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

Julien Marché  Sorbonne Université, IMJ-PRG, 75252 Paris cédex 05, France
<EMAIL_ADDRESS>and  Christopher-Lloyd Simon  Université de Lille,
CNRS, UMR 8524 - Laboratoire Paul Painlevé, F-59000 Lille, France
<EMAIL_ADDRESS>
###### Abstract.

We study the space of measured laminations $\mathrm{ML}$ 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 $\mathrm{ML}$.
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 $g\geq 2$. 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
$t_{\gamma}\colon\rho\mapsto\operatorname{tr}\rho(\gamma)$ 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 $\\{\alpha_{1},\ldots,\alpha_{n}\\}$ 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

(1) $t_{\alpha}=\sum m_{\mu}t_{\mu}$

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 [MS20].

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
$\operatorname{\Delta}(t_{\alpha})\subset\mathrm{MC}$ 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 $\operatorname{Supp}(f)=\\{\mu\in\mathrm{MC},m_{\mu}\neq 0\\}$.

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

The Newton set $\operatorname{\Delta}(f)$ 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 $\alpha=\\{\alpha_{1},\ldots,\alpha_{n}\\}$, the function
$t_{\alpha}$ is unitary in the sense that $m_{\mu}=\pm 1$ for all
$\mu\in\operatorname{\Delta}(t_{\alpha})$.

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)
$\\{t_{\alpha},t_{\beta}\\}=\sum_{p\in\alpha\cap\beta}\epsilon_{p}(t_{\alpha_{p}\beta_{p}}-t_{\alpha_{p}\beta_{p}^{-1}})$

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 $p$.

Intuitively, our second result says that the Newton polytope of $\\{f,g\\}$ is
contained in the Newton polytope of $fg$, and interprets the coefficients of
$\\{f,g\\}$ at the extremal multicurves of $fg$ in terms of the symplectic
structure of the space $\mathrm{ML}$ of measured laminations on $S$.

######  Theorem B (Extremal structure constants for the Poisson Bracket).

Let $\mu$ and $\nu$ be two multicurves. For
$\xi\in\operatorname{\Delta}(t_{\mu}t_{\nu})$ denote $m_{\xi}$ its coefficient
in $\\{t_{\mu},t_{\nu}\\}$.

If $m_{\xi}\neq 0$ then $\xi\in\operatorname{\Delta}(\\{t_{\mu},t_{\nu}\\})$.
In any case, $m_{\xi}=\\{i_{\mu},i_{\nu}\\}(\lambda)$ is the Poisson bracket
of the length functions $i_{\mu}(\cdot)=i(\mu,\cdot)$ on $\mathrm{ML}$ at any
generic measured lamination $\lambda$ satisfying
$i(\xi,\lambda)=i(\mu\cup\nu,\lambda)$.

To rephrase the statement, we may decompose $\mathrm{ML}$ in subsets indexed
by $\operatorname{\Delta}(t_{\mu}t_{\nu})$, defined by
$E_{\xi}=\\{\lambda\in\mathrm{ML}\,|\,i(\xi,\lambda)=i(\mu\cup\nu,\lambda)\\}$
which have disjoint interiors: then the function $\\{i_{\mu},i_{\nu}\\}$ is
constant over $E_{\xi}$ and equal to the coefficient of $\xi$ in
$\\{t_{\mu},t_{\nu}\\}$.

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
$\operatorname{\Delta}(t_{\alpha}t_{\beta})=\\{c_{1}\cup c_{3},c_{2}\cup
c_{4},\gamma,\delta\\}$, whereas
$\Delta(\\{t_{\alpha},t_{\beta}\\})=\\{\gamma,\delta\\}$.

The Newton set of $t_{\alpha}t_{\beta}$ decomposes $\mathrm{ML}$ 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.

$c_{1}$$c_{2}$$c_{3}$$c_{4}$$\alpha$$\beta$$\gamma$$\delta$ Figure 1. Product
and Poisson bracket in a sphere with four punctures.

Strong relations between the symplectic structures on $X$ and $\mathrm{ML}$
had already been observed, for instance in [PP91] or [SB01]. However, the
relation explained here seems to be new.

#####

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 [MS20]. 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 $(t_{\mu})$ 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
$\mathbb{C}[t_{1},\ldots,t_{n}]$ by

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

where $t^{\mu}=t_{1}^{\mu_{1}}\cdots t_{n}^{\mu_{n}}$, and
$\langle\cdot,\cdot\rangle$ stands for the usual scalar product. This degree
(the opposite of) a monomial valuation. For
$P\in\mathbb{C}[t_{1},\ldots,t_{n}]$, a monomial $t^{\mu}$ is an extremal
point of its usual Newton polytope $\operatorname{\Delta}(P)$ if $m_{\mu}\neq
0$ and for some $d\in\mathbb{R}^{n}$ the maximum defining $\deg_{d}$ is
attained uniquely at $t^{\mu}$.

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 $v(f+g)\leq\max(v(f),v(g))$ 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. Specialists will notice that such
valuations centered at infinity of the affine variety $X$ 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 $v(t_{\mu})=i(\lambda,\mu)$ but it was
not clear what should be $v(f)$ for a general element $f\in\mathbb{C}[X]$.

In our previous article [MS20], we showed that the space of measured
laminations $\mathrm{ML}$ can be identified with the space of _simple_
valuations $v:\mathbb{C}[X]\to\\{-\infty\\}\cup\mathbb{R}_{\geq 0}$, simple
meaning monomial with respect to the multicurve basis:

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

This justifies our definition for the Newton set of $f=\sum m_{\mu}t_{\mu}$ as
the set of $\mu\in\operatorname{Supp}(f)$ such that the maximum in (3) is
attained uniquely at $t_{\mu}$ for some $v\in\mathrm{ML}$.

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 $t_{1},t_{2},t_{3}$ the trace functions along
these components, we find that $t_{\alpha}\in\mathbb{Z}[t_{1},t_{2},t_{3}]$.
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
$\operatorname{\Delta}(t_{\alpha})$ from the $\alpha_{i}\in\pi_{1}(S)$, and
the unitarity property is not an obvious one.

