** A simple valuation on $\mathbb{C}[X]$ is a map $v: \mathbb{C}[X] \to \{-\infty\} \cup \mathbb{R}_{\ge 0}$ satisfying:

(1) $v(f) = -\infty$ if and only if $f = 0$.

(2) $v(fg) = v(f) + v(g)$ for all $f, g \in \mathbb{C}[X]$.

(3) If $f = \sum m_\mu t_\mu$ then $v(f) = \max\{v(t_\mu) | m_\mu \neq 0\}$.

The following characterization was of fundamental importance in [MS21]: it yields an homeomorphism between the space of simple valuations and ML, both topologies being defined by simple convergence for the evaluations of multicurves.

**Theorem 4 (M-S).** The simple valuations on $\mathbb{C}[X]$ are precisely the $v_\lambda$ for $\lambda \in \text{ML}$.

In this paper we only consider simple valuations, so we write $v \in \text{ML}$ and $\lambda \in \text{ML}$ interchangeably.

The maximality condition of Definition 2 implies that for any $f, g \in \mathbb{C}[X]$, we have $v(f + g) \le \max\{v(f), v(g)\}$, with equality if $v(f) \neq v(g)$. Given a multiloop $\alpha$ with a self-intersection $p$, the two smoothings at $p$ give multiloops $\alpha_+$ and $\alpha_-$ and the trace relation reads $t_\alpha = \pm t_{\alpha_+} \pm t_{\alpha_-}$. Hence any valuation $v$ satisfies $v(t_\alpha) \le \max\{v(t_{\alpha_+}), v(t_{\alpha_-})\}$. The following lemma was proven by Dylan Thurston in [Thu09], and removes the condition $v(t_{\alpha_+}) \neq v(t_{\alpha_-})$ for the equality to hold. We provide an independent proof in Section 5 which relies on the geometry of real trees.

**Lemma 1 (Smoothing Lemma).** Let $\alpha$ be a taut multiloop, having a self intersection $p$ with smoothings $\alpha_+$ and $\alpha_-$. For any $v \in \text{ML}$ we have $v(t_\alpha) = \max\{v(t_{\alpha_+}), v(t_{\alpha_-})\}$.

Still, it will prove useful to consider valuations $v$ for which we always have $v(t_{\alpha_+}) \neq v(t_{\alpha_-})$. This holds over subsets of full measure in ML, as we now explain.

2.2. Acute valuations. We say that a simple valuation $v = v_\lambda \in \text{ML}$ is positive if $v(f) > 0$ for all non-constant $f \in \mathbb{C}[X]$. It is equivalent to say that $i(\lambda, \alpha) > 0$ for all $\alpha \in \pi_1(S)$, or $i(\lambda, \mu) > 0$ for all simple curves $\mu$. Such measured laminations are called filling or aperiodic in the literature.

We now introduce the notion of acute valuation, which will happen to be equivalent to the notion of maximal measured geodesic lamination, as we will show in Proposition 6.

**Definition 3.** A simple valuation $v \in \text{ML}$ is called acute if it is positive and for any non-trivial $\alpha, \beta \in \pi_1(S)$ we do not have $v(t_{\alpha\beta}) = v(t_\alpha t_\beta) = v(t_{\alpha\beta^{-1}})$.

**Lemma 2 (Unique smoothing).** A positive simple valuation $v_\lambda \in \text{ML}$ is acute if and only if for every taut multiloop $\alpha$, and smoothings $\alpha_\pm$ at a self-intersection, we have:

$$i(\lambda, \alpha_+) \neq i(\lambda, \alpha_-).$$

This justifies the terminology: $v \in \text{ML}$ is acute when for every such a multiloop $\alpha$, we have either $v(\alpha) = v(\alpha_+) > v(\alpha_-)$ or $v(\alpha) = v(\alpha_-) > v(\alpha_+)$, so the values $v(\alpha), v(\alpha_+), v(\alpha_-)$ are the lengths of an acute isosceles triangle with one shortest edge corresponding to either $v(\alpha_-)$ or $v(\alpha_+)$.

*Proof.* Suppose $v \in \text{ML}$ is acute. By decomposing $\alpha$ into connected components, we observe that the smoothing concerns at most two of them, and the proof reduces to the following cases.

(1) Either $\alpha$ is a single loop self-intersecting at $p$. Denote by $\gamma, \delta \in \pi_1(S, p)$ the elements such that $\alpha$ is homotopic to $\gamma\delta$. The tautness assumption implies that $\gamma$ and $\delta$ are non-trivial. Depending on the combinatorics of the intersection, one smoothing is homotopic to $\gamma\delta^{-1}$ and the other to the union $\gamma \cup \delta$. If $v(t_{\gamma\delta^{-1}}) = v(t_\gamma t_\delta)$ then, from the acute property $v(t_{\gamma\delta})$ differs from them, which contradicts the Smoothing Lemma 1.
(2) Otherwise the multiloop $\alpha$ has two components intersecting at $p$. We denote by $\gamma, \delta \in \pi_1(S, p)$ the (non-trivial) homotopy classes of the two components. Again, $\alpha_+$ and $\alpha_-$ are homotopic to $\gamma\delta$ and $\gamma\delta^{-1}$: the reasoning is the same.

Conversely, suppose $\alpha, \beta \in \pi_1(S)$ are non trivial. If they are powers of a same element, say $\alpha = \gamma^n$ and $\beta = \gamma^m$, then $v(t_{\alpha\beta}) = |n+m|v(t_\gamma)$ and $v(t_{\alpha\beta^{-1}}) = |n-m|v(t_\gamma)$. As $v(t_\gamma) > 0$, the equality $v(t_{\alpha\beta}) = v(t_\alpha t_\beta) = v(t_{\alpha\beta^{-1}})$ implies $mn = 0$ which is impossible.

Consider a hyperbolic structure on $S$, so that $\alpha$ and $\beta$ act on $\tilde{S} \simeq \mathbb{H}^2$ by hyperbolic translations along distinct axes $A_\alpha$ and $A_\beta$ respectively.

(1) If $A_\alpha \cap A_\beta = \{p\}$, then $p$ projects to a point on $\alpha \cap \beta$. The smoothings at $p$ are $\alpha\beta$ and $\alpha\beta^{-1}$. The assumption $i(\lambda, \alpha\beta) \neq i(\lambda, \alpha\beta^{-1})$ says that $v$ satisfies the condition $v(t_\alpha t_\beta) \neq v(t_{\alpha\beta^{-1}})$ ensuring that of Definition 3.

(2) If $A_\alpha \cap A_\beta = \emptyset$, then up to replacing $\beta$ with $\beta^{-1}$, we may assume the axes point in the same direction. Now, the axes of $\alpha\beta$ and $\beta\alpha$ intersect in a point $p$. This point projects to a self-intersection of $\alpha\beta$ which, after smoothing, gives alternatively $\alpha \cup \beta$ and $\alpha\beta^{-1}$. The assumption $i(\lambda, \alpha \cup \beta) \neq i(\lambda, \alpha\beta^{-1})$ says that $v$ satisfies the condition $v(t_\alpha t_\beta) \neq v(t_{\alpha\beta^{-1}})$ ensuring that of Definition 3.

□

**2.3. Strict valuations.** A simple valuation $v$ can be extended to $\mathbb{C}(X)$ by $v(f/g) = v(f) - v(g)$. We define its valuation ring $\mathcal{O}_v = \{f \in \mathbb{C}(X) \mid v(f) \le 0\}$ which has a unique maximal ideal $\mathcal{M}_v = \{f \in \mathbb{C}(X) \mid v(f) < 0\}$ and residue field $k_v = \mathcal{O}_v/\mathcal{M}_v$.

**Lemma 3.** A simple valuation $v = v_\lambda$ satisfies $k_v = \mathbb{C}$ if and only if for all distinct multicurves $\mu, \nu$ we have $i(\lambda, \mu) \neq i(\lambda, \nu)$.

Following [MS21], we will refer to them as strict valuations. We showed in [MS21, Lemma 3.4] that the set of non-strict valuations has zero measure in ML.

*Proof.* Suppose that $k_v = \mathbb{C}$ and consider two distinct multicurves $\mu$ and $\nu$. If $v(t_\mu) = v(t_\nu)$ then $t_\mu/t_\nu \in \mathcal{O}_v \setminus \mathcal{M}_v$ so there exists $\lambda \in \mathbb{C}^*$ such that $t_\mu/t_\nu - \lambda \in \mathcal{M}_v$ thus $v(t_\mu/t_\nu - \lambda) < 0$. But this implies $v(t_\mu - \lambda t_\nu) < v(t_\nu)$, which contradicts the third condition in Definition 2.

Conversely, suppose that $v$ takes distinct values on distinct multicurves and pick $f = P/Q \in \mathcal{O}_v \setminus \mathcal{M}_v$. Then $v(P) = v(Q)$, so the decompositions of $P$ and $Q$ in the basis of multicurves must be of the form $P = at_\mu + P'$ and $Q = bt_\mu + Q'$ with $a, b \in \mathbb{C}^*$ and $v(P'), v(Q') < v(t_\mu)$. This gives

$$f = \frac{at_{\mu} + P'}{bt_{\mu} + Q'} = \frac{a + P'/t_{\mu}}{b + Q'/t_{\mu}} = \frac{a}{b} \mod \mathcal{M}_{v}.$$

□

For a simple valuation $v = v_\lambda$, the set of values $\Lambda_v^+ = v(\mathbb{C}[X] \setminus \{0\})$ coincides with $\Lambda_v^+ = \{i(\lambda, \mu) \mid \mu \in \mathrm{MC}\}$ by Condition 3 in Definition 2, and has the structure of an abelian semi-group by Condition 2 in Definition 2. Its associated group is $\Lambda_v = v(\mathbb{C}(X)^*)$ and consists in differences of $\lambda$-lengths.

