# Friedman-Ramanujan functions in random hyperbolic geometry and application to spectral gaps II

Nalini Anantharaman, Laura Monk

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Nalini Anantharaman, Laura Monk. Friedman-Ramanujan functions in random hyperbolic geometry and application to spectral gaps II. 2025. hal-05304152

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Distributed under a Creative Commons Attribution 4.0 International License
FRIEDMAN-RAMANUJAN FUNCTIONS
IN RANDOM HYPERBOLIC GEOMETRY
AND APPLICATION TO SPECTRAL GAPS II

NALINI ANANTHARAMAN¹ AND LAURA MONK²

ABSTRACT. The core focus of this series of two articles is the study of the distribution of the length spectrum of closed hyperbolic surfaces of genus *g*, sampled randomly with respect to the Weil-Petersson probability measure. In the first article [2], we introduced a notion of local topological type **T**, and established the existence of a density function $V_g^T(l)$ describing the distribution of the lengths of all closed geodesics of type **T** in a genus *g* hyperbolic surface. We proved that $V_g^T(l)$ admits an asymptotic expansion in powers of $1/g$. We introduced a new class of functions, called *Friedman-Ramanujan functions*, and related it to the study of the spectral gap $\lambda_1$ of the Laplacian.

In this second part, we provide a variety of new tools allowing to compute and estimate the volume functions $V_g^T(l)$. Notably, we construct new sets of coordinates on Teichmüller spaces, distinct from Fenchel-Nielsen coordinates, in which the Weil-Petersson volume has a simple form. These coordinates are tailored to the geodesics we study, and we can therefore prove nice formulae for their lengths. We use these new ideas, together with a notion of pseudo-convolutions, to prove that the coefficients of the expansion of $V_g^T(l)$ in powers of $1/g$ are Friedman-Ramanujan functions, for any local topological type **T**. We then exploit this result to prove that, for any $\epsilon > 0$, $\lambda_1 \ge \frac{1}{4} - \epsilon$ with probability going to one as $g \to +\infty$, or, in other words, typical hyperbolic surfaces have an asymptotically optimal spectral gap.

CONTENTS

<table><tr><td>1. Introduction</td><td>2</td></tr><tr><td>2. Loops and generalized eights</td><td>14</td></tr><tr><td>3. Coordinate system adapted to a generalized eight</td><td>24</td></tr><tr><td>4. Expression of length functions in the new coordinate system</td><td>43</td></tr><tr><td>5. Expression of the volume functions as pseudo-convolutions of FR functions</td><td>54</td></tr><tr><td>6. Pseudo-convolutions and stability of the FR hypothesis</td><td>58</td></tr><tr><td>7. Erasing of crossing parameters in the length of c</td><td>76</td></tr><tr><td>8. The lengths of boundary curves</td><td>89</td></tr><tr><td>9. Proof of the main theorem for generalized eights</td><td>106</td></tr></table>

<footnote>(1) COLLÈGE DE FRANCE, 11 PLACE MARCELIN BERTHELOT, 75005 PARIS / IRMA, 7 RUE RENÉ DESCARTES, 67084 STRASBOURG CEDEX, FRANCE</footnote>

<footnote>(2) SCHOOL OF MATHEMATICS, UNIVERSITY OF BRISTOL, BRISTOL BS8 1UG, U.K.</footnote>

<footnote>E-mail address: nalini.anantharaman@college-de-france.fr, laura.monk@bristol.ac.uk.</footnote>

<footnote>Date: June 12, 2025.</footnote>

<footnote n="2020">2020 Mathematics Subject Classification. Primary 58J50, 32G15; Secondary 05C80, 11F72.</footnote>

<footnote>Key words and phrases. Random hyperbolic surfaces, Weil-Petersson form, moduli space, spectral gap, closed geodesic, Selberg trace formula.</footnote>
<table><tr><td>10.</td><td>Proof of the crucial comparison estimate (Proposition 9.3)</td><td>113</td></tr><tr><td>11.</td><td>Arbitrary loop topologies</td><td>125</td></tr><tr><td>12.</td><td>Variant of the main theorem for lc-surfaces</td><td>134</td></tr><tr><td>13.</td><td>Proof of the spectral gap result</td><td>146</td></tr><tr><td colspan="2">Appendix A. List of notations</td><td>153</td></tr><tr><td colspan="2">References</td><td>160</td></tr></table>

# 1. INTRODUCTION

**1.1. Main results and motivations.** For an integer $g \ge 2$, let $S_g$ be a fixed compact oriented surface of genus $g$ with no boundary. This is the second part of a two-part article, studying the distribution of the length spectrum of a random hyperbolic metric $X$ on $S_g$. Our probability space is the moduli space $\mathcal{M}_g$ of hyperbolic structures on $S_g$, endowed with the Weil-Petersson measure $\mathbb{P}_g^{\text{WP}}$ normalized to be a probability measure.

**1.1.1. Local topological type and associated density.** In the first part [2], we defined a notion of local topological type for a loop [2, §4]. Intuitively, a local type $\mathbf{T}$ corresponds to the topological datum of a loop $\mathbf{c}$ filling a surface $\mathbf{S}$, so that we know the topology of $\mathbf{c}$ and its regular neighbourhood, but not the way it is embedded in the ambient surface of genus $g$. Examples of local topological types include the type $\mathbf{s}$ “simple” (loops with no self-intersections), or the figure-eight (with exactly one self-intersection). We developed new tools allowing to study the following averages.

**Definition 1.1.** Let $\mathbf{T} = [\mathbf{S}, \mathbf{c}]_{\text{loc}}$ be a local topological type. For any measurable function $F: \mathbb{R}_{\ge 0} \to \mathbb{C}$, bounded and compactly supported, we define the $\mathbf{T}$-average of $F$ to be

$$ (1.1) \qquad \langle F \rangle_g^{\mathbf{T}} := \mathbb{E}_g^{\text{WP}} \left[ \sum_{\substack{\gamma \in \mathcal{G}(X) \\ \gamma \in \mathbf{T}}} F(\ell_X(\gamma)) \right] $$

where $\mathbb{E}_g^{\text{WP}}$ denotes the expectation on the moduli space $\mathcal{M}_g$ when the hyperbolic structure $X \in \mathcal{M}_g$ is chosen randomly according to the normalized Weil-Petersson measure $\mathbb{P}_g^{\text{WP}}$, and

* $\mathcal{G}(X)$ is the set of primitive oriented closed geodesics for the metric $X$;

* for any $\gamma \in \mathcal{G}(X)$, $\ell_X(\gamma)$ is the length of $\gamma$ for the hyperbolic metric $X$;

* $\gamma \in \mathbf{T}$ means that the sum is restricted to closed geodesics of local topological type $\mathbf{T}$.

Taking $F$ to be the indicator function $1_{[a,b]}$ of an interval $[a, b]$, the average $\langle F \rangle_g^{\mathbf{T}}$ is the expected number of closed primitive geodesics of local topological type $\mathbf{T}$ and length in the interval $[a, b]$. In the case of the local topological type “simple”, denoted as $\mathbf{s}$, the condition $\gamma \in \mathbf{s}$ means that the sum is restricted to all simple primitive closed geodesics.

We proved in [2, Prop. 5.11] that the averages $\langle F \rangle_g^{\mathbf{T}}$ can be written in an integral form, against a density function $V_g^{\mathbf{T}}(\ell)$ which is, informally, the “proportion” of surfaces $X \in \mathcal{M}_g$ containing a primitive closed geodesic of length $\ell$ and local topological type $\mathbf{T}$. The formal definition is as follows.
**Proposition 1.2.** For any local topological type **T**, there exists a unique locally integrable function $V_g^T: \mathbb{R}_{>0} \to \mathbb{R}_{\ge 0}$ such that, for any measurable function $F$ bounded with compact support,

$$ \langle F \rangle_g^T = \frac{1}{V_g} \int_0^{+\infty} F(\ell) V_g^T(\ell) \, d\ell. $$

The functions $V_g^T(\ell)$ are fundamental objects for the study of geodesics in random hyperbolic surface. However they have been little studied, except in the case of the local topological type “simple”. For simple closed geodesics, Mirzakhani showed in [8] that the functions $V_g^T(\ell)$ are polynomial in $\ell$ (hence they are called volume polynomials), and that they can be computed explicitly for all genus $g$ by topological recursion. Otherwise, we call $\ell \mapsto V_g^T(\ell)$ the volume function associated with local topological type **T** on surfaces of genus $g$.

The precise nature of the functions $V_g^T$ is not known. One can deduce from [9, Theorem 8.1] that $V_g^T(\ell)$ behaves like a power of $\ell$ to leading order as $\ell \to +\infty$ with $g$ fixed. We believe that, for local topological type other than “simple”, the function $\ell \mapsto V_g^T(\ell)$ is not polynomial, but is exponentially close to one as $\ell \to +\infty$ with $g$ fixed.

1.1.2. Large-genus asymptotic expansion. The core focus of this paper is the large genus behavior of the volume functions $V_g^T$. We proved in the first article [2] that they admit expansions to any order in powers of $g^{-1}$, in the following sense.

**Theorem 1.3 ([2, Theorem 1.15])**. For any local topological type **T** = [S, c]_loc, there exists a unique family of locally integrable functions $(f_k^T)_{k \ge 0}$ such that, for any integers $N \ge 0$ and $g \ge 2$, any $\epsilon > 0$,

$$ (1.2) \quad \left| \langle F \rangle_g^T - \sum_{k=0}^N \frac{1}{g^k} \int_0^{+\infty} F(\ell) f_k^T(\ell) \, d\ell \right| \le C_{N,\chi(S),\epsilon} \frac{\sup_{\ell \ge 0} \{|F(\ell)|e^{(1+\epsilon)\ell}\}}{g^{N+1}} $$

for any measurable function $F: \mathbb{R}_{\ge 0} \to \mathbb{R}$ bounded with compact support.

As the notation indicates, the constant $C_{N,\chi(S),\epsilon}$ depends on the order $N$ of the expansion, the Euler characteristic of the filled surface $S$ and the small parameter $\epsilon$, but not on the specific loop $c$ filling $S$. In forthcoming applications of our results to the spectral gap of the Laplacian, this uniform behavior with respect to all loops filling the same surface is important.

1.1.3. Statement of the main results. The main technical result of this paper may be summarised as follows, answering positively the question asked in [2].

**Theorem 1.4.** Let **T** be a local topological type other than “simple”. For any $k \ge 0$, the function $\ell \mapsto f_k^T(\ell)$ is a Friedman-Ramanujan function in the weak sense.

The notion of Friedman-Ramanujan function was introduced in [2, §3]. A locally integrable function $f: \mathbb{R}_{\ge 0} \to \mathbb{R}$ is said to be a Friedman-Ramanujan function if there exists a polynomial function $p$ such that $f(\ell) - p(\ell)e^\ell$ grows at most like a polynomial multiple of $e^{\ell/2}$. Precise definitions follow in §1.2, together with a quantitative version of our main result (see Theorem 1.15).
**Remark 1.5.** The case of the local topological type **s** is excluded because, in this case, the function $f_k^s$ have pole of order one at 0. We prove that $\ell \mapsto \ell f_k^s(\ell)$ are Friedman-Ramanujan functions in [1] (see [2, Proposition 3.4] for more details).

Theorem 1.4 has already been obtained when **T** is a figure-eight, or more generally when the filled surface **S** has Euler characteristic $-1$, in [2]. For this relatively simple topological situation, we could use almost explicit expressions of the functions $(f_k^T)_{k \ge 0}$.

The present paper deals with general local topological types and uses entirely new methods, involving finding explicit expressions for the Weil-Petersson measure in coordinates other than the Fenchel-Nielsen ones, and for the lengths of non-simple geodesics in those coordinates.

This work is largely motivated by the question of estimating the gap at the bottom of the spectrum of the Laplacian, for typical hyperbolic surfaces, as detailed in [2, 4]. In the first part of this article [2], we used the result when **S** has Euler characteristic $-1$ to prove that typical surfaces have spectral gap at least $2/9 - \epsilon$. More generally, our theorem (together with the variant presented in §12) implies the optimal lower bound $1/4 - \epsilon$ on the spectral gap.

**Theorem 1.6.** Denote by $\lambda_1(X)$ the first non-trivial eigenvalue of Laplacian on the hyperbolic surface $X$. Then, for any $\epsilon > 0$,

$$ (1.3) \qquad \mathbb{P}_g^{\text{WP}} \left( \lambda_1(X) \ge \frac{1}{4} - \epsilon \right) \xrightarrow{g \to +\infty} 1. $$

1.2. **Full statement of the Friedman-Ramanujan property.** Let us introduce a few definitions, and present our main result more precisely.

1.2.1. *Definitions.* We recall from [2, §3] the strong and weak definitions of the class of Friedman-Ramanujan functions, directly inspired by J. Friedman's seminal work on random graphs [6, Definition 2.1] where he proves the Alon conjecture.