We believe that the Newton set $\operatorname{\Delta}(t_{\mu}t_{\nu})$ for two
multicurves $\mu$ and $\nu$ is worth studying as it gives some interesting
information on the structure constants for the multiplication of
$\mathbb{C}[X]$ in the basis of multicurves. As an illustration, we will show
the following proposition. For multicurves $\mu,\nu\in\mathrm{MC}$, let
$L_{\mu}(\nu)$ be the multicurve obtained from $\mu\cup\nu$ by smoothing every
intersection with a left turn as we travel along a segment of $\mu$ which
meets a segment of $\nu$. This “product” has been introduced and studied by
Luo (see [Luo10] and references therein).

######  Proposition A.

For all multicurves $\mu,\nu\in\mathrm{MC}$, if $i(\mu,\nu)>0$ then
$L_{\mu}(\nu)$ and $L_{\nu}(\mu)$ are distinct, and both belong to
$\operatorname{\Delta}(t_{\mu}t_{\nu})$ and
$\operatorname{\Delta}(\\{t_{\mu},t_{\nu}\\})$ simultaneously. Moreover, the
coefficients of $L_{\mu}(\nu)$ and $L_{\nu}(\mu)$ in $\\{t_{\mu},t_{\nu}\\}$
are $\pm i(\mu,\nu)$.

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
$\mathrm{ML}$. However, we can define the dual Newton polytope of a function
$f\in\mathbb{C}[X]$ as $\operatorname{\Delta}^{*}(f)=\\{v\in\mathrm{ML}\mid
v(f)\leq 1\\}$. Moreover, using the order structure we could define the poset
of faces of $\operatorname{\Delta}(f)$, whose 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 $\mathrm{ML}$ provides a notion of volume, thus
we may ask for the topological meaning of the volume
$\operatorname{Vol}\operatorname{\Delta}^{*}(t_{\alpha})$ when $\alpha$ is a
multiloop. It vanishes unless $\alpha$ is filling, meaning it intersects every
simple curve.

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 $\operatorname{Mod}(S)$ as a bound on their
complexity tends to infinity. More precisely, fix $\beta$ another filling
multiloop, and denote $m_{g}>0$ the volume of the moduli space of hyperbolic
metrics on $S$ for the Weil-Petersson form, then :

$\lim_{R\to\infty}\frac{\operatorname{Card}\\{\varphi\in\operatorname{Mod}(S)\mid
i(\varphi(\alpha),\beta)\leq
R\\}}{R^{6g-6}}=\frac{\operatorname{Vol}\operatorname{\Delta}^{*}(t_{\beta})\operatorname{Vol}\operatorname{\Delta}^{*}(t_{\alpha})}{m_{g}}$

The identification between measured laminations and simple valuations implies,
using Equation (3), that the Newton dual polytope
$\operatorname{\Delta}^{*}(f)$ of $f\in\mathbb{C}[X]$ equals the intersection
of $\operatorname{\Delta}^{*}(t_{\mu})$ for $\mu\in\operatorname{\Delta}(f)$.
These “elementary cones” $\\{v\in\mathrm{ML}\mid v(t_{\mu})<1\\}$ are
described by explicit sets of linear inequalities in any PL chart of
$\mathrm{ML}$, and the volume of their intersection is computable. This yields
a constructive procedure to compute Mirzakhani’s constant
$\operatorname{Vol}\operatorname{\Delta}^{*}(t_{\alpha})$, and shows that it
depends only on $\operatorname{\Delta}(t_{\alpha})$. 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
$\gamma_{1},\ldots,\gamma_{6g-6}\in\pi_{1}(S)$ and
$x_{1},\ldots,x_{6g-6}\in\mathbb{C}$. 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
$\operatorname{\mathrm{rat.rk}}(v)+\operatorname{\mathrm{tr.deg}}(k_{v})\leq
6g-6$. Here we will show the following.

######  Proposition B (Characterising strict valuations).

For a valuation $v$ associated to a measured lamination $\lambda$, the
following properties are equivalent.

  1. (1)

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

  2. (2)

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

  3. (3)

The $\mathbb{Q}$-vector space generated by the set of lengths $i(\lambda,\mu)$
for $\mu\in\mathrm{MC}$ 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 $v(f)\leq 0$. 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, 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.

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

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 Morgan-Shalen work, the key notion allowing to relate measured laminations
to valuations is the action of $\pi_{1}(S)$ on real trees. We like to think of
this dynamical point of view as lying in between the two others as in the
following schematic table.

Topological/Geometrical | Dynamical | Functional
---|---|---
measured foliation | action of $\pi_{1}(S)$ | valuation
measured geodesic laminations | on a real tree | intersection function

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: is given by the residual Poisson bracket on
$T_{v}\mathrm{ML}$ 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 $S^{\prime}\to S$ known as the
orientation cover of the lamination. The group $H^{1}(S^{\prime},\mathbb{R})$
splits into a symmetric and antisymmetric part with respect to involution of
the covering $S^{\prime}\to S$. The space $H^{1}(S^{\prime},\mathbb{R})^{-}$
with the cup-product form is the geometrical model for
$T_{\lambda}\mathrm{ML}$.

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
$c\colon V(T)^{2}\to\mathbb{R}$ on the set of pairs of trivalent vertices of
$T$ which satisfy:

  1. (1)

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

  2. (2)

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

  3. (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 a variation on the notion
of transverse cocycles introduced by Bonahon (see [Bon96]). 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
$S^{\prime}$. We may wonder which of these three symplectic identifications
persist for more general actions of Fuchsian groups on real trees.

###### Plan of the paper

  1. 1 Background
  2. 2 Measured laminations and simple valuations
  3. 3 Newton polytopes of trace functions
  4. 4 Residual Poisson structure on $\mathrm{ML}$
  5. 5 Actions of $\pi_{1}(S)$ on real trees
  6. 6 Identifying the symplectic tangent models

#### Acknowledgements

We wish to thank Chris Leininger and Maxime Wolff for useful discussions
around this project, as well as Patrick Popescu-Pampu for reading the
introduction.