When $v$ is strict, the map $\mu \mapsto i(\lambda, \mu)$ is a bijection between MC and $\Lambda_v^+$. It is enlightening to think about the semi-group structure on multicurves obtained by pulling back the addition in $\Lambda_v^+$ in the following way. Let $\mu$ and $\nu$ be two multicurves, viewed as elements of $K(S, \mathbb{C})$. All smoothings of $\mu \cup \nu$ are multicurves $\xi$ with $i(\lambda, \xi) \le i(\lambda, \mu) + i(\lambda, \nu)$ and equality holds for exactly one of them corresponding to the “sum of $\mu$ and $\nu$ with respect to $v$”.
We define the rational rank of $v$ to be rat.rk($v$) = $\dim_{\mathbb{Q}} \Lambda_v \otimes \mathbb{Q}$. It satisfies the following Abhyankar inequality (see [Ota15] and references therein):

$$ \mathrm{rat.rk}(v) + \mathrm{tr. deg}(k_v) \leq \dim X $$

from which we deduce that if a simple valuation has maximal rational rank, that is rat.rk($v$) = $\dim X$, then it is strict.

**Proof of Proposition A.** By Lemma 3, we know that the first two properties of the proposition are equivalent. The Abhyankar inequality gives the implication $\mathrm{rat.rk}(v) = 6g-6 \implies \mathrm{tr. deg}(k_v) = 0$. The reverse implication will follow from the results of the remaining sections. Precisely, given a strict valuation $v$, we will define a tangent space $T_v$ML whose dimension is $\mathrm{rat.rk}(v)$. Then, we will show successively that this tangent space is isomorphic to the Bonahon and Thurston models. It is well-known that the latter has dimension $6g-6$, proving the last step of the proposition. $\square$

### 3. NEWTON POLYTOPES OF TRACE FUNCTIONS

This section relies on the following lemma whose proof is postponed to Section 5.

**Lemma 4.** The set of acute simple valuations has full measure in ML.

**Definition 4.** Let $v \in \mathrm{ML}$ be any simple valuation and $f \in \mathbb{C}[X]$ any function decomposed as $\sum m_\mu t_\mu$ in the multicurve basis.

- The multicurve $\mu \in \mathrm{Supp}(f)$ is $v$-extremal in $f$ if $v(t_\nu) < v(t_\mu)$ for every other $\nu \in \mathrm{Supp}(f)$.

- The multicurve $\mu$ is extremal in $f$ if it is $v$-extremal in $f$ for some $v$.

- The function $f$ is unitary if $m_\mu = \pm 1$ for any extremal multicurve in $f$.

- The Newton set of $f$ is the subset $\Delta(f) \subset \mathrm{MC}$ of extremal curves in $f$.

We observe that if $v$ is strict, then $\mu$ is $v$-extremal in $f$ if and only if $v(f) = v(t_\mu)$. Moreover, as strict valuations are dense in ML, a multicurve is extremal in $f$ if and only if it is $v$-extremal in $f$ for some strict $v$.

#### 3.1. Trace functions are unitary.

**Theorem 5 (Unitarity).** If $\alpha$ is a multiloop in $S$, then $t_\alpha$ is unitary.

**Proof.** Let $v$ be a strict acute valuation and $\mu$ be the unique multicurve such that $v(t_\alpha) = v(t_\mu)$. We must prove that $m_\mu = \pm 1$. We proceed by induction on the number of intersections of $\alpha$. If there are none, then the result is obvious. Otherwise, put $\alpha$ in taut position and consider its smoothings at an intersection. The Lemma 1 and the assumption that $v$ is acute imply that $v(t_{\alpha_+}) \neq v(t_{\alpha_-})$. One can suppose that $v(t_\alpha) = v(t_{\alpha_+}) > v(t_{\alpha_-})$. The coefficient of $t_\mu$ in $t_\alpha$ is the same as in $\pm t_{\alpha_+}$, so the induction hypothesis yields the result. $\square$

**Remark 1.** If we represent a taut multiloop as the projection of a banded link $L$ in $S \times [0, 1]$, we may decompose it in the basis of multicurves $\mu \in K(S, \mathbb{Z}[A^\pm])$ with blackboard framing. Then, the coefficient of $\mu$ in $L$ is equal to $A^{n^+ - n^-}$ where $n^\pm$ count the number of $\pm$-resolutions performed while transforming $L$ into $\mu$. At $A = -1$, we find the sign $(-1)^s$ for the extremal coefficient, where $s$ is the number of self-intersections of $\alpha$. The proof is the same, using inductively the skein relation.

**Remark 2.** We know from [Thu14] that MC indexes another basis $(t'_\mu)$ of $\mathbb{C}[X]$ for which the multiplicative structure constants are positive. The change of basis from $(t_\mu)$ to $(t'_\mu)$ is triangular, in the sense that if $\mu = \{\mu_1, \dots, \mu_k\}$ as a multiset, then $t'_\mu$ is a polynomial in the $t_{\mu_j}$ with leading
monomial $\pm t_{\mu_1} \dots t_{\mu_k}$. In this basis, the analog notion of Newton set will be the same (that is indexed by the same multicurves), and its extremal coefficients will be 1.

**Corollary 1.** *Any strict valuation is acute.*

*Proof.* Let $v$ be a strict valuation and consider a taut multiloop $\alpha$. Suppose $v(t_{\alpha_+}) = v(t_{\alpha_-})$. Then $t_{\alpha_+}$ and $t_{\alpha_-}$ must have the same $v$-extremal multicurve $\mu$. This defines an open condition on $v \in \mathrm{ML}$, namely that $v(t_\mu) > v(t_\nu)$ for all $\nu \in \Delta(t_{\alpha_-} t_{\alpha_+}) \setminus \{\mu\}$. But simple acute valuations are dense in $\mathrm{ML}$ so the same will hold for some acute valuation, contradicting Lemma 2. The conclusion follows from the converse part of that Lemma. $\square$

**3.2. Extremal multicurves of $t_\mu t_\nu$ and $\{t_\mu, t_\nu\}$**. Let $\mu$ and $\nu$ be multicurves in $S$ and consider a taut immersion $\mu \cup \nu$ for their union. Note that such an immersion is unique up to isotopy and permutations of parallel strands. This follows from the methods and results of [HS94], specifically Theorem 2.1 and the discussion ensuing Example 2.4.

We define the embedding $L_\mu(\nu)$ obtained by smoothing all intersections of $\mu \cup \nu$ with a left turn as we travel along a segment of $\mu$ and meet a segment of $\nu$. Smoothing all intersections with a right turn would yield $L_\nu(\mu)$.

This is the product considered by Luo in [Luo10], in particular his Lemma 8.1 shows that $L_\mu(\nu)$ is a multicurve (it has no trivial components) and his Theorem 2.1 describes several of its properties.

**Proposition 1.** Let $\mu, \nu$ be multicurves. The multicurves $L_\mu(\nu)$ and $L_\nu(\mu)$ are extremal for the product $t_\mu t_\nu$, and if $i(\mu, \nu) > 0$ then they are distinct.

*Proof.* If $i(\mu, \nu) = 0$ then $L_\mu(\nu) = \mu \cup \nu = L_\nu(\mu)$ and the statement follows.

Now suppose $i(\mu, \nu) > 0$. We first observe that among all smoothings of the union $\mu \cup \nu$, those which maximise $v_\mu$ are precisely $L_\mu(\nu)$ and $L_\nu(\mu)$. Indeed, we know from [Luo10, Theorem 2.1 (iii)] that $i(\mu, L_\mu(\nu)) = i(\mu, \nu) = i(\mu, L_\nu(\mu))$, but any other smoothing $\xi$ is made of segments of $\mu$ and $\nu$ which somewhere alternate between a left turn and right turn, thus forming a bigon with $\mu$ so that $i(\mu, \xi) < i(\mu, \nu)$. The fact that $L_\mu(\nu) \neq L_\nu(\mu)$ can be obtained from [Luo10, Corollary 8.2] which proves $i(L_\mu(\nu), L_\nu(\mu)) = 2i(\mu, \nu)$.

We deduce from the preceding discussion and the multiplication formula (4) that the distinct multicurves $L_\mu(\nu)$ and $L_\nu(\mu)$ both appear in the decomposition of $t_\mu t_\nu$, and are the only two maximizers of $v_\mu$. The condition $v_\lambda(L_\mu(\nu)) = v_\lambda(L_\nu(\mu))$ defines on $\lambda \in \mathrm{ML}$ a codimension-1 PL-subset (see [MS21, Lemma 1.6] for a proof). Hence a slight perturbation of the valuation $v_\mu$ off that subset in one direction or the other shows that $L_\mu(\nu)$ and $L_\nu(\mu)$ are indeed extremal terms in the product. $\square$

**Corollary 2.** Let $\mu$ and $\nu$ be multicurves such that $i(\mu, \nu) > 0$. Then $L_\mu(\nu)$ and $L_\nu(\mu)$ are extremal terms for the Poisson bracket $\{t_\mu, t_\nu\}$ whose coefficients in the basis of multicurves equals $\pm i(\mu, \nu)$.

*Proof.* We may deduce this using Theorem 3 which derives the Poisson bracket from the commutator in the skein algebra, but let us detail the computation without referring to the skein product.

For this, apply the Goldman formula (2) to the multiloops $\alpha, \beta$ and decompose for each $p \in \alpha \cap \beta$, the terms $t_{\alpha_p \beta_p}$ and $t_{\alpha_p \beta_p^{-1}}$ in the basis of multicurves $\xi \in \mathrm{MC}$, to find

$$ \{t_{\alpha}, t_{\beta}\} = \sum_{\xi} w_{\xi} t_{\xi} = \sum_{\xi} \left( \sum_{\sigma_{\xi}} \prod_{p} \sigma_{\xi}(p) \right) t_{\xi} $$
where $w_\xi = \sum_{\sigma_\xi} \prod_p \sigma_\xi(p)$ is the sum over the smoothings $\sigma_\xi: \alpha \cap \beta \to \{\pm 1\}$ of $\alpha \cup \beta$ yielding the multiloop $\xi$.