**Definition 1.7.** Let $N \ge 1$. We define the class of functions $\mathcal{R}^N$ as the set of locally integrable functions $f: \mathbb{R}_{\ge 0} \to \mathbb{C}$ such that there exists $c > 0$ satisfying

$$ (1.4) \qquad \forall \ell \ge 0, \quad |f(\ell)| \le c(\ell + 1)^{N-1} e^{\frac{\ell}{2}}. $$

We let $\mathcal{R} = \bigcup_{N \ge 1} \mathcal{R}^N$. A locally integrable function $f: \mathbb{R}_{\ge 0} \to \mathbb{C}$ is said to be a *Friedman-Ramanujan function* if there exists a polynomial function $p$ such that

$$ f(\ell) - p(\ell)e^{\ell} \in \mathcal{R}. $$

We denote as $\mathcal{F}$ the set of Friedman-Ramanujan function. For $f \in \mathcal{F}$, the function $\ell \mapsto p(\ell)e^{\ell}$ is uniquely defined, we call it the *principal term* of $f$.

**Notation 1.8.** For an integer $K \ge 0$, we denote as $\mathbb{C}_{K-1}[\ell]$ the space of polynomials of degree strictly less than $K$, and by $\mathbb{C}_{K-1}[\ell] e^{\ell}$ the class of functions of the form $p(\ell) e^{\ell}$ where $p \in \mathbb{C}_{K-1}[\ell]$. Note that these sets are reduced to the constant function equal to 0 if $K = 0$.

**Definition 1.9.** For $K \ge 0$, we denote as $\mathcal{F}^K$ the subset of Friedman-Ramanujan functions of principal term in $\mathbb{C}_{K-1}[\ell] e^{\ell}$, and for $N \ge 1$, $\mathcal{F}^{K,N} := \mathbb{C}_{K-1}[\ell] e^{\ell} \oplus \mathcal{R}^N$.
Note that the space $\mathcal{R}$ can be identified with the subset $\mathcal{F}^0$ of Friedman-Ramanujan functions with principal term $p=0$, and similarly $\mathcal{R}^N = \mathcal{F}^{0,N}$.

In our proof of Theorem 1.4 we will rather use the following weak version of the classes $\mathcal{R}$ and $\mathcal{F}$. Remark that in the work of Friedman, where the functions are defined on $\mathbb{Z}_{>0}$ instead of $\mathbb{R}_{\ge 0}$, this distinction between weak and strong definition does not exist.

**Definition 1.10.** Let $N \ge 1$. We define the class of functions $\mathcal{R}_w^N$ as the set of locally integrable functions $f: \mathbb{R}_{\ge 0} \to \mathbb{C}$ such that there exists $c > 0$ satisfying

$$ (1.5) \qquad \forall \ell \ge 0, \quad \int_0^\ell |f(s)| \, ds \le c(\ell+1)^{N-1} e^{\frac{\ell}{2}}. $$

We let $\mathcal{R}_w = \bigcup_{N \ge 1} \mathcal{R}_w^N$. A locally integrable function $f: \mathbb{R}_{\ge 0} \to \mathbb{C}$ is said to be a Friedman-Ramanujan function in the weak sense if there exists a polynomial function $p$ such that

$$ f(\ell) - p(\ell) e^{\ell} \in \mathcal{R}_w. $$

For $K \ge 0$, we write $\mathcal{F}_w^K := \mathbb{C}_{K-1}[\ell] e^\ell \oplus \mathcal{R}_w$ and for $N \ge 1$, $\mathcal{F}_w^{K,N} := \mathbb{C}_{K-1}[\ell] e^\ell \oplus \mathcal{R}_w^N$.

As the name suggests, the strong definition implies the weak one.

### 1.2.2. Norms on the set of Friedman-Ramanujan functions.
Let us define a norm on the spaces $\mathcal{F}^{K,N}$ and $\mathcal{F}_w^{K,N}$, which captures the size of the principal term and the remainder of the Friedman-Ramanujan function.

**Notation 1.11.** For a polynomial function $p \in \mathbb{C}[\ell]$, we define $\|p\|_{\ell\infty}$ to be the maximum modulus of the coefficients of $p$.

**Definition 1.12.** For $N \ge 1$, we define a norm on $\mathcal{R}^N$ and $\mathcal{R}_w^N$ by

$$ \|f\|_{\mathcal{R}^N} := \sup_{\ell \ge 0} \frac{|f(\ell)|}{(\ell+1)^{N-1} e^{\frac{\ell}{2}}} \quad \text{and} \quad \|f\|_{\mathcal{F}^N}^w := \sup_{\ell \ge 0} \frac{\int_0^\ell |f(s)| \, ds}{(\ell+1)^{N-1} e^{\frac{\ell}{2}}}. $$

If now $K \ge 0$, we equip $\mathcal{F}^{K,N}$ with the norm

$$ \|f\|_{\mathcal{F}^{K,N}} := \|p\|_{\ell\infty} + \|f(\ell) - p(\ell) e^{\ell}\|_{\mathcal{R}^N} $$

where $p(\ell) e^\ell$ is the principal term of $f$. The norm $\|f\|_{\mathcal{F}^{K,N}}^w$ is defined the same way replacing the last norm by its weak version.

It will be convenient to adopt the following convention.

**Notation 1.13.** We denote as $\text{fn}((u_i)_{i \in \mathcal{V}})$ any quantity that is a function solely of some set of variables $(u_i)_{i \in \mathcal{V}}$ (but for which we do not need to write an explicit expression).

**Remark 1.14.** By definition, if $f \in \mathcal{R}^N$, then for all $\ell \ge 0$,

$$ (1.6) \qquad |f(\ell)| \le \|f\|_{\mathcal{R}^N} (1+\ell)^{N-1} e^{\ell/2}. $$

If we rather assume that $f \in \mathcal{F}^{K,N}$, then for $\ell \ge 0$,

$$ (1.7) \qquad |f(\ell)| \le \text{fn}(K, N) \|f\|_{\mathcal{F}^{K,N}} (1+\ell)^{K-1} e^{\ell} $$

for some constant $\text{fn}(K, N) := 1 + \sup_{\ell \ge 0} (1+\ell)^{N-K} e^{-\ell/2}$.
1.2.3. *Precise statement.* The main technical result of this paper is the following precised version of Theorem 1.4.

**Theorem 1.15.** Let $\mathbf{T} = [\mathbf{S}, \mathbf{c}]_{\text{loc}}$ be a local type other than "simple". Then, for any $k \ge 0$, there exist $K, N$ and $C$, depending on $k$ and $\mathbf{S}$ but not on $\mathbf{c}$, such that $f_k^T \in \mathcal{F}_w^{K,N}$ and $\|f_k^T\|_{\mathcal{F}^{K,N}} \le C$.

**Notation 1.16.** We denote as $(g_S, n_S)$ the signature of $\mathbf{S}$, and $\chi(\mathbf{S}) := 2g_S - 2 + n_S$ its absolute Euler characteristic. We denote as $\partial\mathbf{S}$ the set of boundary components of $\mathbf{S}$, which is identified with $\{1, \dots, n_S\}$ through the labelling of the boundary of $\mathbf{S}$.

**Remark 1.17.** The proof gives a seemingly optimal estimate on $K$, and a non-optimal estimate on $N$ and $C$: we may take $K = 2k + 2\chi(\mathbf{S})$ and $N = n_S(a_{k+1} + 1)$, where $a_{k+1}$ is an explicit sequence coming from [1] which is a priori not optimal either. Remark 1.24 shows how we may optimise the value of $K$ further, depending on the topology of the multi-loop $\mathbf{c}$.

1.3. **Outline of the proof.** Let $\mathbf{T} = [\mathbf{S}, \mathbf{c}]_{\text{loc}}$ be a local topological type, i.e. the topological datum of a loop $\mathbf{c}$ filling a surface $\mathbf{S}$. We present prior results on the volume function $V_g^T$ and the coefficients $(f_k^T)_{k \ge 0}$ of its asymptotic expansion in powers of $1/g$. We then explain the key ideas behind the proof of the main theorem.

1.3.1. *Teichmüller spaces and the Weil-Petersson measure.* For $\mathbf{x} = (x_1, \dots, x_{n_S}) \in \mathbb{R}^{n_S}_{>0}$, we denote as $\mathcal{T}_{g_S, n_S}(\mathbf{x})$ the Teichmüller space of hyperbolic surfaces of genus $g_S$ with $n_S$ labelled geodesic boundary components of respective lengths $x_1, \dots, x_{n_S}$. This space is a real-analytic manifold of dimension $6g_S - 6 + 2n_S$. It is equipped with the *Weil-Petersson symplectic form*, which induces a volume that we call the *Weil-Petersson measure* and denote as $\text{Vol}_{g_S, n_S, \mathbf{x}}^{\text{WP}}$.

In this article, it will be more interesting to view the boundary lengths as free variables, and hence introduce the Teichmüller space

$$ \mathcal{T}_{g_S, n_S}^* = \{(\mathbf{x}, Y) : \mathbf{x} \in \mathbb{R}^{n_S}_{>0}, Y \in \mathcal{T}_{g_S, n_S}(\mathbf{x})\}, $$

which is the space of compact hyperbolic metrics on the surface $\mathbf{S}$. The space $\mathcal{T}_{g_S, n_S}^*$ is a real-analytic manifold of dimension $6g_S - 6 + 3n_S = 3\chi(\mathbf{S})$. It may be endowed with the positive measure $d\text{Vol}_{g_S, n_S}^{\text{WP}}(\mathbf{x}, Y) := d\mathbf{x} d\text{Vol}_{g_S, n_S}^{\text{WP}}(Y)$, where $d\mathbf{x}$ is the Lebesgue measure on $\mathbb{R}^{n_S}_{>0}$, which we also call *Weil-Petersson measure*. We will sometimes omit the mention of $\mathbf{x}$ and simply write $Y \in \mathcal{T}_{g_S, n_S}^*$ when it is not useful to emphasize the boundary length of $Y$.

1.3.2. *Level-set desintegration.* The map $\mathcal{T}_{g_S, n_S}^* \ni Y \mapsto \ell_Y(\mathbf{c})$ is a non-constant analytic function on the Teichmüller space $\mathcal{T}_{g_S, n_S}^*$. We can therefore desintegrate the Weil-Petersson measure $d\text{Vol}_{g_S, n_S}^{\text{WP}}$ on its level-sets. This yields a measure $d\text{Vol}_{g_S, n_S}^{\text{WP}}(Y)/d\ell$ defined by the property that, for any integrable function $F: \mathcal{T}_{g_S, n_S}^* \to \mathbb{R}_{\ge 0}$, we have

$$ \int_{\mathcal{T}_{g_S, n_S}^*} F(Y) \, d\text{Vol}_{g_S, n_S}^{\text{WP}}(Y) = \int_0^{+\infty} \left( \int_{\ell_Y(\mathbf{c})=\ell} F(Y) \frac{d\text{Vol}_{g_S, n_S}^{\text{WP}}(Y)}{d\ell} \right) d\ell. $$
In particular, if $F(\mathbf{x}, Y) = \tilde{F}(\ell_Y(\mathbf{c}))\phi(\mathbf{x})$ is a multiplicative function of $\ell_Y(\mathbf{c})$ and $\mathbf{x}$, this equation takes the special form

$$ \int_{T_{g_S, n_S}^*} \tilde{F}(\ell_Y(\mathbf{c}))\phi(\mathbf{x}) \mathrm{dVol}_{g_S, n_S}^{\mathrm{WP}}(\mathbf{x}, Y) = \int_0^{+\infty} \tilde{F}(\ell) \left( \int_{\ell_Y(\mathbf{c})=\ell} \phi(\mathbf{x}) \frac{\mathrm{dVol}_{g_S, n_S}^{\mathrm{WP}}(\mathbf{x}, Y)}{\mathrm{d}\ell} \right) \mathrm{d}\ell. $$

1.3.3. Description of the volume function and coefficients. We have proven in [2, §5] that the volume function $V_g^T(\ell)$ and the coefficients $f_k^T(\ell)$ are all of the form

$$ (1.8) \qquad \ell \mapsto \int_{\ell_Y(\mathbf{c})=\ell} \phi(\mathbf{x}) \frac{\mathrm{dVol}_{g_S, n_S}^{\mathrm{WP}}(\mathbf{x}, Y)}{\mathrm{d}\ell}, $$

with particular choices of $\phi(\mathbf{x})$.

More precisely, the volume function $V_g^T(\ell)$ is obtained by replacing $\phi$ with the following explicit polynomial function:

$$ (1.9) \qquad \Phi_g^T(\mathbf{x}) := \frac{1}{n(\mathbf{T})} \frac{x_1 \dots x_{n_S}}{V_g} \sum_{\mathfrak{R} \in R_g(\mathbf{S})} V_{\mathfrak{R}}(\mathbf{x}), $$

which enumerates all possible embeddings of $\mathbf{S}$ in a surface of genus $g$ (the coefficient $n(\mathbf{T})$ is a combinatorial factor, and $V_{\mathfrak{R}}(\mathbf{x})$ are products of volumes of moduli spaces, enumerated by configurations $\mathfrak{R} \in R_g(\mathbf{S})$).