##  1\. Background

###  1.1. Algebra of functions on the character variety

Let $S$ be a closed connected and oriented surface of genus $g\geq 1$. We
denote by $X$ the character variety of $S$, which is the algebraic quotient of
its representation variety
$\operatorname{Hom}(\pi_{1}(S),\operatorname{SL}_{2}(\mathbb{C}))$ by the
conjugacy action of $\operatorname{SL}_{2}(\mathbb{C})$, defined as the
spectrum of its algebra of functions:

$\mathbb{C}[X]=\mathbb{C}[\operatorname{Hom}(\pi_{1}(S),\operatorname{SL}_{2}(\mathbb{C}))]^{\operatorname{SL}_{2}(\mathbb{C})}.$

A celebrated result of Procesi gives generators and relations for this algebra
(which holds for any finitely generated group). For $\alpha\in\pi_{1}(S)$, we
note $t_{\alpha}\in\mathbb{C}[X]$ the trace function given by
$t_{\alpha}([\rho])=\operatorname{tr}\rho(\alpha)$.

######  Theorem 1 (Procesi).

The algebra $\mathbb{C}[X]$ is generated by the $t_{\alpha}$ for
$\alpha\in\pi_{1}(S)$. The ideal of relations is generated by $t_{1}-2$ and
$t_{\alpha}t_{\beta}-t_{\alpha\beta}-t_{\alpha\beta^{-1}}$ for all
$\alpha,\beta\in\pi_{1}(S)$.

######  Definition 1.

A _multiloop_ on $S$ is a class of continuous maps $f\colon\Gamma\to S$ from
compact $1$-dimensional manifolds $\Gamma$ to $S$ which is not homotopic to a
constant on any component. We consider it modulo the relation declaring $f$
equivalent to $f^{\prime}\colon\Gamma^{\prime}\to S$ when there is a
homeomorphism $\varphi:\Gamma\to\Gamma^{\prime}$ such that
$f^{\prime}\circ\phi$ is homotopic to $f$.

A _multicurve_ is a multiloop which is represented by an embedding, we denote
$\mathrm{MC}$ the set of multicurves.

A multiloop amounts to a finite multiset $\\{\alpha_{1},\ldots,\alpha_{n}\\}$
of non-trivial conjugacy classes in $\pi_{1}(S)$ considered up to inversion:
we define $t_{\alpha}=\prod_{i=1}^{n}t_{\alpha_{i}}$, in particular
$t_{\emptyset}=1$. The components of a multicurve must be non contractible,
simple and pairwise disjoint.

Applying the trace relation recursively to reduce the number of self
intersections in multiloops, one may deduce part of the following theorem
[PS00]. The linear independance requires more work.

######  Theorem 2.

The family $(t_{\mu})_{\mu\in\mathrm{MC}}$ forms a linear basis of the algebra
$\mathbb{C}[X]$.

###  1.2. Deriving the Poisson algebra from the Kauffman algebra

The multiplication and the Poisson bracket on $\mathbb{C}[X]$ appear naturally
as by-products of the Kauffman algebra $K(S,R)$ over some ring $R$ containing
an invertible element $A$. Recall that a banded link in an oriented 3-manifold
is an oriented submanifold diffeomorphic to a finite union of annuli.

As an $R$-module, the Kauffman algebra is the quotient of the free module over
isotopy classes of banded links $L$ in $S\times[0,1]$, by the sub-module
generated by Kauffman’s local skein relations $[\bigcirc\cup
L]=(-A^{2}-A^{-2})[L]$ and $[L_{\times}]=A[L_{+}]+A^{-1}[L_{-}]$ where
$L_{\times},L_{+},L_{-}$ are banded links differing in a ball as shown in
Figure 2.

$L_{\times}$$L_{+}$$L_{-}$ Figure 2. Local skein relation.

The product is given by stacking two banded links one above the other.
Precisely, $[L_{0}][L_{1}]=[\Phi_{0}(L_{0})\cup\Phi_{1}(L_{1})]$ for maps
$\Phi_{i}(x,t)=(x,(t+i)/2)$ of $S\times[0,1]$ into itself.

Any multicurve $\mu$ on $S$ can be seen as a banded link $[\mu]$ in
$S\times[0,1]$ by considering a tubular neighborhood $S\times\\{1/2\\}$ often
called its blackboard framing.

######  Theorem 3 (Przytycki/Turaev).

Using the previous notations:

  1. (1)

The module $K(S,R)$ is a free $R$-module generated by multicurves.

  2. (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. (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:

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

Let us explain more precisely the first point. 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 $\xi\colon C\to\\{\pm 1\\}$, set $w_{\xi}=\sum_{c}\xi(c)$ and
consider the diagram $D_{\xi}$ obtained after smoothing each crossing $c\in C$
according to the sign $\xi(c)$, and removing the $n_{\xi}$ trivial components
which appear in the result. We have in $K(S,R)$ the formula:

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

which, after grouping terms corresponding to a same diagram $[D_{\xi}]$,
yields the decomposition of $[L]$ in the basis of multicurves. We use this
formula in order to understand the product of two multicurves: intuitively,
the product consists in taking the disjoint union and summing over all
possible smoothings.

With the second point, 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: we can replace the notion of banded
link with the simplest notion of multiloop that we defined previously.

The Kauffman algebra is not completely necessary for our purposes. However, we
find it conceptually 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 the third point and the obvious associativity of multiplication the
Kauffman algebra. Finally, in the context of this article, it provides an
alternative formula for the Poisson bracket which enlightens Theorem B: a
smoothing $\xi$ which is extremal for $[\alpha][\beta]$ is also extremal for
the Poisson bracket $\\{t_{\alpha},t_{\beta}\\}$, and its coefficient is $\pm
w_{\xi}$, this integer will be interpreted 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 any simple curve $\gamma$.
This “functional” point of view can be extended to define a map
$v_{\lambda}:\mathbb{C}[X]\to\\{-\infty\\}\cup[0,+\infty)$ satisfying
$v(0)=-\infty$ and for all $f=\sum m_{\mu}t_{\mu}$ decomposed in the
multicurve basis:

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

where $i(\lambda,\mu)=i(\lambda,\mu_{1})+\cdots+i(\lambda,\mu_{n})$ for a
multicurve $\mu$ with components $\mu_{1},\ldots,\mu_{n}$. By [MS20,
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 [MS20, Definition
1.1].