Now suppose that $\alpha = \mu$ and $\beta = \nu$ are multicurves with $i(\mu, \nu) > 0$ and consider the multicurves $\xi$ indexing the sum obtained by smoothing all intersections of $\mu \cup \nu$. Reasoning as in the proof of Proposition 1, we find that $L_\mu(\nu)$ and $L_\nu(\mu)$ both index a term for corresponding to a unique smoothing map $\sigma_\xi$ which is constant, equal to 1 or $-1$. $\square$

**Remark 3.** In the next section, we will prove that extremal coefficients of $\{t_\mu, t_\nu\}$ which are also extremal for $t_\mu t_\nu$ are values of the Thurston Poisson bracket $\{i_\mu, i_\nu\}_\lambda$ for $\lambda \in \text{ML}$, as announced in Theorem B. Our approach consists in reinterpreting the Thurston Poisson bracket $\{i_\mu, i_\nu\}_\lambda$ as a residual value $\{t_\mu, t_\nu\}_v$ of the Goldman Poisson bracket at $v = v_\lambda$.

The previous corollary shows that the (residual) Poisson bracket of multicurves determines their intersection number by the formula:

$$i(\mu, \nu) = \max\{\{i_\mu, i_\nu\}_\lambda \mid \lambda \in \text{ML}\}.$$

### 4. RESIDUAL POISSON STRUCTURE ON ML

**4.1. Tangent space.** Recall that ML embeds in the space of real functions on $\mathbb{C}[X]^* = \mathbb{C}[X] \setminus \{0\}$. We thus define its tangent space at $v$ as the set of maps $\phi = \frac{d}{ds}\big|_{s=0}v_s: \mathbb{C}[X]^* \to \mathbb{R}$, where $v_s$ is a family of simple valuations depending on a parameter $s \in [0, \epsilon]$ starting at $v_0 = v$, such that the map $s \mapsto v_s(t_\gamma)$ is differentiable for every curve $\gamma$.

Observe that the pair $(v, \phi): \mathbb{C}[X]^* \to [0, +\infty) \times \mathbb{R}$ satisfies all the axioms in Definition 2 of simple valuations provided the maximum is taken with respect the lexicographic ordering. When $v$ is a strict valuation, the lexicographic ordering depends only on the first coordinate and everything becomes much easier. As we only deal with the strict case, we consider straight away the following as a definition.

**Definition 5.** Let $v \in \text{ML}$ be a strict valuation. We define $T_v\text{ML}$ to be the set of group homomorphisms $\phi: \mathbb{C}(X)^* \to \mathbb{R}$ satisfying for any function $f \in \mathbb{C}[X]$ decomposed as $f = \sum m_\mu t_\mu$ in the linear basis of multicurves:

$$(6) \qquad \phi(f) = \phi(t_\nu) \text{ where } \nu \text{ is } v\text{-extremal in } f.$$

We will refer to this definition of the tangent space as the Goldman model. In this section, we define a symplectic structure on it, and will relate it to the models of Thurston and Bonahon introduced later on.

**Proposition 2.** For any strict valuation we have a sequence of natural isomorphisms:

$$T_v\text{ML} = \operatorname{Hom}(\Lambda_v^+, \mathbb{R}) = \operatorname{Hom}(\Lambda_v, \mathbb{R}) = \operatorname{Hom}(\Lambda_v \otimes \mathbb{Q}, \mathbb{R}) = \operatorname{Hom}(\mathbb{C}(X)^*/\mathcal{O}_v^\times, \mathbb{R})$$

where $\operatorname{Hom}$ is understood first as the space of semi-group homomorphisms and then as the space of group homomorphisms. In particular, $T_v\text{ML}$ has dimension $\text{rank}(v)$ (which is $\le \dim X$).

*Proof.* Recall that the map $\mu \mapsto v(t_\mu)$ is a bijection between the set of multicurves and $\Lambda_v^+$. We have $v(t_\mu) + v(t_\nu) = v(t_\xi)$ where $\xi$ is the $v$-extremal multicurve in $t_\mu t_\nu$. Given $\phi \in T_v\text{ML}$, the map $v(t_\mu) \mapsto \phi(t_\mu)$ gives by construction a homomorphism of semi-groups $\Lambda_v^+ \to \mathbb{R}$ and this construction can easily be reversed, giving the isomorphism $T_v\text{ML} = \operatorname{Hom}(\Lambda_v^+, \mathbb{R})$. The remaining isomorphisms are purely formal, noticing that $\mathcal{O}_v^\times$ is the kernel of the group homomorphism $v: \mathbb{C}(X)^* \to \mathbb{R}$. $\square$
**Definition 6.** For $f \in \mathbb{C}(X)$, we define the differential of the map $v \mapsto v(f)$ at $v$ by

$$d_v \log f: T_v \mathcal{M}L \to \mathbb{R}, \quad d_v \log f(\phi) = \phi(f).$$

We introduced the log to make the formula $d_v \log(fg) = d_v \log f + d_v \log g$ look more natural.

By Proposition 2, the elements $d_v \log f$ span $T_v^*\mathcal{M}L$. More precisely, we obtain a basis by letting $f$ range over a family of multicurves whose $v$-lengths form a basis of $\Lambda_v \otimes \mathbb{Q}$.

## 4.2. Residual Poisson structure.

**Proposition 3.** For all $f, g \in \mathbb{C}[X]$ and $v \in \mathcal{M}L$ we have: $v(\{f, g\}) \le v(f) + v(g)$.

*Proof.* By linearity of the Poisson bracket, it is sufficient to prove the inequality for $f = t_\mu$ and $g = t_\nu$ where $\mu$ and $\nu$ are multicurves. Then, by the Leibnitz formula, it is sufficient to prove it for curves $\mu$ and $\nu$. Suppose that $\mu$ and $\nu$ are in taut position and apply Goldman's formula (2). It is sufficient to prove that for any $p \in \mu \cap \nu$ we have $v(t_{\mu_p \nu_p} - t_{\mu_p \nu_p^{-1}}) \le v(t_\mu t_\nu)$, but this is a consequence of the Smoothing Lemma 1. $\square$

Given a strict valuation $v \in \mathcal{M}L$, the preceding proposition allows us to define the residual Poisson bracket at $v$ in the following way.

**Definition 7.** For $f, g \in \mathbb{C}[X]$ and $v \in \mathcal{M}L$ strict, we define $\{f, g\}_v \in k_v = \mathbb{C}$ by

$$\{f, g\}_v = \frac{\{f, g\}}{fg} \mod \mathcal{M}_v.$$

**Proposition 4.** There is an element $\pi_v \in \Lambda^2 T_v \mathcal{M}L$ representing this Poisson structure in the sense that for any $f, g \in \mathbb{C}[X]$ we have

$$\{f, g\}_v = \langle \pi_v, d_v \log(f) \wedge d_v \log(g) \rangle.$$

*Proof.* Let us fix $f$ and consider the map $F: \mathbb{C}[X] \to \mathbb{C}$ defined by $F(g) = \{f, g\}_v$. By the Leibnitz identity, this map satisfies $F(g_1g_2) = F(g_1) + F(g_2)$ thus extends to an element of $\mathrm{Hom}(\mathbb{C}(X)^*, \mathbb{C})$, and we must first show that it vanishes on $\mathcal{O}_v^\times$. Any $g \in \mathcal{O}_v^\times$ can be written $g = \alpha + h$ with $\alpha \in \mathbb{C}^*$ and $v(h) < 0$. We compute

$$F(g) = \frac{\{f, \alpha + h\}}{f(\alpha + h)} = \frac{\{f, h\}}{f(\alpha + h)}.$$

Since $v(h) < 0$ we have $v(f(\alpha+h)) = v(f)+v(\alpha+h) = v(f)$ and with Proposition 3, $v(\{f, h\}) < v(f)$ thus $F(g) \in \mathcal{M}_v$ and the claim is proved. What we have shown implies that there exists an element $\phi_f \in T_v \mathcal{M}L$ such that $F(g) = \langle \phi_f, d_v \log g \rangle$. As the Poisson bracket is antisymmetric, the same is true with the variables interchanged, and the conclusion follows. $\square$

## 5. ACTIONS OF $\pi_1(S)$ ON REAL TREES

A real tree is a metric space $T$ such that any two points $x, y \in T$ are joined by a unique injective segment. Recall that $S$ is a closed oriented surface of genus $g \ge 2$. We consider real trees with an action of $\pi_1(S)$ which is minimal, in the sense that the only subtrees $T' \subset T$ satisfying $\gamma T' \subset T'$ for all $\gamma \in \pi_1(S)$ are $\emptyset$ and $T$.

The action of an element $\alpha \in \pi_1(S)$ on $T$ either fixes a point and is called elliptic; otherwise it is a hyperbolic translation along an axis $A_\alpha$ with positive translation length $l(\alpha) = \min\{d(x, \alpha x) | x \in T\}$, and $d(x, \alpha x) = l(\alpha)$ if and only if $x \in A_\alpha$.

We face the following alternative. If all elements of $\pi_1(S)$ act elliptically, then it has a global fixed point; the minimality assumption implies that $T$ is reduced to a point. If at least one element
of $\pi_1(S)$ acts hyperbolically, then the union of all translation axes forms an invariant subtree (see [Pau89]), which equals $T$ by the minimality assumption.

An action of $\pi_1(S)$ is free when only the trivial element of $\pi_1(S)$ has a fixed point, or equivalently when $l(\alpha) > 0$ for all non-trivial $\alpha \in \pi_1(S)$. It is small when the stabilizer of any non-trivial segment in $T$ is cyclic. This condition appears naturally in the following important results.