From this expression, the functions $f_k^T(\ell)$ are then obtained by performing the asymptotic expansion of $\Phi_g^T(\mathbf{x})$ in powers of $1/g$, using the results of [1, 2], see [2, Proposition 5.22]. It follows that $f_k^T(\ell)$ has the form (1.8) for a $\phi(\mathbf{x})$ which is a linear combination of functions

$$ (1.10) \qquad \prod_{i \in V_0} x_i^{k_i} \prod_{i \in V_+} x_i^{k_i} \cosh\left(\frac{x_i}{2}\right) \prod_{i \in V_-} x_i^{k_i} \sinh\left(\frac{x_i}{2}\right) \prod_{j=1}^k x_{i_j} \delta(x_{i_j} - x_{i'_j}), $$

where:

* $V_0, V_+$, $V_-$ are three disjoint subsets of $\partial\mathbf{S}$;

* $m = \bigsqcup_{j=1}^k \{i_j, i'_j\}$ is a perfect matching of $\partial\mathbf{S} \setminus (V_0 \sqcup V_+ \sqcup V_-)$;

* $\delta$ is the usual Dirac mass.

The degree $k_i$ is odd if $i \in V_0 \cup V_+$ and even if $i \in V_-$. In [1] we proved<fnref n="1" /> that $\sum_{i \in V_+ \cup V_-} k_i \le 2k$, and that for $i \in V_0$, we have $k_i \le a_{k+1}$ for a certain sequence $a_{k+1}$ (which is explicit but a priori not optimal).

1.3.4. Brief reminder about the method followed in [1, 2]. It is worth now recalling the main steps in the proof of the asymptotic expansion of $\Phi_g^T(\mathbf{x})$.

**Notation 1.18.** Throughout this article, we shall use the Landau notation $\mathcal{O}$, i.e. we will write $A = \mathcal{O}(B)$ if there exists a constant $c > 0$ such that, for any choice of parameters, $|A| \le cB$. If the constant $c$ depends on some parameters, we shall write them as a subscript of the $\mathcal{O}$.

<footnote n="1">¹It is shown in [1, §5.4] that $\sum_{i \in V_+ \cup V_-} k_i \le 2k$. There is an imprecision in [1, §5.4] in the estimate of $\sum_{i \in V_+ \cup V_-} m_i$ because $m_i$ is either $k_i$ or $k_i - 1$ depending on the parity of $k_i$, but this has no impact on the original estimate of $\sum_{i \in V_+ \cup V_-} k_i$.</footnote>
• First, we note that, although the cardinality of $R_g(\mathbf{S})$ in (1.9) grows with $g$, we can identify a set of embeddings $R_g(\mathbf{S}, N)$ of bounded cardinality, such that

$$ \Phi_g^{\mathbf{T}}(\mathbf{x}) = \frac{1}{n(\mathbf{T})} \frac{x_1 \dots x_{n_S}}{V_g} \sum_{\mathfrak{R} \in R_g(\mathbf{S}, N)} V_{\mathfrak{R}}(\mathbf{x}) + \mathcal{O}_{N, n_S} \left( \frac{(\|\mathbf{x}\| + 1)^{\alpha_N} e^{\frac{x_1 + \dots + x_{n_S}}{2}}}{g^{N+1}} \right) $$

in a weak sense. This is done in [2, Proposition 5.22].

• Then, for each realisation $\mathfrak{R} \in R_g(\mathbf{S}, N)$, we observe that there exist integers $n_1 \le n_S$ and $m \le (N+n_S)/2$ such that the ratio $V_{\mathfrak{R}}(\mathbf{x})/V_{\mathfrak{R}}(0)$ coincides (modulo permutation of the variables) with $V_{g-m,n_1}(x_1, \dots, x_{n_1})/V_{g-m,n_1}$ multiplied by a fixed polynomial in $(x_{n_1+1}, \dots, x_{n_S})$. We recall that, for any integers $g, n$ with $2g - 2 + n > 0$, $V_{g,n}$ denotes the constant coefficient of the volume polynomial $V_{g,n}(\cdot)$, i.e. its value at $(0, \dots, 0)$.

• The main result of [1] says that $V_{g-m,n_1}(x_1, \dots, x_{n_1})/V_{g-m,n_1}$ coincides, modulo a remainder in $\mathcal{O}_{N,n_1}$ $(\|\mathbf{x}\| + 1)^{\alpha_N} e^{\frac{x_1+\dots+x_{n_1}}{2}}/g^{N+1}$, with a linear combination of the functions (1.10), with degree $k_i \le a_{N+1}$, the coefficients being of the form

$$ A_{V_0, V_+, V_-}(n_1, m, N, k_1, \dots, k_{n_1}) \frac{c_{g-m,n_1}(\alpha)}{V_{g-m,n_1}} $$

and $\|\alpha\|_{\infty} \le 2N + a_{N+1}$. Here $a_{N+1}$ is a sequence introduced in [1], and $c_{g-m,n_1}(\alpha)$ is our notation for the coefficients of volume polynomials.

• At the end we inject the result of Mirzakhani-Zograf [10, Theorem 4.1], showing the existence of an asymptotic expansion in powers of $g^{-1}$ for $c_{g-m,n_1}(\alpha)/V_{g-m,n_1}$ and for $V_{\mathfrak{R}}(0)/V_g$, and hence for $c_{g-m,n_1}(\alpha)/V_g$.

**Remark 1.19.** The last step is only necessary if we insist on obtaining an expansion $V_g^{\mathbf{T}}(\ell)$ in powers of $g^{-1}$. The three first steps, together with the fact that $c_{g-m,n_1}(\alpha)/V_{g-m,n_1}$ and $V_{g-m,n_1}(0)/V_g$ are bounded functions of $g$, are actually sufficient to obtain the spectral gap (Theorem 1.6).

Until §12, the surface $\mathbf{S}$ is fixed and we do not need to know how the quantity $A_{V_0, V_+, V_-}(n_1, m, N, k_1, \dots, k_{n_1})$ depends on $n_1$. In §12, we will need to come back to the proof of the asymptotic expansion, in order to have a very rough control of the dependency on $n_1$, $n_S$ of those coefficients, as well as the constants involved in $\mathcal{O}_{N,n_S}$ and $\mathcal{O}_{N,n_1}$.

1.3.5. *Unshielded simple portions.* The following topological quantity, associated to a local topological type, will appear in our abstract statement.

**Definition 1.20.** Let $\mathbf{T} = [\mathbf{S}, \mathbf{c}]_{\text{loc}}$ be a local topological type.

(1) A *simple portion* of $\mathbf{c}$ is a maximal open subsegment of $\mathbf{c}$ which does not contain any self-intersection point of $\mathbf{c}$.

(2) A simple portion is said to be *shielded* if it belongs to the boundary of a contractible component of $\mathbf{S} \setminus \mathbf{c}$, and *unshielded* otherwise.

(3) We say $\mathbf{c}$ is *double-filling* if all simple portions are shielded.

We denote by $\rho[\mathbf{T}]$ the number of unshielded simple portions of $\mathbf{T}$.
The notion of simple portion and double-filling multi-loop appeared in our previous paper [2, Definition 8.3]. We have the inequality $\rho[\mathbf{T}] \le 2\chi(\mathbf{S})$, which means that the number of unshielded simple portions is bounded uniformly in $\mathbf{c}$.

1.3.6. Main abstract statement. We can use (1.10) to reduce the proof of Theorem 1.15 to a general abstract statement, which we shall now state and comment. Because our main theorem requires quite a few notation, we group them below.

**Notation 1.21** (Assumptions for main theorem).

• Let $\mathbf{T} = [\mathbf{S}, \mathbf{c}]_{\text{loc}}$ be a local type of signature $(g_{\mathbf{S}}, n_{\mathbf{S}})$. We denote as $\rho[\mathbf{T}]$ be the number of unshielded simple portions of $\mathbf{T}$.

• Let $V$ be a subset of $\partial\mathbf{S}$ such that $n_{\mathbf{S}} - \#V = 2k$ is even, and $m$ be a perfect matching of $\partial\mathbf{S} \setminus V$, that is, a partition $\{1, \dots, n_{\mathbf{S}}\} \setminus V = \bigsqcup_{j=1}^k \{i_j, i'_j\}$.

• We fix a smooth function $\chi_z$ that vanishes at order 1 at $x=0$ and is constant equal to 1 on $[1, +\infty)$, and such that

$$ (1.11) \qquad \left| \frac{\chi_z(y)^2}{(1 - e^{-y})^2} - 1 \right| \le 8e^{-y}. $$

• For $j \in V$, let $f_j$ be a function such that $\tilde{g}_j(x) := f_j(x) \sinh(\frac{x}{2})/\chi_z(x)^2$ is continuous and lies in $\mathcal{F}^{K_j, N_j}$ for some integers $K_j$ and $N_j$. We assume w.l.o.g. that $K_j \le N_j$. We set

$$ K = \sum_{j \in V} (K_j - 1) + \rho[\mathbf{T}] \quad \text{and} \quad N = \sum_{j \in V} N_j + 3\chi(\mathbf{S}) + 1. $$

**Theorem 1.22.** With the assumptions from Notation 1.21, the function

$$ (1.12) \quad \mathcal{J} = \mathcal{J}_{\mathbf{T},V,m,f} : \ell \mapsto \int_{(\mathbf{x},Y) \in \mathcal{T}_{g_{\mathbf{S}},n_{\mathbf{S}}}^*} \prod_{\ell_Y(\mathbf{c})=\ell} \prod_{j \in V} f_j(x_j) \prod_{j=1}^k x_{i_j} \delta(x_{i_j} - x_{i'_j}) \frac{\mathrm{d}\mathrm{Vol}_{g_{\mathbf{S}},n_{\mathbf{S}}}^{\mathrm{WP}}(\mathbf{x}, Y)}{\mathrm{d}\ell} $$

is a Friedman-Ramanujan function in the weak sense. More precisely, $\mathcal{J} \in \mathcal{F}_w^{K,N}$ and

$$ (1.13) \qquad \|\mathcal{J}\|_{\mathcal{F}^{K,N}}^w \le \mathrm{fn}(g_{\mathbf{S}}, n_{\mathbf{S}}, (K_j, N_j)_{j \in V}) \prod_{j \in V} \|\tilde{g}_j\|_{\mathcal{F}^{K_j, N_j}}. $$

**Remark 1.23.** Since $\rho[\mathbf{T}] \le 2\chi(\mathbf{S})$, our Friedman-Ramanujan exponents $K$ and $N$ can be made uniform amongst all loops filling $\mathbf{S}$.

**Remark 1.24.** In the proof of Theorem 1.22, we have tried to optimise the value of $K$, but not the value of $N$. One can further improve the value of $K$ depending on the loop $\mathbf{c}$, and prove that it can be taken to be $\sum_{j \in V'} (K_j - 1) + \rho[\mathbf{T}]$, where $V' \subseteq V$ is the set of boundary components containing unshielded simple portions of $\mathbf{c}$. See details in Remark 11.24.

We notice that in the special case when the loop $\mathbf{c}$ is double-filling, we have $V' = \emptyset$ and $\rho[\mathbf{T}] = 0$. In other words, the function $\mathcal{J}$ is actually an element of $\mathcal{R}_w$ as soon as $\mathbf{c}$ is double-filling.

Let us now show how to go from Theorem 1.22 to Theorem 1.15.
*Proof of Theorem 1.15.* We simply use the expression (1.10) for the functions $f_k^T$, and apply our main technical result to $V = V_0 \cup V_+ \cup V_-$ and the functions

$$f_j(x) = \begin{cases} x_j^{k_j} & \text{if } j \in V_0 \\ x_j^{k_j} \cosh\left(\frac{x_j}{2}\right) & \text{if } j \in V_+ \\ x_j^{k_j} \sinh\left(\frac{x_j}{2}\right) & \text{if } j \in V_- \end{cases}$$

which clearly satisfy the hypotheses with

$$\begin{cases} K_j = N_j = k_j + 1 & \text{if } j \in V_+ \cup V_- \\ K_j = 0, N_j = a_{k+1} + 1 & \text{if } j \in V_0, \end{cases}$$

using the parity condition on the integers $k_j$ to obtain the needed cancellation at 0. □

**Remark 1.25.** The proof above allows us to see that, in Theorem 1.15, we may take the Friedman-Ramanujan parameters to be

$$K = \sum_{j \in V_+ \cup V_-} k_j + \rho[\mathbf{T}] \le 2k + \rho[\mathbf{T}] \le 2k + 2\chi(\mathbf{S})$$

$$N = \sum_{j \in V_0 \cup V_+ \cup V_-} N_j + 3\chi(\mathbf{S}) + 1 \le n_{\mathbf{S}}(a_{k+1} + 1) + 3\chi(\mathbf{S}) + 1.$$

The proof of Theorem 1.6 about the spectral gap of random surfaces requires variants of Theorem 1.22. A first straightforward variant is that Theorem 1.22 also applies in the case where $\mathbf{c}$ is a multi-loop, defined as a collection of loops (§2.1.1). Another variant will be given in Theorem 9.4 (where some of the loops in $\mathbf{c}$ are constrained to have bounded lengths). The final variant needed for the spectral gap is described in §12.