######  Definition 2.

A simple valuation on $\mathbb{C}[X]$ is a map
$v:\mathbb{C}[X]\to\\{-\infty\\}\cup\mathbb{R}_{\geq 0}$ satisfying:

  1. (1)

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

  2. (2)

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

  3. (3)

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

The following characterization was of fundamental importance in [MS20]: it
yields an homeomorphism between the space of simple valuations and
$\mathrm{ML}$, 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
$\lambda\in\mathrm{ML}$.

The maximality condition of Definition 2 implies that for any
$f,g\in\mathbb{C}[X]$, we have $v(f+g)\leq\max(v(f),v(g))$, 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
$v(t_{\alpha})\leq\max(v(t_{\alpha_{+}}),v(t_{\alpha_{-}}))$. The following
lemma was proven by Dylan Thurston in [Thu09], and removes the condition
$v(t_{\alpha_{+}})\neq v(t_{\alpha_{-}})$ for 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 with a self intersection $p$. We denote as
usual by $\alpha_{+}$ and $\alpha_{-}$ the two smoothings of $\alpha$ at $p$.
For any $v\in\mathrm{ML}$ we have
$v(t_{\alpha})=\max(v(t_{\alpha_{+}}),v(t_{\alpha_{-}}))$.

In fact, the inequality $v(t_{\alpha_{+}})\neq v(t_{\alpha_{-}})$ holds under
certain conditions on $v$ which are satisfied over subsets of full measure in
$\mathrm{ML}$, as we now explain.

###  2.2. Acute valuations

We say that a simple valuation $v=v_{\lambda}\in\mathrm{ML}$ 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.

######  Definition 3.

A simple valuation $v\in\mathrm{ML}$ 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 $v_{\lambda}\in\mathrm{ML}$ 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: $v\in\mathrm{ML}$ is acute when for every such
a multiloop $\alpha$, the values
$i(\lambda,\alpha),i(\lambda,\alpha_{+}),i(\lambda,\alpha_{-})$ are the
lengths of an acute isosceles triangle such that one of the longest edges
corresponds to the multiloop $\alpha$ with greatest self-intersection number.

###### Proof.

Suppose $v\in\mathrm{ML}$ 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. (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. 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
$v(t_{\alpha\beta})=\lvert n+m\rvert v(t_{\gamma})$ and
$v(t_{\alpha\beta^{-1}})=\lvert n-m\rvert v(t_{\gamma})$. As
$v(t_{\gamma})>0$, we have $mn=0$ 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. (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. (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

Given a simple valuation $v$ we can extend it to $\mathbb{C}(X)$ by
$v(f/g)=v(f)-v(g)$. We define its valuation ring
$\mathcal{O}_{v}=\\{f\in\mathbb{C}(X)\mid v(f)\leq 0\\}$ 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 [MS20], we will refer to them as _strict_ valuations. We showed that
the set of non-strict valuations has zero measure in $\mathrm{ML}$.

###### 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 means $v(t_{\mu}-\lambda
t_{\nu})<v(t_{\nu})$, 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 $v(P)=v(Q)$:
decomposing $P$ and $Q$ in the basis of multicurves, they are necessarily of
the form $P=at_{\mu}+P^{\prime}$ and $Q=bt_{\mu}+Q^{\prime}$ with
$a,b\in\mathbb{C}^{*}$ and $v(P^{\prime}),v(Q^{\prime})<v(t_{\mu})$. This
gives

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

∎

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
$\mathrm{MC}$ 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 $i(\lambda,\xi)\leq
i(\lambda,\mu)+i(\lambda,\nu)$ 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
$\operatorname{\mathrm{rat.rk}}(v)=\dim_{\mathbb{Q}}\Lambda\otimes\mathbb{Q}$;
it satisfies the following Abhyankar inequality (see [Ota15] and references
therein):

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

from which we deduce that if a simple valuation has maximal rational rank,
that is $\operatorname{\mathrm{rat.rk}}(v)=\dim X$, then it is strict. We will
prove the converse in Section 4.

##  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 $\mathrm{ML}$.

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

  * -

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
$\operatorname{\Delta}(f)\subset\mathrm{MC}$ 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
$\mathrm{ML}$, 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 $v$ 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. Otherwize, put $\alpha$ in taut position and
consider its smoothings at an intersection: Lemma 1 and the assumption that
$v$ is acute shows that $v(t_{\alpha_{+}})\neq v(t_{\alpha_{-}})$. One can
suppose that $v(t_{\alpha})=v(t_{\alpha_{+}})>v(t_{\alpha_{-}})$. The
coefficient of $\mu$ in $t_{\alpha}$ is the same up to sign as in
$t_{\alpha_{+}}$, and the induction hypothesis gives the result. ∎

######  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 $\mu$. Putting $A=-1$,
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 an ingenious triangular change of basis in
$\mathbb{C}[X]$ make the multiplicative structure constants positive. In this
basis, the Newton polytope will be the same, 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 $v\in\mathrm{ML}$. 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. ∎

###  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 since any
two are related by triangle moves [HS94], but there are no triangles as the
components of $\mu$ and $\nu$ are simple.

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
$i(\mu,\nu)>0$ they are disctinct.

###### Proof.

If $i(\mu,\nu)=0$ then $L_{\mu}(\nu)=\mu\cup\nu=L_{\nu}(\mu)$ and there is
nothing more to say.