**Theorem 6 (Culler-Morgan).** If $T_1, T_2$ are two real trees with a small and minimal action of $\pi_1(S)$, then there is an equivariant isometry $\Phi: T_1 \to T_2$ if and only if $l_1(\alpha) = l_2(\alpha)$ for all $\alpha \in \pi_1(S)$.

**Theorem 7 (Thurston, Skora).** To any measured lamination $\lambda \in \text{ML}$ one can associate a "dual tree" $T_\lambda$ together with a small minimal action of $\pi_1(S)$ on $T_\lambda$ such that $l(\alpha) = 2i(\lambda, \alpha)$ for all $\alpha \in \pi_1(S)$. Conversely, any tree with a small and minimal action of $\pi_1(S)$ is produced in this way.

Let us briefly outline the construction of the dual tree to a measured lamination, in the case where $\lambda$ is filling (or equivalently when the simple valuation $v_\lambda$ is positive).

First represent the filling measured lamination $\lambda$ on $S$ by a measured geodesic lamination for some fixed hyperbolic metric, and lift it in $\tilde{S}$ to obtain a $\pi_1(S)$-invariant measured geodesic lamination $\tilde{\lambda}$. Following [Ota01, Section 2.3], the tree $T_\lambda$ is the quotient of $\tilde{S}$ by the equivalence relation whose classes are given either by the closure of a connected component of $\tilde{S} \setminus \tilde{\lambda}$ or else by a leaf of $\tilde{\lambda}$ which is not contained in the previous classes. The quotient map $f: \tilde{S} \to T_\lambda$ is clearly $\pi_1(S)$-equivariant.

To describe the complement of a point $x \in T_\lambda$, consider its preimage $f^{-1}(\{x\})$. If it consists of a geodesic leaf of $\tilde{\lambda}$, then $T_\lambda \setminus \{x\}$ has 2 connected components. Otherwise it is isometric to the closure of an ideal hyperbolic polygon with $k$-sides, so $T_\lambda \setminus \{x\}$ has $k > 2$ components, and $x$ is called a *branch point* of $T$. In any case the connected components of $T_\lambda \setminus \{x\}$ have a cyclic orientation which is $\pi_1(S)$-invariant. These local cyclic orientations match together to give a global cyclic orientation on the Gromov boundary of $T_\lambda$. See [Wol11] for more details.

The map $f: \tilde{S} \to T_\lambda$ is not proper so does not extend to the Gromov boundary. A non-trivial element $\alpha \in \pi_1(S)$ acts on $\tilde{S} \simeq \mathbb{H}^2$ by hyperbolic translation along an axis which is transverse to $\lambda$ and thus crosses every leaf at most once. Hence the projection $f$ maps it bijectively to a geodesic in $T$ which, by equivariance, coincides with the axis $A_\alpha$. Hence we can associate to the attractive and repulsive points of $\alpha$ in $\partial\mathbb{H}^2 = \partial\pi_1(S)$ the corresponding endpoints of $A_\alpha$ in $\partial T$. This partially defined map between the Gromov boundaries of $\pi_1(S)$ and $T$ is $\pi_1(S)$-equivariant, orientation preserving and independent of the initial hyperbolic metric.

We recall the following proposition from [CP20] that we will use repeatedly.

**Proposition 5.** Let $\gamma, \delta$ be two hyperbolic isometries acting on a real tree $T$ with axes $A_\gamma$ and $A_\delta$. Then one of the following holds.

(1) If $A_\gamma \cap A_\delta = \emptyset$ then $l(\gamma\delta) = l(\gamma) + l(\delta) + 2D$ where $D$ is the distance between $A_\gamma$ and $A_\delta$.

(2) If $A_\gamma \cap A_\delta \neq \emptyset$, we denote by $D \in [0, +\infty]$ the length of the intersection.

(i) If $D > 0$ and the translation directions of $\gamma$ and $\delta$ on $A_\gamma \cap A_\delta$ coincide, or if $D = 0$ then $l(\gamma\delta) = l(\gamma) + l(\delta)$.

(ii) If $D > 0$ and the translation directions of $\gamma$ and $\delta$ on $A_\gamma \cap A_\delta$ are opposite, then $l(\gamma\delta) < l(\gamma) + l(\delta)$.

**Corollary 3.** Let $\gamma, \delta$ be two hyperbolic isometries acting on a real tree $T$ with axes $A_\gamma$ and $A_\delta$. When the segment $A_\gamma \cap A_\delta$ has positive length, we may compare the translation directions of $\gamma$ and
$\delta$: let cosign$(\gamma, \delta) = \pm 1$ be $+1$ if they coincide and $-1$ if they differ. One of the following holds.

$$(\gamma \cup \delta) \qquad l(\gamma) + l(\delta) < l(\gamma\delta) = l(\gamma\delta^{-1}) \qquad \text{if } A_{\gamma} \cap A_{\delta} = \emptyset$$

$$(\text{equil}) \qquad l(\gamma\delta) = l(\gamma) + l(\delta) = l(\gamma\delta^{-1}) \qquad \text{if } A_{\gamma} \cap A_{\delta} \text{ is reduced to a point}$$

$$(\gamma\delta^{-1}) \qquad l(\gamma\delta^{-1}) < l(\gamma) + l(\delta) = l(\gamma\delta) \qquad \text{if } l(A_{\gamma} \cap A_{\delta}) > 0 \text{ and cosign}(\gamma, \delta) = 1.$$

$$(\gamma\delta) \qquad l(\gamma\delta) < l(\gamma) + l(\delta) = l(\gamma\delta^{-1}) \qquad \text{if } l(A_{\gamma} \cap A_{\delta}) > 0 \text{ and cosign}(\gamma, \delta) = -1.$$

To illustrate how we will apply this corollary, let us propose a new proof of the smoothing lemma, independently from the equivalence between measured laminations and simple valuations (which we showed in [MS21] using the smoothing lemma).

Notice that the equivalence between measured laminations $\lambda$ and simple valuations $v$ recovers the smoothing lemma, because for a taut multiloop $\alpha$ we have for obvious geometric reasons $i(\lambda, \alpha) \ge \max\{i(\lambda, \alpha_+), i(\lambda, \alpha_-)\}$, and as $t_\alpha = \pm t_{\alpha_+} \pm t_{\alpha_-}$ we have $v_\lambda(t_\alpha) \le \max\{v_\lambda(t_{\alpha_+}), v_\lambda(t_{\alpha_-})\}$.

**Proof of the Smoothing Lemma 1.** Let us represent our measured lamination $\lambda$ by an action of $\pi_1(S)$ on a tree $T$. Fix a hyperbolic metric on $S$ to identify $\tilde{S} \simeq \mathbb{H}^2$.

Consider a taut multiloop $\alpha$ with a self-intersection point $p$, which may either be a self-intersection of a single component or an intersection between two components. We wish to prove that $i(\lambda, \alpha) = \max\{i(\lambda, \alpha_+), i(\lambda, \alpha_-)\}$.

Suppose first that $p$ is the intersection point between two components which we may write as $\gamma, \delta \in \pi_1(S, p)$. Lift $\gamma, \delta$ in $\tilde{S} \simeq \mathbb{H}^2$ starting from $\tilde{p}$ to obtain geodesics $\tilde{\gamma}, \tilde{\delta}$ which intersect transversely at $\tilde{p}$ and only at $p$. Consequently, their end-points are linked in $\partial\mathbb{H}^2$ with respect to the cyclic orientation. As the same holds for the end-points of $A_\gamma$ and $A_\delta$ in $\partial T$, we must have $A_\gamma \cap A_\delta \neq \emptyset$, so we are not in case $(\gamma \cup \delta)$ of Corollary 3 whence $l(\gamma) + l(\delta) = \max\{l(\gamma\delta), l(\gamma\delta^{-1})\}$.

Suppose now that $p$ is the self-intersection point of a single component of $\alpha$ which we may decompose as $\gamma\delta$ for $\gamma, \delta \in \pi_1(S, p)$. Lift $\gamma, \delta$ in $\mathbb{H}^2$ starting from $\tilde{p}$ to obtain quasi-geodesics $\tilde{\gamma}, \tilde{\delta}$. They intersect only at $\tilde{p}$ because, using the monodromy homomorphism associated to the developing map, another intersection point would imply an equality of the form $\gamma^m = \delta^n$ for some $m, n > 0$, which is impossible. The projection map $\tilde{S} \to S$ is a local diffeomorphism, so the germs of arcs $(\tilde{\gamma} \cup \tilde{\delta}, \tilde{p})$ and $(\gamma \cup \delta, p)$ are topologically equivalent. Hence, up to inversion and exchange of $\gamma$ and $\delta$, the endpoints of $\tilde{\gamma}$ and $\tilde{\delta}$ in $\partial\mathbb{H}^2$ have cyclic order $(\gamma_+, \gamma_-, \delta_+, \delta_-)$ as shown in Figure 3. Consequently we are not in case $(\gamma\delta)$ of Corollary 3, whence $l(\gamma\delta) = \max\{l(\gamma) + l(\delta), l(\gamma\delta^{-1})\}$. $\square$

<figure><img src="image_4.png" /><figcaption>FIGURE 3. Configuration of axes at a self-crossing</figcaption></figure>
**5.1. Trivalent real trees.** Recall that a point $x$ in a real tree is a *branch point* if $T \setminus \{x\}$ has at least three connected components. We will denote by $V(T)$ the set of branch points of $T$. A real tree is *trivalent* if any branch point disconnects it in three connected components.

A measured geodesic lamination $\lambda$ is called maximal if there is no measured geodesic lamination whose support is strictly bigger; or equivalently if the regions in its complement $S \setminus \lambda$ are isometric to the interiors of ideal hyperbolic triangles.

**Proposition 6.** Let $T$ be a real tree with a free minimal action of $\pi_1(S)$, associated to a filling measured lamination $\lambda$, and denote by $v$ the associated positive valuation.