### 1.3.7. Friedman-Ramanujan functions as solutions of an integral equation.
A first ingredient in the proof of Theorem 1.22 is the fact that Friedman-Ramanujan functions are fully characterized as solutions (modulo $\mathcal{R}$) of a certain integral equation. We define two operators acting on locally integrable functions, $\mathcal{P} = \mathcal{P}_x = \int_0^x$ (taking the primitive vanishing at 0) and $\mathcal{L} = \mathcal{L}_x = \text{id} - \mathcal{P}_x$ (where id stands for the identity operator). It is trivial, nevertheless useful, to note that $\mathcal{P}$ and $\mathcal{L}$ preserve $\mathcal{F}, \mathcal{F}_w, \mathcal{R}, \mathcal{R}_w$. More precisely, $\mathcal{L}$ is a continuous operator from $\mathcal{F}^K$ to $\mathcal{F}^{K-1}$ and from $\mathcal{R}$ to itself (and this is also true for the weak spaces).

We will make use of the following characterization of Friedman-Ramanujan functions:

**Proposition 1.26.** For $K \ge 1$, any locally integrable function $f$,

$$\begin{cases} f \in \mathcal{F}^K \Leftrightarrow \mathcal{L}^K f \in \mathcal{R} \\ f \in \mathcal{F}_w^K \Leftrightarrow \mathcal{L}^K f \in \mathcal{R}_w. \end{cases}$$

Furthermore, if $N \ge 1$ is so that $\mathcal{L}^K f \in \mathcal{R}_w^N$, then

$$(1.14) \qquad \|f\|_{\mathcal{F}^{K,N}}^w \le \text{fn}(K, N) \|\mathcal{L}^K f\|_{\mathcal{R}^N}^w.$$

The “only if” part is a straightforward consequence of the fact that $\mathcal{L}$ sends $\mathcal{R}_w$ to itself, $\mathcal{R}$ to itself, and that $\mathcal{L}^K$ sends $\mathbb{C}_{K-1}[t]e^t$ to $\mathbb{C}_{K-1}[t] \subset \mathcal{R}$. The “if” part is proven by induction on $K$, iterating the following lemma:
**Lemma 1.27.** Let $K \ge 1$ be an integer. If $\mathcal{L}f = b + c$, with $b \in \mathbb{C}_{K-1}[t]e^t$ and $c \in \mathcal{R}$ or $\mathcal{R}_w$, then there exists a constant $C$ such that

$$ (1.15) \qquad f(t) = Ce^t + b(t) + c(t) + \int_0^t b(s)e^{t-s} ds + \int_t^{+\infty} c(s)e^{t-s} ds. $$

As a consequence, if $c \in \mathcal{R}_w^N$, then $f$ belongs in $\mathcal{F}_w^{K,N}$ and

$$ (1.16) \qquad \|f\|_{\mathcal{F}_w^{K,N}} \le \text{fn}(K, N) (\|b\|_{\mathcal{F}} + \|c\|_{\mathcal{R}_w^N}). $$

The same holds with the strong forms.

*Proof.* The lemma is proven by the usual method for solving linear ODEs: let $F = \mathcal{P}f$, then $F$ solves $F' - F = b + c$, so there exists a constant $C$ such that

$$ F(t) = Ce^t + \int_0^t b(s)e^{t-s} ds + \int_t^{+\infty} c(s)e^{t-s} ds. $$

Here, because of the different behaviors of $b$ and $c$ at infinity, we have chosen different boundary conditions for the primitives of $b(s)e^{-s}$ and $c(s)e^{-s}$. Since $f$ is the derivative of $F$, this implies (1.15). Assume that $c \in \mathcal{R}_w^N$: then $\int_0^t b(s)e^{t-s} ds \in \mathbb{C}_{K-1}[t]e^t$, $\int_t^{+\infty} c(s)e^{t-s} ds \in \mathcal{R}^N$, so $f \in \mathcal{F}_w^{K,N}$, and

$$ \|f\|_{\mathcal{F}_w^{K,N}}^w \le \text{fn}(K, N) (\|c\|_{\mathcal{R}^N}^w + |C| + \|b\|_{\mathcal{F}}). $$

Evaluation at $t = 0$ yields $C = - \int_0^{+\infty} c(s)e^{-s} ds$ so

$$ |C| \le \left( \int_0^{+\infty} (1+s)^{N-1} e^{-s/2} ds \right) \|c\|_{\mathcal{R}^N}^w. $$

Inequality (1.16) follows. $\square$

**Remark 1.28.** We can extend Remark 1.14 to the images of a Friedman-Ramanujan function $f \in \mathcal{F}^{K,N}$ by $\mathcal{L}$ and obtain a constant $M_{K,N}$ such that, for all $\ell \in \mathbb{R}_{\ge 0}$ and any $0 \le m < K$,

$$ |\mathcal{L}^m f(\ell)| \le M_{K,N} \|f\|_{\mathcal{F}^{K,N}} (1+\ell)^{K-1} e^{\ell} $$

$$ |\mathcal{L}^K f(\ell)| \le M_{K,N} \|f\|_{\mathcal{F}^{K,N}} (1+\ell)^{N-1} e^{\frac{\ell}{2}}. $$

Here $M_{K,N} = \max_{m \le K} (\text{fn}(K-m, N)) \|\mathcal{L}^m\|_{\mathcal{F}^{K,N} \to \mathcal{F}^{K-m,N}}$ with $\text{fn}(K-m, N)$ the constant in (1.7).

**1.3.8. Stability under convolution.** Another idea behind our proof is the stability of the classes $\mathcal{F}$, $\mathcal{F}_w$, $\mathcal{R}$, $\mathcal{R}_w$ under convolution. This was proven in [2, Proposition 3.6] by direct algebraic calculation, following [6, Theorem 7.2]. Another way to understand this is to refer to the usual property of convolution (classically known for differential operators, but valid also for the operator $\mathcal{L}$): given two functions $f_1, f_2$ and integers $K_1, K_2$, we have

$$ (1.17) \qquad \mathcal{L}^{K_1+K_2}(f_1 * f_2) = \mathcal{L}^{K_1} f_1 * \mathcal{L}^{K_2} f_2. $$

If $f_i \in \mathcal{F}^{K_i}$, then $\mathcal{L}^{K_i} f_i \in \mathcal{R}$. Thus, using the stability of $\mathcal{R}$ under convolution (a straightforward upper bound), we obtain that $\mathcal{L}^{K_1+K_2}(f_1 * f_2) \in \mathcal{R}$, implying $f_1 * f_2 \in \mathcal{F}^{K_1+K_2}$. The same method holds with $\mathcal{F}_w$ instead of $\mathcal{F}$.
In the proof of Theorem 1.22, we show that the function $\mathcal{J}$ defined by the integral (1.12) can be expressed in terms of “convolutions” of Friedman-Ramanujan functions. This requires to express the measure $d\text{Vol}_{g_S,n_S}^{\text{WP}}(Y)$ in a well chosen set of coordinates on $\mathcal{T}_{g_S,n_S}^*$, adapted to the loop $\mathbf{c}$. These coordinates are not the well known Fenchel-Nielsen coordinates: their description will be the focus of §3.

The reason why (1.12) looks roughly like a convolution is that the loop $\mathbf{c}$ can be decomposed into a union of simple curves $\beta = (\beta_1, \dots, \beta_n)$, by “opening” all the intersections (this operation is described in §2.2). In the case of curvature $-\infty$ (i.e. graphs) we would have

$$ \ell_Y(\mathbf{c}) = \ell_Y(\beta_1) + \dots + \ell_Y(\beta_n), $$

and (1.12) would exactly be a convolution. But we are in finite negative curvature, and $\ell_Y(\mathbf{c})$ differs from $\ell_Y(\beta_1) + \dots + \ell_Y(\beta_n)$: the expression (1.12) is not exactly a convolution. We will nevertheless see that $\ell_Y(\mathbf{c})$ is close to $\ell_Y(\beta_1) + \dots + \ell_Y(\beta_n)$ in a certain topology. In §6, we define a notion of pseudo convolutions and we develop a sophisticated variant of (1.17), first in an abstract setting – with the aim of later applying it to the study of the integral (1.12).

### 1.3.9. Geometric comparison estimates.
A key ingredient of the proof of Theorem 1.22 is the use of geometric comparison estimates, which we shall now comment on.

The following straightforward bound, comparing the length of a loop $\mathbf{c}$ and the boundary of the surface $\mathbf{S}$ it fills, is a simple case of a geometric comparison estimate.

**Proposition 1.29.** Let $\mathbf{T} = [\mathbf{S}, \mathbf{c}]_{\text{loc}}$ be a local type. For any $(\mathbf{x}, Y) \in \mathcal{T}_{g_S,n_S}^*$,

$$ (1.18) \qquad \sum_{i=1}^{n_S} x_i \le \ell(\mathbf{c}) + \sum_{\substack{\mathcal{I} \text{ unshielded} \\ \text{simple portion}}} \ell(\mathcal{I}) \le 2\ell(\mathbf{c}). $$

The estimate $\sum_{i=1}^{n_S} x_i \le 2\ell(\mathbf{c})$ is a classical consequence of the fact that $\mathbf{c}$ is assumed to fill the surface $\mathbf{S}$, see e.g. [2, Lemma 4.4]. In the special case of double-filling loops, Proposition 1.29 tells us that this can be improved by dropping the factor of 2, which was the crucial observation made in [2, Lemma 8.6]. Proposition 1.29 is a straightforward generalization of this lemma.

Comparison estimates are a crucial step in the proof of our main results: they give upper bounds for terms such as $\prod_{j \in V} f_j(x_j)$, appearing in the definition of $\mathcal{J}(\ell)$ in Theorem 1.22. Proposition 1.29 also holds for multi-loops, which will be defined as collections of loops (§2.1.1).

The advantage of gaining a factor of 2 in the comparison estimate can clearly be seen from the statement of Theorem 1.22: the function $\prod_{j \in V} f_j(x_j)$ grows like $\exp(\frac{1}{2} \sum_{j \in V} x_j)$ (up to some polynomial factors, allowed in the definition of Friedman-Ramanujan functions), so the bound $\sum_{j=1}^{n_S} x_j \le 2\ell(\mathbf{c})$ implies the upper bound

$$ (1.19) \qquad \prod_{j \in V} |f_j(x_j)| \le C(1 + \ell(\mathbf{c}))^{\alpha} e^{\ell(\mathbf{c})} $$

for some $C, \alpha \ge 0$. This ensures that $|\mathcal{J}(\ell)| \le C'(1+\ell)^{\alpha'} e^{\ell}$ in a weak sense (see (3.28) for a precise bound). However, the Friedman-Ramanujan property requires an *exact expansion* rather than an exponential upper bound. Therefore, we cannot directly
deduce the Friedman-Ramanujan property from the trivial inequality $\sum_{j=1}^{n_S} x_j \le 2\ell(\mathbf{c})$. The fundamental challenge addressed in this article consists in refining the approach to study $\mathcal{J}(\ell)$, hence providing an exact expansion instead of the straightforward upper bound above.

In the special case when the loop $\mathbf{c}$ is double-filling, then (1.19) may be replaced by

$$ (1.20) \qquad \prod_{j \in V} |f_j(x_j)| \le (1 + \ell(\mathbf{c}))^{\alpha} e^{\frac{\ell(\mathbf{c})}{2}}, $$

which is enough to ensure that $\mathcal{J}(\ell)$ is in the class $\mathcal{R}_w$, and thus in particular is a weak Friedman-Ramanujan function (see Corollary 3.27). In other words, Theorem 1.22 can be obtained by a straightforward upper bound in the case of double-filling loops. In general, the presence of shielded segments should imply less work to prove Theorem 1.22, thanks to Proposition 1.29. On the other end of the spectrum, the worse possible scenario should be the case when none of the simple portions are shielded: we call such loops generalized eights. Most of the paper is dedicated to proving Theorem 1.22 in the case of generalized eights. For other multi-loops we will indicate how to modify our discussion at the end of the paper (§11).

### 1.3.10. Plan of the paper.
In §2 we “open” all the intersections of $\mathbf{c}$ to obtain a new representative $\mathbf{c}^{\text{OP}}$ of the same homotopy class, decomposed into simple curves, joined by “bars” replacing the intersections. The whole discussion applies to multi-loops, defined as finite collections of loops. We define a class of multi-loops called generalized eights. They admit a rather simple geometric description but offer the highest degree of difficulty, as far as the proof of Theorem 1.22 is concerned.