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 multiplication formula (4) that the distinct multicurves
$L_{\mu}(\nu)$ and $L_{\nu}(\mu)$ both appear in the decomposition of
$t_{\mu}t_{\nu}$ as the unique 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 [MS20, 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. ∎

######  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
$\pm i(\mu,\nu)$.

##  4\. Residual Poisson structure on $\mathrm{ML}$

###  4.1. Tangent space

Recall that $\mathrm{ML}$ embeds in the space of real functions on
$\mathbb{C}[X]^{*}$. We thus define its tangent space at $v$ as the set of
maps $\phi=\frac{d}{dt}\big{|}_{t=0}v_{t}:\mathbb{C}[X]^{*}\to\mathbb{R}$,
where $v_{t}$ is a family of simple valuations depending on a parameter
$t\in[0,\epsilon[$ starting at $v_{0}=v$, such that the map $t\mapsto
v_{t}(t_{\gamma})$ 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 $v\in\mathrm{ML}$ be a strict valuation. We define $T_{v}\mathrm{ML}$ to
be the set of group morphisms $\phi:\mathbb{C}(X)^{*}\to\mathbb{R}$ satisfying
for any function $f\in\mathbb{C}[X]$ decomposed as $f=\sum m_{\mu}t_{\mu}$ in
the linear basis of multicurves:

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

We can give equivalent descriptions, several of which serve in the sequel.

######  Proposition 2.

For any strict valuation we have a sequence of natural isomorphisms:

$T_{v}\mathrm{ML}=\operatorname{Hom}(\Lambda_{v}^{+},\mathbb{R})=\operatorname{Hom}(\Lambda_{v},\mathbb{R})=\operatorname{Hom}(\Lambda_{v}\otimes\mathbb{Q},\mathbb{R})=\operatorname{Hom}(\mathbb{C}(X)^{*}/\mathcal{O}_{v}^{\times},\mathbb{R})$

where homomorphisms are understood first as semi-group morphisms and then as
group morphisms. In particular, $T_{v}\mathrm{ML}$ has dimension
$\operatorname{\mathrm{rat.rk}}(v)$ (which is $\leq\dim X$).

###### 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 $\phi\in
T_{v}\mathrm{ML}$, the map $v(t_{\mu})\mapsto\phi(t_{\mu})$ gives by
construction a morphism of semi-groups $\Lambda_{v}^{+}\to\mathbb{R}$ and this
construction can easily be reversed, giving the isomorphism
$T_{v}\mathrm{ML}=\operatorname{Hom}(\Lambda_{v}^{+},\mathbb{R})$. The
remaining isomorphisms are purely formal, noticing that
$\mathcal{O}_{v}^{\times}$ is the kernel of the group morphism
$v:\mathbb{C}(X)^{*}\to\mathbb{R}$. ∎

For any $f\in\mathbb{C}(X)$, we can consider the differential of the map
$v\mapsto v(f)$ at $v$. We will denote it by $d_{v}\log
f:T_{v}\mathrm{ML}\to\mathbb{R}$ and define it by $d_{v}\log f(\phi)=\phi(f)$.
We introduced the “$\log$” in order to make the following formula look more
natural:

$d_{v}\log(fg)=d_{v}\log f+d_{v}\log g.$

By Proposition 2, the elements $d_{v}\log f$ span $T_{v}^{*}\mathrm{ML}$: 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 $v\in\mathrm{ML}$ we have: $v(\\{f,g\\})\leq
v(f)+v(g)$.

###### 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 of Theorem 2. It is sufficient to prove that for any $p\in\mu\cap\nu$
we have $v(t_{\mu_{p}\nu_{p}}-t_{\mu_{p}\nu_{p}^{-1}})\leq v(t_{\mu}t_{\nu})$,
but this is a consequence of the Smoothing Lemma 1. ∎

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

######  Definition 6.

For $f,g\in\mathbb{C}[X]$ and $v\in\mathrm{ML}$ 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 $\pi_{v}\in\Lambda^{2}T_{v}\mathrm{ML}$ 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 $\Psi:\mathbb{C}[X]\to\mathbb{C}$ defined
by $\Psi(g)=\\{f,g\\}_{v}$. By the Leibnitz identity, this map satisfies
$\Psi(g_{1}g_{2})=\Psi(g_{1})+\Psi(g_{2})$ thus extends to an element of
$\operatorname{Hom}(\mathbb{C}(X)^{*},\mathbb{C})$, 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

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

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 $\Psi(g)\in\mathcal{M}_{v}$ and the
claim is proved. What we have shown implies that there exists an element
$\psi_{f}\in T_{v}\mathrm{ML}$ such that $\Psi(g)=\langle\psi_{f},d_{v}\log
g\rangle$. As the Poisson bracket is antisymmetric, the same is true with the
variables interchanged, and the conclusion follows. ∎

##  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, and this segment is geodesic. We
consider real trees on which $\pi_{1}(S)$ acts in a minimal way, meaning
without any invariant proper subtree.

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 $l(\alpha)=\min\\{d(x,\alpha
x)\mid x\in T\\}$, and $d(x,\alpha x)=l(\alpha)$ if and only if $x\in
A_{\alpha}$.

We face the following alternative. If all of $\pi_{1}(S)$ acts 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, which equals $T$ by the minimality assumption.

An action of $\pi_{1}(S)$ is free if only the trivial element of $\pi_{1}(S)$
has a fixed point, or equivalently if $l(\alpha)>0$ for all non-trivial
$\alpha\in\pi_{1}(S)$. It is small if 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.

To any measured lamination $\lambda$ in $\mathrm{ML}$ one can associate a
“dual tree” $T_{\lambda}$ together with a small minimal action of $\pi_{1}(S)$
on $T_{\lambda}$. Moreover (Skora theorem), 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 by a leaf of $\tilde{\lambda}$ which is
not contained in the previous classes. The quotient map $f\colon\tilde{S}\to
T_{\lambda}$ 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 in a geodesic leaf of $\lambda$,
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 end points 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 hold.

  1. (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. (2)

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

    1. (i)

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

    2. (ii)

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

As a first corollary of this proposition, we obtain a proof of the smoothing
lemma.

###### Proof of the Smoothing Lemma 1.

Let us represent our measured lamination $\lambda$ by an action of
$\pi_{1}(S)$ on a tree $T$. Consider a multiloop $\alpha$ with a self-
intersection $p$: we wish to prove that
$v(t_{\alpha})=\max(v(t_{\alpha_{+}}),v(t_{\alpha_{-}}))$. Again, we consider
two cases.