The following are equivalent:

(1) $v$ is acute

(2) $T$ is trivalent

(3) $\lambda$ is maximal

*Proof.* (1) $\iff$ (2). Suppose $T$ is trivalent. Let $\alpha, \beta$ be non-trivial elements in $\pi_1(S)$ and consider their translation axes $A_\alpha, A_\beta \subset T$. From Proposition 3, we find that $l(\alpha\beta) = l(\alpha\beta^{-1}) = l(\alpha) + l(\beta)$ holds only when $A_\alpha$ and $A_\beta$ meet in exactly one point, which is forbidden by the trivalence assumption. Thus $v$ is acute.

Conversely, suppose $T$ is not trivalent. Consider a branch point $x \in T$ with valency $k > 3$. We denote by $C_1, \dots, C_k$ the components of $T \setminus \{x\}$. They decompose the Gromov boundary of $T$ into disjoint open subsets $\partial C_1, \dots, \partial C_k$. It is known that the set of pairs of ends of axes $A_\gamma$ for $\gamma \in \pi_1(S)$ form a dense subset of $\partial T \times \partial T$. One proof consists in considering the sequence of fixed points for the elements $\alpha\beta^n$: the attractive points converge to the image by $\alpha$ of the attractive point of $\beta$ and the repulsive points to the repulsive point of $\beta$. By minimality, the set of repulsive points of all $\beta$'s is dense in $\partial T$, and again by minimality the images of a given attractive point by all $\alpha$'s is dense in $\partial T$. Thus we can find two axes $A_\alpha, A_\beta$ whose ends are respectively in $\partial C_1 \times \partial C_3$ and $\partial C_2 \times \partial C_4$. These two axes meet exactly in $x$, and Proposition 3 implies that $l(\alpha\beta) = l(\alpha\beta^{-1}) = l(\alpha) + l(\beta)$, showing that $v$ is not acute.

(3) $\iff$ (2). Recall from the construction of the dual tree $T$ to a filling measured geodesic lamination $\lambda \subset S$ that the valency of a branch point in $T$ is equal to the number of sides to the corresponding hyperbolic ideal polytope in $\tilde{S} \setminus \tilde{\lambda}$. Hence $T$ is trivalent if and only if $\lambda$ is maximal. $\square$

It is well-known that the set of maximal laminations has a full measure in ML, see [LM08, Lemma 2.3]. Hence Proposition 6 implies the following corollary.

**Corollary 4.** The set of acute simple valuations has a full measure in ML.

**5.2. Bonahon cycles.** Let $T$ be a trivalent real tree with a free and minimal action of $\pi = \pi_1(S)$. To define its tangent space in the “moduli space” of such objects, imagine the combinatorial structure as being fixed while the distance function undergoes an infinitesimal deformation. Restricting attention to the variation of the distance between branch points, we obtain a symmetric map $c: V(T)^2 \to \mathbb{R}$ which is $\pi_1(S)$-invariant and satisfies $c(x, y) = c(x, z) + c(z, y)$ whenever $z$ belongs to the geodesic joining $x$ to $y$. We will refer to these maps as Bonahon cocycles and introduce them formally using a dual approach.

**Definition 8.** We define the space $\mathcal{B}(T)$ as the real vector space generated by pairs $(x, y)$ of elements in $V(T)$ subject to the relations:

(1) $(x, y) = (y, x)$ for all $x, y \in V(T)$

(2) $(x, y) = (x, z) + (z, y)$ if $z$ belongs to the geodesic joining $x$ to $y$.
The group $\pi = \pi_1(S)$ acts linearly on $\mathcal{B}(T)$ by $g(x, y) = (gx, gy)$ and Bonahon cocycles are the elements of $\text{Hom}_{\pi}(\mathcal{B}(T), \mathbb{R}) = \text{Hom}(\mathcal{B}(T)_{\pi}, \mathbb{R})$ where $\mathcal{B}(T)_{\pi}$ is the space of coinvariants.

**Proposition 7.** There is a unique alternating bilinear form $\cdot$ on $\mathcal{B}(T)$ such that for all pairs $(x, y), (z, t) \in V(T)^2$ we have:

(1) $(x, y) \cdot (z, t) = 0$ if the geodesics from $x$ to $y$ and from $z$ to $t$ are disjoint.

(2) $(x, z) \cdot (z, y) = \frac{\epsilon}{2}$ if $z$ belongs to the geodesic from $x$ to $y$, where $\epsilon = \pm 1$ is the cyclic order of the components $(h_x, h, h_y)$ of $T \setminus \{z\}$ such that $x \in h_x$, $y \in h_y$.

*Proof.* The intersection of $(x, y)$ and $(z, t)$ is either empty or has the form $(a, b)$. Decomposing $(x, y)$ and $(z, t)$ into segments involving $a$ and $b$ as in Figure 4, we are reduced by bilinearity and antisymmetry, to cases 1 or 2. This proves both uniqueness and existence. $\square$

<figure><img src="image_5.png" /><figcaption>FIGURE 4.</figcaption></figure>

It is an amusing exercise to show that this pairing is non-degenerate. Instead we will deduce it from Poincaré duality in Thuston's model in Section 6. Indeed, we are interested in the space $\mathcal{B}(T)_{\pi}$ endowed with the following pairing obtained by averaging the previous one, whose non-degeneracy will thus follow from standard arguments in cohomology.

**Proposition 8.** The following sum is finite, and it defines an alternating bilinear pairing on $\mathcal{B}(T)_{\pi}$:

$$ (x, y) \cdot_{\pi} (z, t) = \sum_{g \in \pi_1(S)} (x, y) \cdot g(z, t). $$

*Proof.* We only have to prove finiteness of the sum. For that, we view $T$ as the dual tree to a maximal measured lamination $\lambda$ on $S$. The vertices $x, y, z, t$ correspond to ideal triangles in $\tilde{S} \simeq \mathbb{H}^2$: choose $x_0, y_0, z_0, t_0$ in each one of them. Since $\pi_1(S)$ acts properly on $\mathbb{H}^2$, the geodesics $[x_0, y_0]$ and $g[z_0, t_0]$ are disjoint for all but a finite number of $g \in \pi_1(S)$. When they are disjoint, their projections in the tree are disjoint or meet as in the middle case of Figure 4, so their intersection vanishes. $\square$

We shall prove in Section 6 that $\mathcal{B}(T)_{\pi}$ is the antisymmetric part of $H_1(\tilde{S}, \mathbb{R})$ where $\tilde{S}$ is the orientation covering of the measured lamination $\lambda$, thus recovering Thurston's original point of view on the tangent space $T_{\lambda}ML$.

**5.3. The symplectomorphism theorem.** Fix a strict valuation $v \in ML$, and recall it identifies the set of multicurves with $\Lambda_v^+$. Let $T$ be a real tree with a free and minimal action of $\pi_1(S)$ representing $v$, so that $l(\alpha) = 2v(t_{\alpha})$ for all $\alpha \in \pi_1(S)$.

**Lemma 5.** The distance between two branch points in $T$ belongs to $\Lambda_v$.
**Proof.** This lemma can be deduced from repeated applications of Proposition 5. For instance, the distance between two disjoint axes $A_\gamma, A_\delta$ can be written $D = \frac{1}{2}(l(\gamma\delta) - l(\gamma) - l(\delta))$ and hence belongs to $\Lambda_v$. Instead, we may prove it as a direct consequence of a more conceptual construction for $T_v$ using Bass-Serre theory: we refer to Formula (3) in [Ota15, Section 4.1]. □

Given $\phi \in T_v\text{ML} = \text{Hom}(\Lambda_v, \mathbb{R})$, we define a corresponding $c_\phi \in \text{Hom}_\pi(\mathcal{B}(T), \mathbb{R})$ by setting $c_\phi(x, y) = \frac{1}{2}\phi(d(x, y))$ where $d$ is the distance in $T$. As $d$ is $\pi_1(S)$-invariant, $c$ is also, and the identity $c(x, z) = c(x, y) + c(y, z)$ for $y$ between $x$ and $z$ follows from the triangular equality satisfied by $d$. In other words, there is a well-defined map

$$ (7) \qquad \Psi: \left\{ \begin{array}{l} T_v\text{ML} \to \text{Hom}_\pi(\mathcal{B}(T), \mathbb{R}) \\ \phi \mapsto c_\phi \end{array} \right. $$

**Proposition 9.** The map $\Psi$ induces an isomorphism $T_v\text{ML} \simeq \text{Hom}_\pi(\mathcal{B}(T), \mathbb{R})$.

**Proof.** The linearity of $\Psi$ is obvious. We first prove injectivity: suppose $c_\phi = 0$. For any non-trivial $\alpha \in \pi_1(S)$, choose a branch point $x$ on its axis $A_\alpha$ so that the translation length satisfies $l(\alpha) = 2v(t_\alpha) = d(x, \alpha x)$. As $c_\phi(x, \alpha x) = \frac{1}{2}\phi(d(x, \alpha x))$ we get $\phi(v(t_\alpha)) = 0$, but $\Lambda_v$ is generated by the $v(t_\alpha)$ for $\alpha \in \pi_1(S)$ so $\phi = 0$.

This suggests the construction of the inverse, but this time we think of $\phi$ as a map $\phi: \mathbb{C}(X)^*/\mathcal{O}_v^\times \to \mathbb{R}$. Given $c \in \text{Hom}_\pi(\mathcal{B}(T), \mathbb{R})$, we define $\phi(t_\alpha) = c(x, \alpha x)$ for any simple curve $\alpha$, where $x$ is any branch point in $A_\alpha$ (by additivity of $c$, this does not depend on the branch point). We extend $\phi$ to any multicurve by linearity. Finally for any $f \in \mathbb{C}[X]^*$ we set $\phi(f) = \phi(t_\mu)$ where $\mu$ is the $v$-extremal multicurve in $f$. The point is to show that $\phi$ indeed belongs to $T_v\text{ML}$: as it satisfies Equation (6) by construction, it remains to prove that it is multiplicative.