In §3 we define a coordinate system on $\mathcal{T}_{g_S,n_S}^*$, adapted to the study of the function $Y \mapsto \ell_Y(\mathbf{c})$, and we express the Weil-Petersson measure in those new coordinates. There are few results expressing the Weil-Petersson measure in coordinates other than the Fenchel-Nielsen ones, so we believe that Theorem 3.11 has an intrinsic interest, and probably deserves further investigation.

In §4, we use the representative $\mathbf{c}^{\text{OP}}$ to find an algebraic expression for $\ell_Y(\mathbf{c})$, as well as the lengths of all boundary curves of $\mathbf{S}$, in the new coordinates. After this change of coordinates, it becomes apparent that the integral (1.12) looks like a convolution of Friedman-Ramanujan functions, see §5. As already mentioned, the fact that we do not have exactly a convolution is related to the fact that the curvature is finite. As a result, we need to develop a theory of pseudo-convolutions in an abstract setting, which is the focus of §6.

In order to apply our theoretical results on pseudo-convolutions to our particular problem, additional geometric and analytic estimates on the function $Y \mapsto \ell_Y(\mathbf{c})$ are required. They are provided in §7 and §8. Most of §7 can be skipped in the purely non-crossing case (defined in §2.6). The proof is finished in §9, except for the “Crucial comparison estimate” which is a variant of Proposition 1.29, proven in §10. We indicate in §11 how to treat arbitrary topologies (i.e. multi-loops which are not generalized eights). In §12 we state a variant of Theorem 1.22, and use it in §13 to deduce Theorem 1.6 about the optimal spectral gap.
**Acknowledgements.** Discussions with J. Friedman were of invaluable help to surmount several obstacles in this project.

This research has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No. 101096550), the EPSRC Standard Grant EP/W007010/1 and the Royal Society Dorothy Hodgkin Fellowship.

## 2. LOOPS AND GENERALIZED EIGHTS

This section starts by setting a number of conventions related to loops on surfaces, segments, orientations and geodesics on the hyperbolic plane. We then construct in §2.2 a procedure allowing to open the intersections of a loop, and following on this some associated concepts (bars, simple portions, set $\Theta$, diagrams, cyclic orderings, u-turns and crossings). Importantly, in §2.3, we introduce a class of loops called *generalized eights*, which will be the focus of the bulk of the paper (§3-10).

**2.1. Definitions.** Let $\mathbf{S}$ be a smooth surface (possibly with boundary). Throughout this article, we will assume that $\mathbf{S}$ is connected and has negative Euler characteristic. Indeed, the case where $\mathbf{S}$ is a cylinder has already been addressed in [2, Proposition 3.4]. In applications, the surface $\mathbf{S}$ we apply Theorem 1.22 to will often not be connected; however, the proof for disconnected $\mathbf{S}$ only requires minor modifications from the proof in the connected case.

### 2.1.1. *Multi-loops*. Let us clarify what we mean by a loop or a multi-loop $\mathbf{c}$.

**Definition 2.1.** A loop is a piecewise smooth map $\mathbf{c} : \mathbb{R}/\mathbb{Z} \to \mathbf{S}$ with nowhere vanishing derivative (we allow loops to be contractible). A loop is called a *curve* if it is non-contractible and has no self-intersection.

From now on we extend the discussion to *multi-loops*. This is not just for the sake of maximal generality, but also because, when we later “remove” some self-intersections from $\mathbf{c}$, this operation will inevitably turn a loop into a multi-loop.

**Definition 2.2.** A *multi-loop* is a collection of loops $\mathbf{c} = (\mathbf{c}_1, \dots, \mathbf{c}_n) : \mathbb{R}/\mathbb{Z} \to \mathbf{S}^n$. The loops $\mathbf{c}_i$ are called the *components of* $\mathbf{c}$.

(1) A multi-loop is said to be *in minimal position* if it has only transverse self-intersections, and minimal number of self intersections in its free homotopy class.

(2) A multi-loop is said to be *simple* if it has no self-intersections.

(3) A multi-loop is called a *multi-curve* if it is simple, its components are non-contractible and, for $i \neq j$, $\mathbf{c}_i$ is not freely homotopic to $\mathbf{c}_j$ nor $\mathbf{c}_j^{-1}$.

**Remark 2.3.** Although the multi-loop $\mathbf{c}$ is originally assumed to be in minimal position, the reader will notice that we will work with representatives of its homotopy class that do not have this property, but are $\mathcal{C}^0$-limits of multi-loops in minimal position.

We will always assume that the multi-loop $\mathbf{c}$ fills the surface $\mathbf{S}$, that is to say, if $\mathcal{N}$ is a regular neighbourhood of $\mathbf{c}$, then all the connected components of $\mathbf{S} \setminus \mathcal{N}$ are disks or annular regions bordered by a boundary component of $\mathbf{S}$ (see [2, Definition 2.3]).

Multi-loops are parameterized and oriented, but often, we will only be interested in their geometric image as subsets of $\mathbf{S}$. Importantly, when we say that two multi-loops
c = (c₁, ..., cₙ) and c' = (c'₁, ..., c'ₙ') are freely homotopic, we take the numberings and the orientations into account: we mean that they have the same number of components (n = n') and that cᵢ is freely homotopic to c'ᵢ for every i. Homotopies must respect orientation, and in particular a loop is not a priori homotopic to its inverse.

2.1.2. Segments and concatenation. Throughout this article, we will decompose our loops in successions of segments.

**Definition 2.4.** We call $J \subset S$ an (oriented) segment if it is of the form $J = p([0, T])$, where $p: [0, T] \to S$ is a smooth oriented path with nowhere vanishing derivative.

**Notation 2.5 (Origin and terminus).** Let $J = p([0, T])$ an oriented segment. The origin of $J$ is the point $o(J) := p(0)$, and its terminus $t(J) := p(T)$.

**Notation 2.6 (Concatenation of paths).** If $p: [0, T] \mapsto S$ and $q: [0, T'] \mapsto S$ are two piecewise smooth paths such that $p(T) = q(0)$, we define a piecewise smooth path $p \cdot q: [0, T + T'] \to S$ by

$$p \cdot q(t) := \begin{cases} p(t) & \text{if } t \in [0, T] \\ q(t - T) & \text{if } t \in [T, T + T'] \end{cases}.$$

This is an associative and non-commutative operation. Most of the time, we shall only be interested in the geometric images of the paths, and not in their actual parametrizations.

**Definition 2.7.** Let $\beta$ be a multi-loop on $S$, and $J = p([0, T])$ be an oriented segment with endpoints on $\beta$. We say an oriented segment $K = q([0, T])$ is homotopic to $J$ with gliding endpoints if there exists a continuous function $h: [0, T] \times [0, 1] \to S$ such that $h(\cdot, 0) = p$, $h(\cdot, 1) = q$, and the endpoints $h(0, t)$ and $h(T, t)$ lie on $\beta$ for any $t \in [0, 1]$.

**Remark 2.8.** In the following, $\beta$ will (for the most part) be a multi-curve. In this case, when we equip the surface $S$ with a metric $Y$, we will systematically do so by picking a representative in the Teichmüller space $\mathcal{T}_{g_s, n_S}^*$ so that $\beta$ is a simple multi-geodesic. In this case, any non-trivial homotopy class of segments $J$ with endpoints gliding on $\beta$ contains a uniquely defined length-minimizing element $\bar{J}$, which is an orthogeodesic (a geodesic segment orthogonal to $\beta$ at its endpoints). In pictures, in an effort to simplify the usual notation, we will often mark the endpoints of orthogeodesics with a black dot to emphasise the fact that there is a right angle there.

2.1.3. Left and right. We take the following conventions in terms of the orientation of segments.

**Definition 2.9.** Let $J = p([0, T])$ an oriented segment on $S$.

* A tangent vector $v \in T_x S$ with $x = p(t) \in J$ is said to be on the left (resp. right) of $J$ if the angle from $\dot{p}(t)$ to $v$ belongs to $[0, \pi]$ (resp. $[-\pi, 0]$).

* Let $K = q([0, T'])$ be an oriented segment. We say that $K$ leaves $J$ from the left (resp. right) if the origin of $K$ lies in $J$ and if the tangent vector $\dot{q}(0)$ is on the left (resp. right) of $J$. Similarly, we say that $K$ arrives on $J$ from the left (resp. right) if the terminus of $K$ lies on $J$ and $\dot{q}(T')$ is on the right (resp. left) of $J$.
2.1.4. *Infinite geodesics on $\mathbb{H}^2$.* We now briefly focus on infinite geodesics in the hyperbolic plane $\mathbb{H}^2$. For two infinite geodesics in the hyperbolic plane, with endpoints at infinity $(a, b)$ and $(c, d)$, the cross-ratio $(a : b : c : d)$ can be used as a convenient way of describing their respective positions:

• A cross-ratio $(a : b : c : d) \in (0, 1)$ means that the two geodesics intersect.

• Whenever $(a : b : c : d) \notin (0, 1)$ we shall speak of *parallel geodesics*.

– A cross-ratio in $[1, +\infty)$ means that the two geodesics are parallel and oriented “head-to-tail”.

– On the opposite, $(a : b : c : d) \le 0$ means that the two geodesics are parallel and are oriented “in the same direction”. We will then say that there are *aligned*.

Since the hyperbolic plane is oriented, an infinite oriented geodesic determines two half-spaces, one on its left and one on its right.

**Definition 2.10 (Algebraic distance $\bar{d}$).** If two points $x, y$ lie on an oriented geodesic $\gamma$, we will say $y$ is on the right of $x$ along $\gamma$ to mean that $y$ is after $x$ along $\gamma$. We denote by $\bar{d}(x, y)$ the signed distance between $x$ and $y$ along $\gamma$, measured positively if $y$ is on the right of $x$ along $\gamma$ and negatively otherwise.

This notion is named according to the tradition of orienting lines from left to right. Although it would be more rigourous to denote this distance by $\bar{d}_{\gamma}(x, y)$, the notation will always be used in the universal cover $\mathbb{H}^2$ and the reference geodesic will always be clear from the context.

The following key observation, illustrated in Figure 1, will be used on multiple occasions throughout this article.

<figure><img src="image_3.png" /><figcaption>FIGURE 1. Illustration of the Useful Remark.</figcaption></figure>

**Remark 2.11 (The Useful Remark).** Take 3 distinct non-intersecting geodesics $(\gamma_i)_{0\le i\le 2}$ in the hyperbolic plane $\mathbb{H}^2$. Assume that for $i = 1, 2$ we have a piecewise smooth oriented path $p_i$ going from $\gamma_0$ to $\gamma_i$, intersecting $\gamma_0$ and $\gamma_i$ only at its endpoints. Therefore, the complement $\mathbb{H}^2 \setminus (\gamma_0 \cup \gamma_i \cup p_i)$ has four connected components, that we respectively call north of $\gamma_i$, south of $\gamma_0$ and east/west of $p_i$, following the compass. Assume in addition that $p_2$ only visits the east of $p_1$, and $p_1$ only visits the west of $p_2$. Then:
(1) the oriented orthogeodesics $\bar{p}_i$ from $\gamma_0$ to $\gamma_i$ are disjoint;

(2) $\bar{p}_2$ entirely lies east of $\bar{p}_1$;

(3) $\bar{p}_1$ entirely lies west of $\bar{p}_2$.

**2.1.5. Length of a path or a geodesic.** If $\gamma$ is a 1-dimensional Riemannian manifold, we will denote by $\ell(\gamma)$ its length. If now $\gamma$ is a smooth path in a Riemannian manifold $Y$, we will denote by $\ell(\gamma)$ its length in $Y$ (which coincides with its length for the metric on $\gamma$ induced by $Y$). If $\gamma$ is a loop in a hyperbolic surface $Y$, we will denote by $\ell_Y(\gamma)$ the minimal length of a representative of the free homotopy class of $\gamma$ (if $\gamma$ is contractible, $\ell_Y(\gamma) = 0$, otherwise it is the length of the unique geodesic representative of the free homotopy class of $\gamma$). In general, $\ell(\gamma)$ and $\ell_Y(\gamma)$ are different, but they are of course equal if $\gamma$ is a closed geodesic in $Y$.

If $S$ is endowed with a hyperbolic structure $Y$, then each component $c_i$ of $c$ is freely homotopic to a periodic geodesic, which is unique if $c_i$ is non-contractible, but may be reduced to a point otherwise. We denote $\ell_Y(c) = \sum_{i=1}^n \ell_Y(c_i)$ the total length of a geodesic representative.

**2.2. Opening the intersections of c.** In this paragraph, we explain how to *open the self-intersections of c to replace them by bars*. Our construction yields a new picture which consists of a simple multi-loop $\beta$ and a collection of bars $B$ with endpoints on $\beta$, replacing the intersections.