In the first case, $\alpha$ consists in a single component that we can write
$\alpha=\gamma\delta$ in the fundamental group based at $p$. Then after
inspection of all possibilities in Proposition 5, we see that
$l(\gamma\delta)=\max(l(\gamma)+l(\delta),l(\gamma\delta^{-1}))$, as desired.

In the second case, $\alpha=\gamma\cup\delta$ and we must prove
$l(\gamma)+l(\delta)=\max(l(\gamma\delta),l(\gamma\delta^{-1}))$. This holds
if and only if the axes of $\gamma$ and $\delta$ intersect (Case 2 of
Proposition 5).

Base the fundamental group at $p\in\gamma\cap\delta$ and fix a hyperbolic
metric on $S$. Now lift $\gamma$ and $\delta$ in $\tilde{S}$ starting from
$p$, to obtain quasi-geodesic curves $\tilde{\gamma}$ and $\tilde{\delta}$ in
$\tilde{S}\simeq\mathbb{H}^{2}$ intersecting transversely at $\tilde{p}$.
Since $\gamma\cup\delta$ is taut, their lifts $\tilde{\gamma}$ and
$\tilde{\delta}$ intersect only at $p$ ([Thu09, see Definition 8 to Lemma
10]). Their end-points should therefore be linked in $\partial\mathbb{H}^{2}$
with respect to the cyclic ordering. As the same holds for the end-points of
$A_{\gamma}$ and $A_{\delta}$ in $\partial T$, these axes must also intersect,
and the lemma is proved. ∎

###  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.

######  Proposition 6.

Let $T$ be a real tree with a free minimal action of $\pi_{1}(S)$ associated
with a filling measured lamination $\lambda$ and denote by $v$ the associated
positive valuation. The following are equivalent:

  1. (1)

$v$ is acute

  2. (2)

$T$ is trivalent

  3. (3)

$\lambda$ is maximal

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.

###### 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 5, 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 $(D=0)$, 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 $C_{1},\ldots,C_{k}$ the components of
$T\setminus\\{x\\}$. They decompose the Gromov boundary of $T$ into disjoint
open subsets $\partial C_{1},\ldots,\partial C_{k}$. It is known that the set
of ends of axes $A_{\gamma}$ is dense in $\partial T\times\partial T$ for
$\gamma\in\pi_{1}(S)$. 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 5, case $2$ $(i)$
$(D=0)$ yields $l(\alpha\beta)=l(\alpha\beta^{-1})=l(\alpha)+l(\beta)$,
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. ∎

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

######  Corollary 3.

The set of acute simple valuations has full measure in $\mathrm{ML}$.

###  5.2. Bonahon cycles

Let $T$ be a trivalent real tree with a free and minimal action of
$\pi_{1}(S)$. To define its tangent space in the “moduli space” of such
objects, imagine the combinatorial structure as being fixed while the distance
function $d$ 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 7.

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. (1)

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

  2. (2)

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

The group $\pi_{1}(S)$ acts linearly on $\mathcal{B}(T)$ by $g(x,y)=(gx,gy)$
and Bonahon cocycles are the elements of
$\operatorname{Hom}_{\pi}(\mathcal{B}(T),\mathbb{R})=\operatorname{Hom}(\mathcal{B}(T)_{\pi},\mathbb{R})$
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. (1)

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

  2. (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 3, we are reduced by bilinearity, to cases 1 or 2. This proves
both uniqueness and existence. ∎

$x$$y$$z$$t$$a$$b$$x$$y$$z$$t$$a$$b$$x$$y$$z$$t$$a$$b$$+1$$0$$-1$ Figure 3.

It is an amusing exercise to show that this pairing is non-degenerate. Instead
we shall deduce it from Poincaré duality in Thuston’s model, see 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 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 $g\in\pi_{1}(S)$.
If they are disjoint, they project in the tree either as disjoint geodesics,
either as in the middle case of Figure 3 and their intersection vanishes. ∎

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 $T_{\lambda}\mathrm{ML}$.

###  5.3. The symplectomorphism theorem

Fix a strict valuation $v\in\mathrm{ML}$, 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 $v$, that is such that
$l(\alpha)=2v(t_{\alpha})$ for all $\alpha\in\pi_{1}(S)$.

The next lemma can be deduced from down-to-earth methods (repeated
applications of Proposition 5). It is a direct consequence of a more
conceptual construction for $T_{v}$ using Bass-Serre theory: we refer to
[Ota15, Section 4].

######  Lemma 5.

The distance between two branch points in $T$ belongs to $\Lambda_{v}$.

Given $\phi\in T_{v}\mathrm{ML}=\operatorname{Hom}(\Lambda_{v},\mathbb{R})$,
we define a corresponding
$c_{\phi}\in\operatorname{Hom}_{\pi}(\mathcal{B}(T),\mathbb{R})$ by setting
$c_{\phi}(x,y)=\frac{1}{2}\phi(d(x,y))$ where $d$ is the distance on $T$. 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
$\Psi:T_{v}\mathrm{ML}\to\operatorname{Hom}_{\pi}(\mathcal{B}(T),\mathbb{R})$
given by $\Psi(\phi)=c_{\phi}$.

######  Proposition 9.

The map $\Psi$ induces an isomorphism
$T_{v}\mathrm{ML}\simeq\operatorname{Hom}_{\pi}(\mathcal{B}(T),\mathbb{R})$.

###### 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 suggest 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
$c\in\operatorname{Hom}_{\pi}(\mathcal{B}(T),\mathbb{R})$, 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 $T_{v}\mathrm{ML}$: 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
$v(t_{\alpha})+v(t_{\beta})<v(t_{\alpha\beta^{-1}})=v(t_{\alpha\beta})$.

In the first case, the axes $A_{\alpha}$ and $A_{\beta}$ intersect along a
segment $xy$ and both isometries push $x$ in the direction of $y$ with a
translation length greater than $d(x,y)$. Then by [Pau89, Proposition 1.6],
$l(\alpha\beta)=d(x,\alpha\beta x)=d(x,y)+d(y,\alpha y)+d(\alpha y,\alpha\beta
x)$. This gives $c(x,\alpha\beta x)=c(x,y)+c(y,\alpha y)+c(y,\beta
x)=c(y,\alpha y)+c(x,\beta x)$. Since both $x$ and $y$ belong simultaneously
to $A_{\alpha}$ and $A_{\beta}$ we establish, by the induction hypothesis,
that the two definitions for $\phi(t_{\gamma})$ coincide.