We first show that the defining property $\phi(t_\gamma) = c(x, \gamma x)$ extends to all loops $\gamma \in \pi_1(S)$ by induction on the number of self intersections. Suppose $\gamma$ has $n > 0$ intersections. Let $p$ be one of them and denote by $\alpha, \beta$ the two elements of $\pi_1(S, p)$ such that $\gamma = \alpha\beta$. Since $v$ is acute, we have either $v(t_{\alpha\beta^{-1}}) < v(t_\alpha) + v(t_\beta) = v(t_{\alpha\beta})$ or $v(t_\alpha) + v(t_\beta) < v(t_{\alpha\beta^{-1}}) = v(t_{\alpha\beta})$, and we apply either case (2, i) or case (1) of [Pau89, Proposition 1.6] (which are unmodified in [CP20]).

In the first case, the axes $A_\alpha$ and $A_\beta$ intersect along a segment $xy$ such that both isometries push $x$ in the direction of $y$, and we have $x \in A_{\alpha\beta}$. If $l(\beta) \ge d(x, y)$ then $l(\alpha\beta) = d(x, \alpha\beta x) = d(x, y) + d(y, \alpha y) + d(\alpha y, \alpha\beta x)$ whence $c(x, \alpha\beta x) = c(x, y) + c(y, \alpha y) + c(y, \beta x) = c(y, \alpha y) + c(x, \beta x)$, with $x \in A_\beta$ and $y \in A_\alpha$. If $l(\beta) \le d(x, y)$ then $d(x, \alpha\beta x) = d(x, \beta x) + d(\beta x, \alpha\beta x)$ whence $c(x, \alpha\beta x) = c(x, \beta x) + c(z, \alpha z)$ with $z = \beta x \in A_\alpha$. Each time, the induction hypothesis applies, showing that both definitions of $\phi(t_\gamma)$ coincide.

In the second case, the axes $A_\alpha$ and $A_\beta$ are disjoint: let $xy$ be the geodesic joining them, and note that $x$ also belongs to the axes of $\alpha\beta$ and $\alpha\beta^{-1}$. By the induction hypothesis, $\phi(t_{\alpha\beta^{-1}})$ is equal to $c(x, \alpha\beta^{-1}x)$. Then $d(x, \alpha\beta^{-1}x) = d(x, \alpha\beta x)$, whence $c(x, \alpha\beta^{-1}x) = c(x, \alpha\beta x)$ and $2\phi(t_{\alpha\beta}) = c(x, \alpha\beta x)$ as claimed.

To finish the proof, we must consider $f, g \in \mathbb{C}[X]$ and show that $v(fg) = v(f) + v(g)$. If $\mu$ and $\nu$ are the $v$-extremal multicurves of $f$ and $g$, then the $v$-extremal multicurve of $fg$ is that of $t_\mu t_\nu$, denoted by $\xi$. We must show that $\phi(t_\mu t_\nu) = \phi(t_\xi) = \phi(t_\mu) + \phi(t_\nu)$. Let us prove more generally that if $\alpha = \alpha_1 \cup \cdots \cup \alpha_n$ is a multiloop then $\phi(t_\alpha) = \phi(t_{\alpha_1}) + \cdots + \phi(t_{\alpha_n})$, reasoning by induction on the self-intersection number of $\alpha$.

If the components $\alpha_j$ are disjoint, we may replace each one of them by its $v$-extremal smoothing, which remain disjoint, and the result follows from the definition of $\phi$. Hence suppose that $\alpha_1$ and $\alpha_2$ intersect at $p$. Up to changing the orientation of $\alpha_2$, we can suppose that $v(t_{\alpha_1\alpha_2}) = v(t_{\alpha_1}) + v(t_{\alpha_2})$.
The computation in the first case at the beginning of the proof shows that $\phi(t_{\alpha_1\alpha_2}) = \phi(t_{\alpha_1})+\phi(t_{\alpha_2})$. We also have $\phi(t_{\alpha_1\alpha_2}t_{\alpha_3}\cdots t_{\alpha_n}) = \phi(t_{\alpha_1\alpha_2})+\phi(t_{\alpha_3})+\cdots+\phi(t_{\alpha_n})$ by the induction hypothesis. $\square$

**Theorem 8.** The isomorphism $\Psi^*: \mathcal{B}(T)_\pi \to T_v^*\text{ML}$ preserves the symplectic form.

Explicitly, $\Psi^*(x, \alpha x) = d_v \log t_\alpha$ for all $\alpha \in \pi_1(S)$ and any branch point $x \in A_\alpha$. Indeed for all $\phi \in T_v\text{ML}$, equation (7) and Definition 6 yield:

$$ \Psi(\phi)(x, \alpha x) = \frac{1}{2}\phi(d(x, \alpha x)) = \phi(t_\alpha) = d_v \log(t_\alpha)(\phi). $$

*Proof.* Let $\alpha, \beta \in \pi_1(S)$ represent two simple curves in $S$. We must prove that $\{t_\alpha, t_\beta\}_v = \langle\pi_v, d_v \log t_\alpha \wedge d_v \log t_\beta\rangle$ equals $(x, \alpha x) \cdot_\pi (y, \beta y)$ for $x \in A_\alpha$ and $y \in A_\beta$. If $i(\alpha, \beta) = 0$ then both quantities are null. Otherwise, put $\alpha \cup \beta$ in taut position.

We first compute the sum defining $\{t_\alpha, t_\beta\}_v$, in which every intersection $p \in \alpha \cap \beta$ contributes to a term $\epsilon_p(t_{\alpha_p\beta_p} - t_{\alpha_p\beta_p^{-1}})t_\alpha^{-1}t_\beta^{-1} \mod \mathcal{M}_v$. The set $\alpha \cap \beta$ is in bijection with pairs of intersecting lifts $(\tilde{\alpha}, \tilde{\beta}) \subset \tilde{S} \times \tilde{S}$ modulo the diagonal action of $\pi_1(S)$. These lifts correspond bijectively to axes of the form $(A_{\tilde{\alpha}}, A_{\tilde{\beta}})$ in $T$ through the equivariant map $f: \tilde{S} \to T$ which preserves the cyclic orientations on the boundaries. Fixing representatives $\alpha, \beta \in \pi_1(S)$, every such pair is represented by some $(A_\alpha, gA_\beta)$ for a unique $g \in \langle\alpha\rangle\backslash\pi/\langle\beta\rangle$. Using again Proposition 5, we can rewrite

$$ (8) \qquad \{t_\alpha, t_\beta\}_v = \sum_{g \in \langle\alpha\rangle \backslash \pi/\langle\beta\rangle} \epsilon(A_\alpha, gA_\beta) = \sum_{g \in \langle\alpha\rangle \backslash \pi/\langle\beta\rangle} \epsilon(A_\alpha, A_{g\beta g^{-1}}) $$

where $\epsilon(A_\alpha, A_\beta) = \pm 1$ if $A_\alpha$ and $A_\beta$ are like in Figure 5 and $\epsilon(A_\alpha, A_\beta) = 0$ in any other configuration. Notice that this formula does not depend on the orientations of the axes, but on their cyclic orders at the branch points of the tree.

<figure><img src="image_6.png" /><figcaption>FIGURE 5. Sign rule for the axes.</figcaption></figure>

To end the proof, we fix $x \in A_\alpha$ and $y \in A_\beta$ to compare Formula (8) with $\sum_{g \in \pi}(x, \alpha x) \cdot (gy, g\beta y)$. Grouping them depending on the class of $g$ in $\langle\alpha\rangle\backslash\pi/\langle\beta\rangle$, we are reduced to the following equality, which is easily checked:

$$ \epsilon(A_\alpha, A_\beta) = \sum_{m, n \in \mathbb{Z}} (\alpha^n x, \alpha^{n+1} x) \cdot (\beta^m y, \beta^{m+1} y). $$

$\square$
# 6. IDENTIFYING THE SYMPLECTIC TANGENT MODELS

Following [PH92, Section 3.2], we recall Thurston's description for the tangent space to ML at a maximal measured lamination $\lambda$. We start with an orientation covering $p: S' \to S$, which is a ramified covering of degree 2 with one ramification point in each triangle of the complement $S \setminus \lambda$, and such that the preimage $p^{-1}(\lambda)$ is naturally co-oriented (meaning that its normal bundle is oriented). By the Gauss-Bonnet theorem, the set $R$ of ramification points has $4g - 4$ elements and the monodromy of the covering is a homomorphism $\rho: \pi_1(S \setminus R) \to \{\pm 1\}$ which is non-trivial around each ramification point. For later purposes, it will be useful to consider the orbifold $S^o$ where ramification points are thought as conical points of order 2.

Let $H_1(S', \mathbb{R})^{\pm}$ be the symmetric and antisymmetric part of $H_1(S', \mathbb{R})$ with respect to the in-
volution of the covering: they are orthogonal for the intersection form. Hence (half) the intersec-
tion form restricted to $H^1(S', \mathbb{R})^{-}$ is non-degenerate. We shall refer to this symplectic space as
Thurston's model for $T_{\lambda}^{*}$ ML. We can avoid introducing the covering by considering instead the
homology group $H_1(S^o, \mathbb{R}^{-})$ with coefficients in the $\pi_1(S^o)$-module $\mathbb{R}$ together the action given
by $\gamma.x = \rho(\gamma)x$. The twisted intersection product $H_1(S^o, \mathbb{R}^{-}) \times H_1(S^o, \mathbb{R}^{-}) \to H_0(S^o, \mathbb{R}) = \mathbb{R}$
coincides with the previous definition for Thurston's model. We will stick to this point of view in
the sequel.