**Notation 2.12.** Throughout the paper, the letter $r$ stands for the number of self-intersection points of the multi-loop $c$. We identify the set of intersection points of the multi-loop $c$ with the set $\{1, \dots, r\}$ by picking an arbitrary numbering.

**2.2.1. The construction.** If $r=0$ then $c$ is a simple loop and we simply take $\beta = c$, and no bars. Now assume $r \ge 1$. Let $a_1, \dots, a_r$ be the intersection points of $c$ and $D_1, \dots, D_r$ be disjoint open disks around them. We can take the disks small enough so that, for each $j$, $c \cap D_j$ is the union of two oriented segments $I_j, I'_j$ intersecting transversally at $a_j$. We open the intersection at each $a_j$ and replace the two intersecting segments $I_j, I'_j$ by two smooth disjoint segments, joined by a transversal segment $B_j$ traversed twice, replacing the intersection. There are two distinct ways to perform this operation, represented in Figure 2b and 2c respectively.

<figure><img src="image_4.png" /><figcaption>(A) The initial intersection. (B) Opening the first way. (C) Opening the second way.</figcaption></figure>

**FIGURE 2.** Opening an intersection to replace it by a bar, two ways.
Opening all of the intersections (having made an arbitrary choice between the two ways for each intersection) yields a new representative $c^{\text{op}}$ of the homotopy class of $c$, represented in Figure 3. The operation of going from $c$ to the new representative $c^{\text{op}}$ will be referred to as *opening the intersections of c* (to replace them by bars).

<figure><img src="image_5.png" /><figcaption>FIGURE 3. A close deformation of the multi-loop $c^{\text{op}}$. Notice that each bar is traversed twice (in the directions we will later denote $B_j^+$ and $B_j^-$).</figcaption></figure>

By convention, the bars $B_j$ are closed, i.e. contain their endpoints. If $\dot{B}_j$ is the open segment, i.e. the segment without its endpoints, we call $\beta$ the connected components of $c^{\text{op}} \setminus (\cup_{j=1}^r \dot{B}_j)$. We observe that $\beta$ is a simple multi-loop. The geometric image of $c^{\text{op}}$ is the simple multi-loop $\beta$ joined by the collection of bars $B = (B_1, \dots, B_r)$.

Note that $\beta$ is not necessarily a multi-curve – see Figure 4b for an example where one of the components of $\beta$ is contractible. When we later specialise our discussion to the class of multi-loops called *generalized eights* in §2.3, the family $\beta$ will actually be a multi-curve (see Lemma 2.22).

**Remark 2.13.** The bars are naturally numbered as $(B_1, \dots, B_r)$ thanks to our choice of numbering of the intersection points of $c$. However, there is no canonical way to number the components of $\beta$ based on our information so far, and because picking a numbering is not very useful for our purposes, we do not do so. We will denote as $\Lambda^\beta$ the set of components of $\beta$ (this notation will make more sense after §3.3), so that $\beta = (\beta_\lambda)_{\lambda \in \Lambda^\beta}$.

2.2.2. *Orientation of the bars.* Let us define orientations $B_j^{\pm}$ on the bars $B_j$ for $1 \le j \le r$. The convention will depend on the type of opening.

We first observe that the endpoints of the bars $B_1, \dots, B_r$ partition the multi-loop $\beta$ into $2r$ segments, reflecting the partition of $c$ into the simple portions defined in Definition 1.20. We will call *simple portions of $c^{\text{op}}$* these subsegments of $\beta$. Each simple portion is naturally oriented by the original orientation of $c^{\text{op}}$. Each bar $B_j$ touches four (non necessarily distinct) simple portions: two at each endpoint.

We then settle on the following convention, illustrated in Figure 3.

(1) In the *first way*, we choose the orientation $B_j^+$ of $B_j$ such that $B_j^+$ arrives on the right of the two simple portions touching its terminus $t(B_j^+)$: one of these simple portions originates in $t(B_j^+)$ and the other one terminates in $t(B_j^+)$. Similarly, $B_j^+$ leaves from the left of the two simple portions touching its origin $o(B_j^+)$; one
<figure><img src="image_6.png" /><figcaption>FIGURE 4. An example of the construction for three loops filling a surface of signature (0, 4). Here, we opened all the intersections the first way.</figcaption></figure>

of them originates in $o(B_j^+)$ and the other one terminates in $o(B_j^+)$. In this case we denote by $B_j^-$ the opposite orientation of $B_j$.

(2) In the second way, we choose the orientation $B_j^+$ of $B_j$ such that the two simple portions touching its terminus $t(B_j^+)$ originate in $t(B_j^+)$, and the two simple portions touching its origin $o(B_j^+)$ terminate in $o(B_j^+)$. In this case, we define the orientation $B_j^-$ to be identical to $B_j^+$.

2.2.3. Labelling of simple portions. Let us label the simple portions and define a few useful notations. Our most important set of indices will be the set

$$ (2.1) \qquad \Theta = \{1, \dots, r\} \times \{+, -\}, $$

which will naturally label the simple portions. Indeed, for $(j, \epsilon) \in \Theta$, let $\mathcal{I}_j^\epsilon$ be the oriented simple portion originating from the terminus of $B_j^\epsilon$, so that $B_j^+$ arrives on $\mathcal{I}_j^+$ from the right at the point $o(\mathcal{I}_j^+) = t(B_j^+)$ and $B_j^-$ arrives on $\mathcal{I}_j^-$ from the left at the point $o(\mathcal{I}_j^-) = t(B_j^-)$, as represented in Figure 3.

**Remark 2.14.** Note that, out of the four simple portions touching the bar $B_j$, only two are labelled $\mathcal{I}_j^\pm$ by the above convention. The other two simple portions are denoted as $\mathcal{I}_{j'}^\pm$ for another (possibly equal) integer $j'$.

We shall most often refer to the elements of $\Theta$ by the letter $q$, so that if $q = (j, \epsilon)$ is an elements of $\Theta$, then the bar $B_q$ stands for $B_j^\epsilon$, $\mathcal{I}_q$ for $\mathcal{I}_j^\epsilon$, etc.

The signs in $\Theta$ will play an important role throughout the paper.
**Notation 2.15.** We define the sign function on $\Theta$ by:

$$ (2.2) \qquad \forall q = (j, \epsilon) \in \Theta, \quad \mathrm{Sign}(q) = \epsilon. $$

We denote as $\iota: \Theta \to \Theta$ the involution defined by $\iota(j, \epsilon) = (j, -\epsilon)$.

It will be helpful to relate the elements of $\Theta$ and the components of the multi-loop $\beta$.

**Notation 2.16.** For $q \in \Theta$, let $\lambda(q) \in \Lambda^\beta$ denote the component of $\beta$ on which $B_q$ terminates, i.e. such that the terminus $t(B_q)$ belongs to the loop $\beta_{\lambda(q)}$.

For $\lambda \in \Lambda^\beta$ (i.e. for a component $\beta_\lambda$ of the multi-loop $\beta$), we define the set of indices

$$ (2.3) \qquad \Theta_t(\lambda) := \{q \in \Theta : t(B_q) \in \beta_\lambda\} \subseteq \Theta. $$

In other words, this is the set of indices $q \in \Theta$ such that $\lambda(q) = \lambda$, or, equivalently, such that the bar $B_q$ terminates on $\beta_\lambda$. We extend this notation to any subset $W \subseteq \Lambda^\beta$ by letting

$$ (2.4) \qquad \Theta_t(W) := \bigcup_{\lambda \in W} \Theta_t(\lambda) = \{q \in \Theta : \lambda(q) \in W\}. $$

**Remark 2.17.** We make the key observation that, since the multi-loop $\beta$ is decomposed in the simple portions $\mathcal{I}_q$ with $q \in \Theta$, the sets $(\Theta_t(\lambda))_{\lambda \in \Lambda^\beta}$ generate a partition of $\Theta$:

$$ \Theta = \bigsqcup_{\lambda \in \Lambda^\beta} \Theta_t(\lambda). $$

**Remark 2.18.** It will be useful to keep in mind from the construction that each component $\beta_\lambda$ of $\beta$ is freely homotopic to an “initial” piecewise smooth curve $\beta_{\lambda, \mathrm{init}}$, included in the original multi-loop $\mathbf{c}$, made of smooth segments joining some of the intersection points of $\mathbf{c}$. The partition of $\beta_\lambda$ into the union of a collection of segments $(\mathcal{I}_q)_{q \in \Theta_t(\lambda)}$ is inherited from a partition of $\beta_{\lambda, \mathrm{init}}$ into segments $(\mathcal{I}_{q, \mathrm{init}})_{q \in \Theta_t(\lambda)}$ which are the simple portions of Definition 1.20.

2.2.4. *Diagrams.* The result of opening all of the intersections of the multi-loop $\mathbf{c}$ is what we call a diagram.

**Definition 2.19.** A diagram $\mathbf{D}$ in $\mathbf{S}$ is the data (modulo isotopy) of a simple multi-loop $\beta$, together with a collection of disjoint bars $B = (B_1, \dots, B_r)$ defined as closed, mutually disjoint segments, having their endpoints in $\beta$ and disjoint from $\beta$ otherwise.

Starting with a multi-loop $\mathbf{c}$ and opening all intersections (either way), we obtain a diagram $\mathbf{D}$, which is said to *represent* $\mathbf{c}$. Conversely, the multi-loop $\mathbf{c}$ is said to *stem from* $\mathbf{D}$ if $\mathbf{D}$ represents $\mathbf{c}$.

Different representatives of the same homotopy class may give rise to non-isotopic diagrams: see Figure 4b. The paper by Graaf and Schrijver [7] precisely describes when two multi-loops can be homotopic without being isotopic: this may happen if and only if one can be sent to the other by an isotopy followed by a finite succession of Reidemeister moves of type III.
2.2.5. *First v.s. second way.* Throughout this article, we will be in particular interested in the case when we open all of the intersections the first way. We denote by $\mathbf{D}^{(1)}$ the diagram obtained by opening all intersections the first way. One possible reason for preferring the first way is the fact that the multi-loop $\beta$ then inherits a global orientation that fits with the orientation of the simple portions $(\mathcal{I}_q)_{q \in \Theta}$, in other words with the initial orientation of $\mathbf{c}$. If we open the intersections the second way, the orientations of the simple portions $(\mathcal{I}_q)_{q \in \Theta}$ do not fit into a global orientation of $\beta$, which is not a problem, but can sometimes be slightly cumbersome in terms of notations.

Unless explicitly stated, for the sake of simplicity, we invite the reader to consider that the construction above has been done by *opening every intersection the first way*. However, many of our results, notably our change of variable presented in §3, hold for whichever choice of opening is made – different openings yielding different interesting results. We will be careful to point out the few places where the choice of opening actually does matter.

2.3. **Generalized eights.** We introduce a class of multi-loops, called *generalized eights*, on which we focus attention for most of the paper. Their name comes from the fact that they are generalizations of the basic figure-eight loop, the loop with exactly one self-intersection ($r = 1$).

**Definition 2.20.** Let $\mathcal{N}$ be a regular neighbourhood of $\mathbf{c}$ in $\mathbf{S}$. We say that $\mathbf{c}$ is a *generalized eight* if no connected components of $\partial\mathcal{N}$ is contractible in $\mathbf{S}$.

**Remark 2.21.** We recall that we have assumed at the beginning of §2.1 that $\mathbf{S}$ is a connected surface, and that it is not a cylinder. In particular, simple loops are not generalized eights. The definition above still holds verbatim in the case where $\mathbf{S}$ has several connected components (provided none of them is a cylinder).

This definition has the following consequence on the surface $\mathbf{S}$ and the multi-loop $\beta$.

**Lemma 2.22.** If $\mathbf{c}$ is a *generalized eight*, then $\chi(\mathbf{S}) = r$ and $\beta$ is a *multi-curve*.

*Proof.* Let $\mathcal{N}$ be a regular neighbourhood of $\mathbf{c}$ in $\mathbf{S}$. Since $\mathbf{c}$ fills $\mathbf{S}$ and no boundary component of $\mathcal{N}$ is contractible, $\mathbf{S} \setminus \mathcal{N}$ is a union of cylinders, and hence $\chi(\mathbf{S}) = \chi(\mathcal{N})$. We then conclude by observing that the regular neighbourhood $\mathcal{N}$ retracts on the 4-regular graph with vertices the $r$ intersections points of $\mathbf{c}$, and edges its $2r$ simple portions, of Euler characteristic $r - 2r = -r$.