In the second case, the axes $A_{\alpha}$ and $A_{\beta}$ are disjoint: let
$xy$ be the geodesic joining them. Following again [Pau89], $x$ belongs to the
axes of both $\alpha\beta$ and $\alpha\beta^{-1}$. By the induction
hypothesis, $\phi(t_{\alpha\beta^{-1}})$ is equal to $c(x,\alpha\beta^{-1}x)$.
The first case in Proposition 5 shows that
$d(x,\alpha\beta^{-1}x)=d(x,\alpha\beta x)$, hence
$c(x,\alpha\beta^{-1}x)=c(x,\alpha\beta x)$ and
$2\phi(t_{\alpha\beta})=c(x,\alpha\beta x)$ as claimed.

To finish the proof, consider $f,g\in\mathbb{C}[X]$, we must 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
$\phi(t_{\alpha})=\phi(t_{\alpha_{1}})+\dots+\phi(t_{\alpha_{n}})$, 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. ∎

######  Theorem 8.

The isomorphism $\Psi^{*}:\mathcal{B}(T)_{\pi}\to T_{v}^{*}\mathrm{ML}$
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
$\phi\in T_{v}\mathrm{ML}$: $\Psi(\phi)(x,\alpha x)=\frac{1}{2}\phi(d(x,\alpha
x))=\phi(t_{\alpha})=d_{v}\log(t_{\alpha})(\phi)$.

###### 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
$\epsilon_{p}(t_{\alpha_{p}\beta_{p}}-t_{\alpha_{p}\beta_{p}^{-1}})t_{\alpha}^{-1}t_{\beta}^{-1}\bmod{\mathcal{M}_{v}}$.
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 orders 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

(7)
$\\{t_{\alpha},t_{\beta}\\}_{v}=\sum_{g\in\langle\alpha\rangle\backslash\pi/\langle\beta\rangle}\epsilon(A_{\alpha},gA_{\beta})=\sum_{g\in\langle\alpha\rangle\backslash\pi/\langle\beta\rangle}\epsilon(A_{\alpha},A_{g\beta
g^{-1}})$

where $\epsilon(A_{\alpha},A_{\beta})=\pm 1$ if $A_{\alpha}$ and $A_{\beta}$
are like in Figure 4 and $\epsilon(A_{\alpha},A_{\beta})=0$ in any other
configuration. Notice that - as it should - this formula do not depend on the
orientations of the axes, but on the local orientation of the tree at the
branch points.

$\epsilon=-1$$A_{\alpha}$$A_{\beta}$$A_{\beta}$$A_{\alpha}$$\epsilon=1$$A_{\beta}$$A_{\alpha}$$A_{\alpha}$$A_{\beta}$
Figure 4. Sign rule for the axes.

To end the proof, we fix $x\in A_{\alpha}$ and $y\in A_{\beta}$ to compare
Formula (7) 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:

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

∎

##  6\. Identifying the symplectic tangent models

Let us recall Thurston’s description for the tangent space to $\mathrm{ML}$ at
a maximal measured lamination $\lambda$. We start with an orientation covering
$p\colon S^{\prime}\to S$, 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. By the
Gauss-bonnet theorem, the set $R$ of ramification points has $4g-4$ elements
and the monodromy of the covering is a morphism $\rho\colon\pi_{1}(S\setminus
R)\to\\{\pm 1\\}$ 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 $H_{1}(S^{\prime},\mathbb{R})^{-}$ be the antisymmetric part of
$H_{1}(S^{\prime},\mathbb{R})$ with respect to the involution of the covering:
it supports a non-degenerate antisymmetric form obtained by restricting half
the intersection form. We shall refer to this symplectic space as Thurston’s
model for $T^{*}_{\lambda}\mathrm{ML}$. 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 $\mathcal{B}(T)_{\pi}$ and Thurson
model $H_{1}(S^{o},\mathbb{R}^{-})$ are naturally isomorphic symplectic vector
spaces.

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

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 $T$.

##### Intuition.

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
consists in 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. What we obtain in the
limit is Milnor’s join construction $E\mathbb{Q}$, 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 $E\mathbb{Q}$
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
$\mathbb{R}\mathbb{P}^{\infty}$. It remains to blow up the preceding
construction at every branch point and insert an infinite dimensional space.
This construction may look complicated but we shall do it in one shot and few
lines below.

##### 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 $T^{o}$ 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 $\mathbb{R}\mathbb{P}^{1}$
through which we attach a copy of $\mathbb{R}\mathbb{P}^{\infty}$. For the
moment, $T^{o}$ 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 5. 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 $\operatorname{\Delta}_{W}=\\{(r_{x})_{x\in
W}\in[0,1-\epsilon]^{W},\sum_{x\in W}r_{x}=1\\}$ is a truncated simplex, and
there is an obvious inclusion $\Delta_{W^{\prime}}\subset\Delta_{W}$ when
$W^{\prime}\subset W$. 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
$T^{o}$ 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 5. The $1$-cells $\Delta_{\\{x,y\\}}$ will be called
edges and denoted $(x,y)$.

Figure 5. Attaching a 3-cell in $T^{o}$

As promised, the action of $\pi_{1}(S)$ on $T^{o}$ is now proper so that we
may form the quotient $\Sigma^{o}=T^{o}/\pi_{1}(S)$. The following lemma shows
that $\Sigma^{o}$ and $S^{o}$ are homotopic. Interestingly, the proof consists
in constructing an equivariant map $F:T^{o}\to\tilde{S}^{o}$, which plays the
role of a (non-existing) retraction for the map $f:\tilde{S}\to T$.