Let $T$ be a trivalent real tree endowed with a free minimal action of $\pi_1(S)$, which is dual to a measured geodesic lamination $\lambda$. In the next section we first recover a model for $S^o$ which depends only on $T$: our space will be an infinite dimensional CW-complex homotopic to $S^o$. As a consequence, its fundamental group is canonically attached to $T$ and its homology will be easy to compute from $T$. We will use it extensively to prove that the Bonahon model $B(T)_\pi$ and Thurston model $H_1(S^o, \mathbb{R}^-)$ are naturally isomorphic symplectic vector spaces.

## 6.1. A homotopical construction of the orbifold tree.

6.1.1. *Idea of the construction.* We first construct a space corresponding to the tree $T$ with an orbifold singularity of order 2 at every branch point. As the topology of $T$ induced by the metric is not given by a cell structure, our first task is to build a cellular model of $T$.

To motivate our construction, let us begin with the following analogy: suppose we wish to replace the real line $\mathbb{R}$, with its usual topology, by a CW-complex whose 0-cells consist of the set $\mathbb{Q}$ of rationals with the discrete topology. We may first add a 1-cell between every pair of distinct 0-cells to make the space connected. This creates a 1-cycle for every triple of distinct rational points, so we attach a 2-cell to each of those in order to make the space simply connected. Now every 4-tuple of rationals form the vertices of a 2-cycle, to which we attach a 3-cell, and so on. In the limit, we obtain Milnor’s join construction $EQ$, which is a space homotopic to $\mathbb{R}$ endowed with a free and proper action of $\mathbb{Q}$.

We shall play a similar game, replacing $\mathbb{R}$ by the real tree $T$, and $\mathbb{Q}$ by its set of branch points $V(T)$. We first attach a 1-cell to every pair of distinct branch points. However, we close the triangle $(x, y, z)$ only if $x, y, z \in V(T)$ belong to a same geodesic in $T$. Then we go on similarly in higher dimensions, so that our space will resemble $EQ$ in restriction to any geodesic of $T$. At this stage, we have a space on which $\pi_1(S)$ acts freely and properly. As it is contractible, its quotient by $\pi_1(S)$ is homotopic to $S$. Next comes the orbifold singularity: in homotopy theory, this is represented by a $K(\mathbb{Z}/2, 1)$-space, that is $\mathbb{RP}^\infty$. It remains to blow up the preceding construction at every branch point and insert an infinite dimensional projective space. This construction may look complicated but we shall do it in one shot and few lines below.
6.1.2. *Formal construction.* A half-edge of $T$ is a pair $(x, h)$ consisting in a branch point $x$ of $T$ and a connected component $h$ of $T \setminus \{x\}$; we sometimes just write $h$ as it determines $x$. Let us construct a CW-complex $T^0$ whose 0-skeleton is the set of half-edges of $T$. First, we attach a 1-cell denoted $(h, k)$ between every pair of half-edges incident to the same branch point $x \in V(T)$. Now at every branch point $x$, the incident half-edges $h_1, h_2, h_3$ form a triangle homeomorphic to $\mathbb{R}P^1$ through which we attach a copy of $\mathbb{R}P^\infty$. For the moment, $T^0$ is a disjoint union of infinite projective spaces indexed by the set of branched points $V(T)$, we call it the *orbifold part*.

Now, we add a *connecting part*, as suggested in Figure 6. Fix $\epsilon > 0$ small enough, say $1/3$. Consider a finite set $W$ of branch points $\{x_0, \dots, x_n\}$ aligned on a geodesic of $T$, and denote $h_i, k_i$ the half-edges incident to $x_i$ containing (a non-empty) part of that geodesic. The $n$-cell $\Delta_W = \{(r_x)_{x \in W} \in [0, 1-\epsilon]^W, \sum_{x \in W} r_x = 1\}$ is a truncated simplex, and there is an obvious inclusion $\Delta_{W'} \subset \Delta_W$ when $W' \subset W$. The face of $\Delta_W$ truncated at $x_i$ corresponds to the set $\Delta_W^{x_i}$ of families $(r_x)$ satisfying $r_{x_i} = 1 - \epsilon$. We attach $\Delta_W^{x_i}$ to the orbifold part of $T^0$ through the map $W \setminus \{x_i\} \to \{h_i, k_i\}$ sending the branch point $x_j$ to the half-edge based at $x_i$ which contains $x_j$, as in Figure 6. The 1-cells $\Delta_{\{x,y\}}$ will be called edges and denoted $(x, y)$.

<figure><img src="image_7.png" /><figcaption>FIGURE 6. Attaching a 3-cell in $T^0$</figcaption></figure>

As promised, the action of $\pi_1(S)$ on $T^0$ is now proper so that we may form the quotient $\Sigma^0 = T^0/\pi_1(S)$. The following lemma shows that $\Sigma^0$ and $S^0$ are homotopic. Interestingly, the proof consists in constructing an equivariant map $F: T^0 \to \tilde{S}^0$, which plays the role of a (non-existing) retraction for the map $f: \tilde{S} \to T$.

**Lemma 6.** Let $\tilde{S}^0$ be the covering of the orbifold $S^0$ corresponding to the kernel of the natural map $\pi_1(S^0) \to \pi_1(S)$. There exists a $\pi_1(S)$-equivariant map $F: T^0 \to \tilde{S}^0$ which induces a homotopy equivalence between $\Sigma^0$ and $S^0$.

*Proof.* To define $F$, represent $T$ as the dual tree to a measured geodesic lamination $\lambda$ and consider the collection of circles inscribed in each triangle of the complement $S \setminus \lambda$: they lift to a collection of circles $C_x$ in $\tilde{S} \simeq \mathbb{H}^2$ indexed by $x \in V(T)$. Moreover, the half-edges incident to $x$ correspond bijectively to the three intersection points of $C_x$ with the leaves of the lamination, see Figure 7.

The covering $\tilde{S}^0$ is obtained from $\mathbb{H}^2$ by drilling out the interior of $C_x$ and gluing back a copy of $\mathbb{R}P^\infty$ along $\mathbb{R}P^1 \simeq C_x$ for every $x \in V(T)$. By construction, the orbifold $S^0$ is homotopic to the quotient $\tilde{S}^0/\pi_1(S)$.

We now proceed to the construction of an equivariant map $F: T^0 \to \tilde{S}^0$. There is already an identification between the orbifold parts of both spaces, so that we are left to define the map $F$ on the connecting part.
<figure><img src="image_8.png" /><figcaption>FIGURE 7. Lifting a geodesic to $\mathbb{H}^2$ (done with Geogebra)</figcaption></figure>

For every pair $(x, y) \in V(T)^2$, we must define a path $F(x, y)$ in $\tilde{S}^o$ connecting the points of $C_x$ and $C_y$ identified to the endpoints $h_x, h_y$ of $(x, y)$ in $T^o$. A first guess would be to consider the geodesic path $\gamma$ between the points $h_x$ and $h_y$. This path actually projects to the geodesic joining $x$ to $y$ in $T$. However it may intersect a forbidden circle $C_z$, in which case it enters its circumscribed ideal triangle $\Delta_z$ by one side and leaves it by another. Call $p_z$ the ideal vertex at the intersection of these two sides. We can homotope $\gamma$ inside $\Delta_z$ to a path avoiding $C_z$ which stays on the side containing $p_z$, see Figure 7.

Moreover, we can choose those paths in such a way that $F$ is $\pi_1(S)$-equivariant. Let us now consider a triple of points $x, z, y$ lying on a geodesic of $T$ in that order. We have defined $F(x, z)$, $F(z, y)$ and $F(x, y)$: it is not hard to see that the region enclosed by the three arcs and the boundary of $C_z$ does not contain any other circle, hence it can be filled by a triangle: this extends $F$ to the 2-skeleton of $T^o$. This procedure can be continued to define an equivariant map $F: T^o \to \tilde{S}^o$, which induces a map $\bar{F}: \Sigma^o \to S^o$.

We would like to show that $\bar{F}$ is a homotopy equivalence. The space $\tilde{S}^o$ is Eilenberg-MacLane and Lemma 7 below shows that so is $T^o$, hence it is sufficient to prove that $\bar{F}$ induces an isomorphism between fundamental groups. Behold the following commutative diagram, and observe that the five lemma reduces the statement to showing that $F_*$ is an isomorphism.

$$ \begin{tikzcd} 0 \arrow[r] \arrow[d, "F_*"] & \pi_1(T^o) \arrow[r] \arrow[d, "\tilde{F}_*"'] & \pi_1(S) \arrow[r] \arrow[d, "="] & 0 \\ 0 \arrow[r] & \pi_1(\tilde{S}^o) \arrow[r] & \pi_1(S^o) \arrow[r] & 0 \end{tikzcd} $$

This last statement is clear from the fact that $\pi_1(T^o)$ and $\pi_1(\tilde{S}^o)$ are both isomorphic to a free product of copies of $\mathbb{Z}/2\mathbb{Z}$ indexed by $V(T)$ (see again Lemma 7). $\square$

6.2. **Homology of $T^o$**. The homology of $T^o$ can be computed from its finite sub-complexes, which are easy to understand thanks to the following lemma. For a finite set $W \subset V(T)$, let $T^o_W$ to be the union of cells involving $W$ only: a cell belongs to $T^o_W$ when all its 0-faces are of the form $(x, h)$ for $x \in W$. We define $T^o_W$ to be the sub-complex of $T^o_W$ whose connecting part reduces to the 1-cells $(x, y)$ for $x, y \in W$ such that there is no other element in $W$ on the geodesic joining them.
In more intuitive terms, $T_W^o$ is a collection of $\mathbb{RP}^\infty$ indexed by $W$, connected in a tree-like fashion given by the embedding of $W$ in $T$.