The definition of generalized eight directly implies that none of the components of $\beta$ is contractible. To prove that $\beta_i$ cannot be freely homotopic to $\beta_j$ or its inverse for $i \neq j$ two components of $\beta$, we remark that if $\beta_i$ is freely homotopic to $\beta_j$ or its inverse, then there is a cylinder in $\mathbf{S}$ bordered by $\beta_i$ and $\beta_j$. This cylinder may contain in its interior other components $\beta_k$ homotopic to $\beta_i^{\pm 1}$. We pick $k$ such that the cylinder $C$ bordered by $\beta_i$ and $\beta_k$ does not contain in its interior any other component of the multi-loop $\beta$. Because $\mathbf{c}$ fills $\mathbf{S}$, the cylinder $C$ must contain bars $B_l$ going from $\beta_i$ to $\beta_k$. These bars, together with $\beta_i$ and $\beta_k$, cut $C$ into connected components homeomorphic to disks, contradicting the fact that $\partial\mathcal{N}$ does not have contractible components. $\square$

**Remark 2.23.** If $\mathbf{c}$ is a generalized eight, Reidemeister moves of type III are not possible and hence any multi-loop homotopic to $\mathbf{c}$ is also isotopic. As a consequence, the diagram $\mathbf{D}^{(1)}$ obtained by opening the intersections the first way is uniquely defined up to isotopy.
2.4. Removing bars and intersections. We define two ways of removing a bar from a diagram: either by just deleting it (we call this the first way of removing a bar), or the second way, shown on Figure 5a.

<figure><img src="image_7.png" /><figcaption>FIGURE 5. The two ways of removing a bar on a diagram or an intersection point on a multi-loop.</figcaption></figure>

If the diagram was obtained by opening all intersections of a multi-loop c, removing a bar corresponds to removing an intersection from c. Correspondingly, there are two ways to remove an intersection from c (the first and the second way), shown on Figure 5b.

We make the following straightforward observation, which will enable us to argue by induction on r in the proof of Theorem 3.11.

**Lemma 2.24.** If c is a generalized eight, then the multi-loop $c'$ obtained by removing an intersection point is still a generalized eight. It fills a surface of absolute Euler characteristic $r - 1$.

Note that the number of components of the multi-loop $c'$ is, a priori, different from the number of components of c; this is one of the motivations for studying multi-loops instead of mere loops.

2.5. **Reconstitution of $c^{\text{OP}}$ and relabelling of $\Theta$.** Let us explain how to retrieve the multi-loop $c$ from the diagram, and define some convenient notations to express the representant $c^{\text{OP}}$ of the homotopy class of $c$ in terms of a succession of bars and simple portions.

2.5.1. *Cycle-decomposition and rewriting of $c^{\text{OP}}$. For $q \in \Theta$, we define*

$$ (2.5) \qquad p(q) := B_q \cdot \mathcal{I}_q $$

to be the concatenation of the oriented segments $B_q$ and $\mathcal{I}_q$ (which do, by definition, connect). The representative $c^{\text{OP}}$ of the homotopy class of $c$ is the concatenation of the $2r$ paths $p(q)$ for $q \in \Theta$. More precisely, there is a unique permutation $\sigma$ of the set $\Theta$,
decomposed into n disjoint cycles $\sigma = \prod_{i=1}^{n} (q_1^i q_2^i \dots q_{n_i}^i)$ such that, for all $i \in \{1, \dots, n\}$, the $i$th component $c_i^{\text{op}}$ of $c^{\text{op}}$ coincides with the concatenation

$$ (2.6) \qquad c_i^{\text{op}} = p(q_1^i) \cdot p(q_2^i) \cdot \dots \cdot p(q_{n_i}^i) $$

(or a circular permutation thereof).

The length of the $i$-th cycle is denoted as $n_i$, and we have $\sum_{i=1}^{n} n_i = 2r$. Because of the invariance of the free homotopy class of (2.6) under circular permutation, it is convenient to consider that the elements $q_k^i = (j_k^i, \epsilon_k^i) \in \Theta$ are numbered by $k \in \mathbb{Z}_{n_i}$ where $\mathbb{Z}_{n_i} := \mathbb{Z}/n_i\mathbb{Z}$.

**Remark 2.25.** By construction, we have the following orientation rules in (2.6), see Figure 3:

* if $\epsilon_k^i = +$ then $B_{q_k^i}$ arrives at $\mathcal{I}_{q_k^i}$ from the right and leaves from $\mathcal{I}_{q_{k-1}^i}$ from the left;

* otherwise, $B_{q_k^i}$ arrives at $\mathcal{I}_{q_k^i}$ from the left and leaves from $\mathcal{I}_{q_{k-1}^i}$ from the right.

**2.5.2.** *Re-labelling of $\Theta$.* For $q \in \Theta$, let $\text{Comp}(q) = i \in \{1, \dots, n\}$ be the label of the component $c_i$ of $c$ in which $q$ appears in the decomposition (2.6). We set

$$ \Theta^i = \{q \in \Theta, \text{Comp}(q) = i\}. $$

Then, for any $q \in \Theta^i$, we can write using the permutation $\sigma$:

$$ (2.7) \qquad c_i^{\text{op}} = p(q) \cdot p(\sigma q) \cdot p(\sigma^2 q) \cdot \dots \cdot p(\sigma^{n_i-1} q). $$

We now consider the set

$$ \Theta' = \bigsqcup_{i=1}^{n} \mathbb{Z}_{n_i} = \{(i, k), i \in \{1, \dots, n\}, k \in \mathbb{Z}_{n_i}\}. $$

The map

$$ (2.8) \qquad \phi: \begin{cases} \Theta' & \to \Theta \\ (i, k) & \mapsto q_k^i = (j_k^i, \epsilon_k^i) \end{cases} $$

is a bijection, and the cyclic ordering of $\mathbb{Z}_{n_i}$ is compatible with the cycling ordering of $q_k^i$ in the path (2.6). We shall identify $\Theta$ with $\Theta'$ through the map $\phi$, thus writing $(i, k)$ for $1 \le i \le n$, $k \in \mathbb{Z}_{n_i}$ to enumerate the elements of $\Theta$. Doing so, $\Theta^i$ can be identified with $\{(i, k), k \in \mathbb{Z}_{n_i}\}$, that is to say, with $\mathbb{Z}_{n_i}$. After this identification, the permutation $\sigma$ is given by $(i, k) \mapsto (i, k+1)$.

**2.5.3.** *Periodic extensions.* Similarly define

$$ \Theta = \bigsqcup_{i=1}^{n} \mathbb{Z} = \{(i, k), i \in \{1, \dots, n\}, k \in \mathbb{Z}\} = \bigsqcup_{i=1}^{n} \Theta^i, $$

where $\Theta^i = \{(i, k), k \in \mathbb{Z}\}$. When needed, the sequence $(i, k) \mapsto q_k^i$ can also be defined for $(i, k) \in \Theta$ by pre-composing with the projection map $\mathbb{Z} \to \mathbb{Z}_{n_i}$. In accordance with the previous discussion, we will denote $\sigma : \Theta \to \Theta$ the shift map $(i, k) \mapsto (i, k+1)$.
2.6. **U-turns and crossings.** The following definition will be useful to our analysis.

**Definition 2.26.** For $q \in \Theta$, we say that $q$ is a *crossing index* if $\text{Sign}(q)\text{Sign}(\sigma q) = +1$, and a *U-turn index* otherwise. We denote by $C \subseteq \Theta$ the set of crossing indices and by $\mathcal{U} \subseteq \Theta$ the set of U-turn indices. If $\Theta = \mathcal{U}$, we say that we are in the *purely non-crossing case*.

<figure><img src="image_8.png" /><figcaption>FIGURE 6. A U-turn v.s. crossing index (here $\text{Sign}(q) = +$).</figcaption></figure>

The terminology is motivated by the fact that if $q$ is a crossing index, then to go from $B_q$ to $B_{\sigma q}$ we have to cross the component $\beta_{\lambda(q)}$ of $\beta$, while in the other case, we make a U-turn when arriving at $\beta_{\lambda(q)}$. See Figure 6.

3. COORDINATE SYSTEM ADAPTED TO A GENERALIZED EIGHT

**We assume until §11 that the multi-loop $c$ is a generalized eight filling a surface $S$.** We open the intersections of $c$ by the procedure defined in the previous section (in this section, we allow every intersection to be opened in the first or second way). By hypothesis, the simple multi-loop $\beta$ is a multi-curve. Let us show how the bars $B_1, \dots, B_r$ and the simple portions $(\mathcal{I}_q)_{q \in \Theta}$ can be used to *define new coordinates* on the Teichmüller space $\mathcal{T}_{g_S, n_S}^*$ of marked hyperbolic metrics on $S$, well-adapted to expressing the length function of $c$.

First, we notice that the dimension of $\mathcal{T}_{g_S, n_S}^*$ is equal to $6g_S - 6 + 3n_S = 3\chi(S)$. We proved in Lemma 2.22 that, for a generalized eight, $\chi(S) = r$. As a consequence, the dimension of the Teichmüller space $\mathcal{T}_{g_S, n_S}^*$ is $3r$, so this is the number of coordinates we expect to need to describe a point on $\mathcal{T}_{g_S, n_S}^*$.

**3.1. Coordinates** $\vec{L} = (L_1, \dots, L_r)$ and $\vec{\theta} = (\theta_1^{\pm}, \dots, \theta_r^{\pm})$ **on the Teichmüller space.** Consider a point $Y \in \mathcal{T}_{g_S, n_S}^*$, i.e. a compact hyperbolic metric on $S$. As explained in Remark 2.8, we take a representative of $Y$ in the Teichmüller space such that the multi-curve $\beta$ is a simple multi-geodesic.

For $q \in \Theta$, we replace the bar $B_q$ by the representative $\bar{B}_q$ of minimal length in its homotopy class with endpoints gliding on $\beta$. While the endpoints $t(\bar{B}_q)$ glide to the new endpoints $t(\bar{B}_q)$ along $\beta$, it is important to note that the segments $\mathcal{I}_q$ glide to new segments $\bar{\mathcal{I}}_q$ of $\beta_{\lambda(q)}$. Although the original segments $\mathcal{I}_q$ were positively oriented along
$\beta_{\lambda(q)}$, it can happen that $\bar{I}_q$ goes in the reverse direction, or that $\bar{I}_q$ wraps an arbitrary number of times around $\beta_{\lambda(q)}$.

We are now ready to define our new coordinates on the Teichmüller space $\mathcal{T}_{g_S, n_S}^*$.

**Definition 3.1** (Coordinates $(\vec{L}, \vec{\theta})$). Let $Y \in \mathcal{T}_{g_S, n_S}^*$.

(1) For $j \in \{1, \dots, r\}$, we denote by $L_j = L_j(Y)$ the (positive) length of the orthogeodesic $\bar{B}_j$. We write $\vec{L} = (L_1, \dots, L_r)$.

(2) For $q \in \Theta$, we denote by $\theta_q = \theta_q(Y)$ the *algebraic length* of $\bar{I}_q$: positive if it goes in the same direction as $\mathcal{I}_q$, negative otherwise. We write $\vec{\theta} = (\theta_q)_{q \in \Theta}$.

<figure><img src="image_9.png" /><figcaption>FIGURE 7. An example of coordinates $(\vec{L}, \vec{\theta})$ in a case when $r=2$. We have additionally represented the pair of pants decomposition constructed in §3.2 as dashed curves (here, $\#\Lambda^\beta = 3$ and $\#\Lambda^\Gamma = 2$).</figcaption></figure>

**Remark 3.2** (Crossing and U-turn variables). If $q \in C$ is a crossing index, then we say that $\theta_q$ is a *crossing variable*; if $q \in U$ is a U-turn variable, then we say that $\theta_q$ is a *U-turn variable*. By the Useful Remark (Remark 2.11), U-turn variables are always positive: if $q \in U$, then $\theta_q \ge 0$. Thus, the discussion concerning U-turn variables is a little simpler than for crossing variables.

**Notation 3.3.** It will often be convenient to use the notation $L(q) = L_j$ for $q = (j, \epsilon) \in \Theta$ (note that $L(q)$ does not depend on the sign component of $q$). We will also benefit from introducing $\epsilon L(q) := \text{Sign}(q)L(q)$. The vector $(\epsilon L(q))_{q \in \Theta}$ then contains each length $L_j$ exactly twice, once positively and once negatively. This corresponds to the fact that each bar is explored exactly twice when going along $c^{\text{op}}$, following its two orientations, as illustrated in Figure 3.

We can already note that $(\vec{L}, \vec{\theta})$ belongs to $\mathbb{R}^r_+ \times \mathbb{R}^\Theta \subset \mathbb{R}^{3r}$, a space of same dimension as the Teichmüller space $\mathcal{T}_{g_S, n_S}^*$. The remainder of this section is devoted to proving that $(\vec{L}, \vec{\theta})$ is a coordinate system on $\mathcal{T}_{g_S, n_S}^*$, finding the expression of the Weil-Petersson measure in these coordinates, and describing its range.

We do not start from scratch, but use the fact that the expression for the Weil-Petersson measure is already known in Fenchel-Nielsen coordinates, thanks to Wolpert’s theorem [13]. In order to apply Wolpert’s result, we use an auxiliary decomposition of
S into pairs of pants, determined by the multi-curve $\beta$ and the bars $B_1, \dots, B_r$. We find relations between the Fenchel-Nielsen coordinates and the parameters $(\vec{L}, \vec{\theta})$, and calculate the determinant.