######  Lemma 6.

Let $\tilde{S}^{o}$ be the covering of the orbifold $S^{o}$ corresponding to
the kernel of the natural map $\pi_{1}(S^{o})\to\pi_{1}(S)$. There exists a
$\pi_{1}(S)$-equivariant map $F:T^{o}\to\tilde{S}^{o}$ which induces a
homotopy equivalence between $\Sigma^{o}$ and $S^{o}$.

###### 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
6.

Figure 6. Lifting a geodesic to $\mathbb{H}^{2}$ (done with Geogebra)

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

We now proceed to the construction of an equivariant map $F\colon
T^{o}\to\tilde{S}^{o}$. 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.

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
end points $h_{x},h_{y}$ 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 6.

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 $\overline{F}:\Sigma^{o}\to
S^{o}$.

We would like to show that $\overline{F}$ 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 $\overline{F}$ 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.

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}(T^{o})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{*}}$$\textstyle{\pi_{1}(\Sigma^{o})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{F}_{*}}$$\textstyle{\pi_{1}(S)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}(\tilde{S}^{o})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}(S^{o})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}(S)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

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). ∎

###  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, $T^{o}_{W}$ is a collection of
$\mathbb{R}\mathbb{P}^{\infty}$ 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 $T^{o}_{(W)}$ retracts by
deformation on $T^{o}_{W}$ .

###### Proof.

We define the retraction by induction on the maximal dimension of the
truncated simplices $\Delta_{U}\subset T^{o}_{(W)}$. 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 $\Delta_{U^{\prime}}$ for $U^{\prime}\subset U$ 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 $W$. ∎

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
$\gamma.x=\rho(\gamma)x$. 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 $T^{o}=\varinjlim T^{o}_{(W)}$ as $W$ exhausts the finite subsets
of the countable set of branch points $V(T)$ and by Lemma 7, $T^{o}_{(W)}$
retracts by deformation on $T^{o}_{W}$, thus
$H_{*}(T^{o},\mathbb{R}^{\pm})=\varinjlim H_{*}(T^{o}_{W},\mathbb{R}^{\pm})$.

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

We now return to the twisted homology of $T^{o}_{W}$. For this we consider the
double cover $T^{\prime}_{W}\to T^{o}_{W}$ 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
$H_{*}(T^{o}_{W},\mathbb{R}^{\pm})$ respectively. The space $T^{\prime}_{W}$
is homotopy equivalent to a graph with vertex set $W$, and two edges above
each edge $e$ of $T^{o}_{W}$ connecting its end points with opposite
orientations, as shown below.

$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{x\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

It follows that $H_{0}(T^{\prime}_{W},\mathbb{R})=\mathbb{R}$ and
$H_{k}(T^{\prime}_{W},\mathbb{R})=0$ if $k>1$. Moreover
$H_{1}(T^{\prime}_{W},\mathbb{R})$ has a basis formed by the cycles $c(x,y)\in
H_{1}(T^{\prime}_{W},\mathbb{R})$ indexed by the edges $(x,y)$ of
$T^{\prime}_{W}$, which consist in making a round trip from $x$ to $y$,
following the arrows. The Galois involution of $T^{\prime}_{W}$ exchanges the
orientation of $c(x,y)$, so $H_{1}(T^{o}_{W},\mathbb{R}^{-})$ 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 $H_{1}(T^{o}_{W},\mathbb{R}^{-})$. ∎

###  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 morphism $\pi_{1}(\Sigma^{o})\to\\{\pm 1\\}$
that we also denote $\rho$. The $\pi$-equivariant homotopy equivalence between
$\Sigma^{o}$ and $S^{o}$ thus yields a morphism $\pi_{1}(S^{o})\to\\{\pm
1\\}$. By the remark following Lemma 7, this morphism is the co-orientation
monodromy of $\lambda$, so its kernel corresponds to the covering
$S^{\prime}\to S^{o}$. 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 $S^{\prime}$ of $S^{o}$ ramified over $R$ satisfies
$\chi(S^{\prime})=2\chi(S)-(4g-4)=8-8g$ by the Riemann-Hurwitz formula. As
$H_{*}(S^{o},\mathbb{R}^{\pm})=H_{*}(S^{\prime},\mathbb{R})^{\pm}$ 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 $E^{2}_{p,q}=H_{p}(\pi,H_{q}(T^{o},\mathbb{R}^{\pm}))$ 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
$H_{1}(\Sigma^{o},\mathbb{R}^{-})=H_{0}(\pi,\mathcal{B}(T))=\mathcal{B}(T)_{\pi}$
and $H_{k}(\pi,\mathcal{B}(T))=0$ 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
$H_{1}(\tilde{S}^{\prime},\mathbb{R})^{-}=H_{1}(\tilde{S}^{o},\mathbb{R}^{-})=H_{1}(T^{o},\mathbb{R}^{-})=\mathcal{B}(T)$
and
$H_{1}(S^{\prime},\mathbb{R})^{-}=H_{1}(S^{o},\mathbb{R}^{-})=\mathcal{B}(T)_{\pi}$.

$\textstyle{\tilde{S}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{p}}$$\scriptstyle{\pi}$$\textstyle{\tilde{S}^{o}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{S^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{S^{o}}$

######  Proposition 10.

The isomorphisms $H_{1}(\tilde{S}^{\prime},\mathbb{R})^{-}=\mathcal{B}(T)$ and
$H_{1}(S^{\prime},\mathbb{R})^{-}=\mathcal{B}(T)_{\pi}$ 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
$H_{1}(\tilde{S}^{\prime},\mathbb{R})$ are represented geometrically by
$c(x,y)=\tilde{p}^{-1}(F(x,y))\subset\tilde{S}^{\prime}$. Notice that these
cycles have a natural orientation (given by the co-orientation of the lifted
lamination $\tilde{\lambda}^{\prime}$).

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 7. 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 7. Double covering over a branching point

Let us now consider the quotient. We showed in Section 6.3 that
$H_{1}(S^{\prime},\mathbb{R})=H_{1}(\tilde{S}^{\prime},\mathbb{R})_{\pi}$. The
result follows from the fact that the intersection form on
$H_{1}(S^{\prime},\mathbb{R})$ coincides with the averaged intersection form
on $H_{1}(S^{\prime},\mathbb{R})$. ∎

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