**Lemma 7.** For all finite $W \subset V(T)$, the cell-complex $T_{(W)}^o$ retracts by deformation on $T_W^o$.

*Proof.* We define the retraction by induction on the maximal dimension of the truncated simplices $\Delta_U \subset T_{(W)}^o$. Let $U = \{x_0, \dots, x_n\}$ correspond to one of them, it is the intersection of $W$ with a geodesic in $T$. We retract $\Delta_U$ by deformation onto the union of $\Delta_{U'}$ for $U' \subset U$ ranging over all subsets which do not contain both $x_0$ and $x_n$. This procedure stops when $U = \{x, y\}$ and $x, y$ are closest neighbours in $W$. $\square$

We define a 1-cochain $\rho \in C^1(T^o, \{\pm 1\})$ sending every 1-cell of $T^o$ to $-1$. It is a cocycle because the 2-cells of $T^o$, being either hexagons (orbifold part) or squares (contained in some $\Delta_W$ for $W$ of cardinal 3), have an even number of 1-faces. The geometric idea underlying this definition is that any half-edge stands for a local coorientation of the lamination $\tilde{\lambda}$, say pointing to the closest singular point. Following an edge $e$ in $T^o$ (transverse to $\tilde{\lambda}$), we arrive at the other end with the opposite co-orientation, giving $\rho(e) = -1$.

This cocycle defines a homomorphism $\rho: \pi_1(T^o) \to \mathbb{R}$ and we denote by $\mathbb{R}^-$ the vector space $\mathbb{R}$ with the action $\gamma \cdot x = \rho(\gamma)x$. Our first task is to compute the homology of $T^o$ with coefficients in $\mathbb{R}$ and $\mathbb{R}^-$.

**Lemma 8.** We have $H_k(T^o, \mathbb{R}) = 0$ if $k \neq 0$, $H_k(T^o, \mathbb{R}^-) = 0$ if $k \neq 1$ and

$$H_1(T^o, \mathbb{R}^-) \simeq \mathcal{B}(T).$$

*Proof.* Observe that $T^o = \lim_{\to} T_{(W)}^o$ as $W$ exhausts the finite subsets of the countable set of branch points $V(T)$ and by Lemma 7, $T_{(W)}^o$ retracts by deformation on $T_W^o$, thus $H_*(T^o, \mathbb{R}^\pm) = \lim_{\to} H_*(T_W^o, \mathbb{R}^\pm)$.

We may forget about the cocycle $\rho$ while computing the untwisted real homology, and further retract the space $T_W^o$ on a wedge of infinite projective spaces. Thus $H_0(T_W^o, \mathbb{R}) = \mathbb{R}$ and $H_k(T_W^o, \mathbb{R}) = 0$ for $k > 0$ so the same goes for $T^o$.

We now return to the twisted homology of $T_W^o$. For this we consider the double cover $T'_W \to T_W^o$ corresponding to $\rho$ and compute the untwisted homology of the total space: it splits into the $\pm 1$-eigenspaces of the involution which coincide with $H_*(T_W^o, \mathbb{R}^\pm)$ respectively. The space $T'_W$ is homotopy equivalent to a graph with vertex set $W$, and two edges above each edge $e$ of $T_W^o$ connecting its endpoints with opposite orientations, as shown below.

<figure><img src="image_9.png" /></figure>

It follows that $H_0(T'_W, \mathbb{R}) = \mathbb{R}$ and $H_k(T'_W, \mathbb{R}) = 0$ if $k > 1$. Moreover $H_1(T'_W, \mathbb{R})$ has a basis formed by the cycles $c(x, y) \in H_1(T'_W, \mathbb{R})$ indexed by the edges $(x, y)$ of $T'_W$, which consist in making a round trip from $x$ to $y$, following the arrows. The Galois involution of $T'_W$ exchanges the orientation of $c(x, y)$, so $H_1(T'_W, \mathbb{R}^-)$ is freely generated by pairs $(x, y)$ where $x, y$ are closest neighbours in $W$.
Taking the limit as W converges to V(T), we obtain $H_k(T^o, \mathbb{R}^-) = 0$ for $k = 0$ and $k > 1$. If an edge $(x, y)$ gets subdivided into $(x, z)$ and $(z, y)$ as W increases, we have $c(x, y) = c(x, z) + c(z, y)$ which is compatible with the equality $(x, y) = (x, z) + (z, y)$, and provides the desired isomorphism for the inductive limit of $H_1(T_W^o, \mathbb{R}^-)$. □

**6.3. Homology of the quotient** $\Sigma^o = T^o/\pi$. Let us write $\pi = \pi_1(S)$ for short. The cocycle $\rho$ on $T^o$ is $\pi$-invariant, so it induces a homomorphism $\pi_1(\Sigma^o) \to \{\pm 1\}$ that we also denote $\rho$. The $\pi$-equivariant homotopy equivalence between $\Sigma^o$ and $S^o$ thus yields a homomorphism $\pi_1(S^o) \to \{\pm 1\}$. By the remark following Lemma 7, this homomorphism is the co-orientation monodromy of $\lambda$, so its kernel corresponds to the covering $S' \to S^o$. Consequently, we may deduce the homology of $S^o$ with coefficients in $\mathbb{R}^\pm$ from that of $\Sigma^o$ with the same coefficients.

The 2-fold covering $S'$ of $S^o$ ramified over $R$ satisfies $\chi(S') = 2\chi(S) - (4g - 4) = 8 - 8g$ by the Riemann-Hurwitz formula. As $H_*(S^o, \mathbb{R}^\pm) = H_*(S', \mathbb{R})^\pm$ we get that $H_*(S^o, \mathbb{R}) = H_*(S, \mathbb{R})$ whereas $H_k(S^o, \mathbb{R}^-) = 0$ if $k \neq 1$ and $\dim H_1(S^o, \mathbb{R}^-) = 6g - 6$.

On the other hand, we can compute $H_*(T^o/\pi, \mathbb{R}^\pm)$ from $H_*(T^o, \mathbb{R}^\pm)$ using the Cartan-Leray spectral sequence. Its second page is $E_{p,q}^2 = H_p(\pi, H_q(T^o, \mathbb{R}^\pm))$ and converges to $H_{p+q}(\Sigma^o, \mathbb{R}^\pm)$. Lemma 8 implies that, with both coefficients, the second page has only one line, whence the isomorphisms:

$$H_*(\Sigma^o, \mathbb{R}) = H_*(\pi, \mathbb{R}) = H_*(S, \mathbb{R}), \quad H_*(\Sigma^o, \mathbb{R}^-) = H_{*-1}(\pi, \mathcal{B}(T)).$$

This yields the proposition that we are after.

**Proposition 10.** Given a maximal measured lamination $\lambda$ with associated covering $S' \to S$ and corresponding tree $T$, there is a natural isomorphism

$$H_1(S', \mathbb{R})^- = H_1(\Sigma^o, \mathbb{R}^-) = H_0(\pi, \mathcal{B}(T)) = \mathcal{B}(T)_\pi.$$

We also have $H_k(\pi, \mathcal{B}(T)) = 0$ for $k = 1, 2$. Observe that from Poincaré duality we get $H_2(\pi, \mathcal{B}(T)) = H^0(\pi, \mathcal{B}(T)) = \mathcal{B}(T)^\pi = 0$. It is not surprising that $\mathcal{B}(T)$ has no invariant cycles as $\pi$ acts freely on $V(T)$. We do not have a similar explanation for the vanishing of $H^1(\pi, \mathcal{B}(T))$.

**6.4. Intersection form.** In the commutative diagram below, the first colon is a Galois covering of surfaces with group $\pi$. We have the identifications $H_1(\tilde{S}', \mathbb{R})^- = H_1(\tilde{S}^o, \mathbb{R}^-) = H_1(T^o, \mathbb{R}^-) = \mathcal{B}(T)$ and $H_1(S', \mathbb{R})^- = H_1(S^o, \mathbb{R}^-) = \mathcal{B}(T)_\pi$.

<figure><img src="image_10.png" /></figure>

**Proposition 11.** The isomorphisms $H_1(\tilde{S}', \mathbb{R})^- = \mathcal{B}(T)$ and $H_1(S', \mathbb{R})^- = \mathcal{B}(T)_\pi$ preserve the symplectic forms.

*Proof.* Let us begin with the first isomorphism. Recall that we defined an equivariant map $F: T^o \to \tilde{S}^o$: it sends the cell $(x, y)$ to a path $F(x, y)$ joining the orbifold points corresponding to $x$ and $y$ and avoiding all other orbifold points. As the homology of the orbifold part of $T^o$ with coefficients $\mathbb{R}^-$ vanishes identically, these paths actually define cycles in $H_1(\tilde{S}^o, \mathbb{R}^-)$, which in $H_1(\tilde{S}', \mathbb{R})$ are represented geometrically by $c(x, y) = \tilde{p}^{-1}(F(x, y)) \subset \tilde{S}'$. Notice that these cycles have a natural orientation (given by the co-orientation of the lifted lamination $\tilde{\lambda}'$).
Recalling the definition of the pairing in $\mathcal{B}(T)$ given in Proposition 7, it suffices to compute $c(x, y) \cdot c(z, t)$ in the case where $(x, y)$ and $(z, t)$ are disjoint or consecutive.

In the first case, the cycles $c(x, y)$ and $c(z, t)$ are also disjoint so their intersection vanishes. In the second case, the cycles $c(x, z)$ and $c(z, y)$ only intersect in a neighborhood of $z$ which looks like in the right hand side of Figure 8. The lifted cycles $c(x, z)$ and $c(z, y)$ go straight through the intersection point, oriented as shown. Analyzing the two possible cases, we find that the signs coincide.

<figure><img src="image_11.png" /><figcaption>FIGURE 8. Double covering over a branching point</figcaption></figure>

Let us now consider the quotient. We showed in Section 6.3 that $H_1(S', \mathbb{R}) = H_1(\tilde{S}', \mathbb{R})_\pi$. The result follows from the fact that the intersection form on $H_1(S', \mathbb{R})$ coincides with the averaged intersection form on $H_1(S', \mathbb{R})$. $\square$

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Email address: christopher-lloyd.simon@ens-lyon.fr