**Remark 3.4.** This approach seems a bit artificial, and it would be interesting to study directly the Weil-Petersson measure in coordinates $(\vec{L}, \vec{\theta})$, without resorting to an auxiliary system of Fenchel-Nielsen coordinates.

**3.2. Decomposition into pairs of pants.** Using the multi-curve $\beta$ and the bars $B_1, \dots, B_r$, we define a decomposition of $S$ into pairs of pants (3-holed spheres) that will serve as an auxiliary tool in several places.

**3.2.1. Pair of pants determined by two simple curves joined by a segment.** Let $(C_+, C_-)$ be a multi-curve on $S$, and $B$ be a simple segment going from $C_-$ to $C_+$, not intersecting $C_- \cup C_+$ except at its endpoints. The topological pair of pants determined by $C_-$, $C_+$ and $B$ is defined as a one-sided regular neighbourhood of $C_- \cup C_+ \cup B$; that is to say, a regular neighbourhood $\mathcal{N}$ of $C_- \cup C_+ \cup B$, from which we remove the cylinders bordered by one of the curves $C_\pm$ and one component of $\partial\mathcal{N}$.

This also applies when $C = C_+ = C_-$ as soon as $B$ is not homotopically trivial with gliding endpoints along $C$. We have either a pair of pants if $B$ leaves and arrives on the same side of $C$, or a once-holed torus otherwise.

The three cases are shown in Figure 8.

<figure><img src="image_10.png" /><figcaption>FIGURE 8. The three possible pairs of pants (or once-holed tori) determined by two simple curves joined by a segment.</figcaption></figure>

**3.2.2. Construction of the pair of pants decomposition.** The (arbitrary) numbering of $B_1, \dots, B_r$ determines a decomposition $(\mathbf{P}_1, \dots, \mathbf{P}_r)$ of $S$ into connected surfaces of Euler characteristic $-1$, defined inductively below. This decomposition is non-canonical, as it depends on the numbering of the bars $B_1, \dots, B_r$.

The construction provides us with an increasing family of multi-curves $\Sigma_0 \subset \Sigma_1 \subset \dots \subset \Sigma_r$, and an increasing family of surfaces $S_j = S_{j-1} \cup P_j$ for $1 \le j \le r$. It will also produce an auxiliary family of sets $\tilde{S}_j$.

* To initiate, we let $S_0 = \emptyset$ and $\tilde{S}_0 = \Sigma_0$ be the multi-curve $\beta$.

* Let $j \in \{1, \dots, r\}$. The intersection of the $j$-th bar $B_j$ with $\tilde{S}_{j-1}$ is the union of two non-empty closed segments $I_+$, $I_-$ (possibly reduced to single points) containing the endpoints of $B_j$. The complement of $I_+ \cup I_-$ in $B_j$ is a non-empty sub-segment $B'_j$ of $B_j$ which meets one or two components of $\partial\tilde{S}_{j-1}$ at
its endpoints; call $C_+, C_-$ these components. We let $\mathbf{P}_j$ be the pair of pants determined by $B'_j$, $C_+$ and $C_-$.

• We then let $\Sigma_j = \Sigma_{j-1} \cup \partial \mathbf{P}_j$ (it is a multi-curve), $\tilde{\mathbf{S}}_j = \tilde{\mathbf{S}}_{j-1} \cup \mathbf{P}_j$ (the disjoint union of a surface and a multi-curve) and $\mathbf{S}_j = \mathbf{S}_{j-1} \cup \mathbf{P}_j = \mathbf{P}_1 \cup \dots \cup \mathbf{P}_j$ (a bona-fide surface). They are related by $\tilde{\mathbf{S}}_j = \mathbf{S}_j \cup \beta$.

By an argument similar to Remark 2.22, because $\mathbf{c}$ is a generalized eight, the boundary of each pair of pants $\mathbf{P}_j$ is a multi-curve in $\mathbf{S}$. The construction finishes for $j=r$, and we have $\tilde{\mathbf{S}}_r = \mathbf{S}_r = \mathbf{S}$, because the multi-loop $\mathbf{c}$ fills the surface $\mathbf{S}$.

**Remark 3.5.** For an integer $1 \le j \le r$, one can consider the multi-loop $c'_0$ obtained by removing the intersections $j, \dots, r$ from $\mathbf{c}$ according to the procedure described in §2.4 (here we remove each intersection the same way as we did when replacing them by bars). This will be particularly useful for reasoning by induction, in the case $j=r$. Note that $c'_0$ is a disjoint union of a (possibly disconnected) generalized eight $\mathbf{c}'$ and a (possibly empty) multi-curve. The generalized eight $\mathbf{c}'$ fills the surface $\mathbf{S}_j$. The pairs of pants decomposition $\mathbf{P}_1, \dots, \mathbf{P}_j$ associated to the multi-loops $\mathbf{c}$ and $\mathbf{c}'$ are the same.

**3.2.3. Reduction to Case (a) and (b).** We shall assume (without loss of generality) that the numbering of $B_1, \dots, B_r$ is such that the first bars $B_1, \dots, B_{r_0}$ are in the third situation of Figure 8 (that is, depart and arrive on one unique component of $\beta$, on opposite sides) and the remaining $B_{r_0+1}, \dots, B_r$, either join two different components of $\beta$, or depart and arrive on one component of $\beta$ on the same side, or leave from a component of $\beta$ already touched by $B_1, \dots, B_{r_0}$. Then, $\mathbf{P}_1, \dots, \mathbf{P}_{r_0}$ are disjoint once-holed tori, and $\mathbf{P}_{r_0+1}, \dots, \mathbf{P}_r$ are bona fide pairs of pants falling into one of the first two cases of Figure 8.

As a consequence, for $j > r_0$, there are only two possible scenarios that can happen at the $j$-th step in the construction of the pair of pants decomposition, which we shall call Case (a) and Case (b); they are represented in Figure 9.

(a) $C_-$ and $C_+$ are distinct (i.e. the bar $B'_j$ has endpoints on different components of $\partial \tilde{\mathbf{S}}_j$), and hence $\mathbf{S}_j$ has one more genus and one less boundary component than $\mathbf{S}_{j-1}$.

(b) $C_- = C_+ =: C$ (i.e. the bar $B'_j$ has endpoints on one unique component of $\partial \tilde{\mathbf{S}}_j$), and hence $\mathbf{S}_j$ has one more boundary component than $\mathbf{S}_{j-1}$, and the same genus.

We will refer to these two cases in proofs by induction.

### 3.3. Set of boundary components of the pair of pants decomposition.

**3.3.1. Definition.** Let us denote as $\Gamma = (\Gamma_\lambda)_{\lambda \in \Lambda}$ the multi-curve $\Sigma_r$ in the construction above, i.e. the multi-curve obtained by taking the boundary of the pair of pants decomposition $\mathbf{P}_1, \dots, \mathbf{P}_r$ of $\mathbf{S}$. It is indexed by a set $\Lambda$ with $3g_{\mathbf{S}} - 3 + 2n_{\mathbf{S}}$ elements.

**Remark 3.6.** The construction above provides a special representative of the homotopy class of each boundary component $\Gamma_\lambda$, as a concatenation of bars $\bar{B}_q$ and simple portions of $\beta$ (see the forthcoming Figure 19 for an example). More precisely, we will show later that the $\Gamma_\lambda$ are what we call *polygonal curves*, as defined in Definition 8.1. This property will play a key role in §4.4 and §8.
<figure><img src="image_11.png" /><figcaption>FIGURE 9. The two possible scenarios when constructing $\mathbf{P}_j$.</figcaption></figure>

**Remark 3.7.** Extending Remark 2.18, it will also be convenient to keep in mind that each boundary component $\Gamma_\lambda$ is freely homotopic to a piecewise smooth curve $\Gamma_{\lambda, \text{init}}$, included in the original c, made of smooth segments joining some of the intersection points of c.

3.3.2. *Partitions of* $\Lambda$. Some of the elements of $\Lambda$ are the boundary components of $S$ – we denote as $\Lambda_{BC}$ this set, which is in natural bijection with $\partial S = \{1, \dots, n_S\}$. The other elements are inner curves, and we denote this set as $\Lambda_{\text{in}} := \Lambda \setminus \Lambda_{BC}$. We have $\#\Lambda_{\text{in}} = 3g_S - 3 + n_S$.

We notice that the multi-curve $\beta$ is part of this pair of pants decomposition (because $\beta = \Sigma_0 \subset \Sigma_r$). To reflect this, we define $\Lambda^\beta \subseteq \Lambda$ to be the set of components of $\Gamma$ which are also a component of the multi-curve $\beta$. This motivates the writing of the multi-curve $\beta$ as $(\beta_\lambda)_{\lambda \in \Lambda^\beta}$ introduced before, where $\Gamma_\lambda = \beta_\lambda$ for $\lambda \in \Lambda^\beta$.

The components of $\Gamma$ which are *not* part of the multi-loop $\beta$, and have therefore been created in the construction above, will be denoted as $\Lambda^\Gamma := \Lambda \setminus \Lambda^\beta$. The partition $\Lambda = \Lambda_{BC} \sqcup \Lambda_{\text{in}}$ between boundary/inner components naturally induces two partitions of $\Lambda^\beta$ and $\Lambda^\Gamma$, defined by

$$ \Lambda_{BC}^{\beta} := \Lambda_{BC} \cap \Lambda^{\beta} $$

$$ \Lambda_{BC}^{\Gamma} := \Lambda_{BC} \cap \Lambda^{\Gamma} $$

$$ \Lambda_{\text{in}}^{\beta} := \Lambda_{\text{in}} \cap \Lambda^{\beta} $$

$$ \Lambda_{\text{in}}^{\Gamma} := \Lambda_{\text{in}} \cap \Lambda^{\Gamma}. $$

3.3.3. *Order on* $\Lambda$. We do not fix a numbering of the set $\Lambda$. However, it will be convenient to equip $\Lambda$ with an order relation in the following way.

**Definition 3.8.** For $\lambda \in \Lambda$, we denote as step($\lambda$) the step of the induction when the component $\Gamma_\lambda$ is created, i.e.

$$ \text{step}(\lambda) := \min\{j \in \{1, \dots, r\} : \Gamma_\lambda \text{ is a boundary component of } \mathbf{P}_j\}. $$

This induces an order on $\Lambda$, defined by writing, for $\lambda, \lambda' \in \Lambda$,

$$ \lambda < \lambda' \Leftrightarrow \text{step}(\lambda) < \text{step}(\lambda'). $$

The relation $\lambda < \lambda'$ simply means that $\Gamma_\lambda$ appeared earlier than $\Gamma_{\lambda'}$ in our recursive construction of the pants decomposition.
**Remark 3.9.** This order is strongly connected (for any $\lambda, \lambda'$, either $\lambda \le \lambda'$ or $\lambda' \le \lambda$) but a priori not anti-symmetric (there would typically be distinct elements $\lambda \ne \lambda'$ with step($\lambda$) = step($\lambda'$), since we often create several boundary components at a given step of the induction).

**Remark 3.10.** In the proof by induction of §3.4.4, it will be convenient to assume that the arbitrary choice of the numbering $\{1, \dots, n_S\}$ of the boundary of $S$ has been made so that the function $\lambda \mapsto \text{step}(\lambda)$ is non-decreasing as a function of $\lambda \in \Lambda_{BC} = \{1, \dots, n_S\}$, i.e. for every $1 \le i < j \le n_S$, the $j$-th boundary component of $\partial S$ appears after (or at the same time as) the $i$-th boundary component in the construction of the pair of pants decomposition. In particular, the new components appearing in the last step, when we add the final pair of pants $P_r$, are labelled $n_S$ in Case (a), or $n_S$ and $n_S - 1$ in Case (b).

**3.4. Expression of the Weil-Petersson measure.** We are now ready to state and prove our new expression for the Weil-Petersson measure in the coordinates $(\vec{L}, \vec{\theta})$.

**3.4.1. Expression in Fenchel-Nielsen coordinates.** For given boundary lengths $x$, once a decomposition of $S$ into pairs of pants $P_1, \dots, P_r$ is given, it is standard to consider Fenchel-Nielsen coordinates $(y_\lambda, \alpha_\lambda)_{\lambda \in \Lambda_{in}}$ on $\mathcal{T}_{g_S, n_S}(x)$. The coordinates $y_\lambda$ are the lengths $\ell_Y(\Gamma_\lambda)$ of the closed geodesics homotopic to $\Gamma_\lambda$ for the metric $Y$. The coordinates $\alpha_\lambda$ are twist parameters, defined up to translation. We will explain in the proof of Theorem 3.11 how to adequately choose the origin of twist parameters in relation with the multi-loop $c$ that we want to study. By work of Wolpert [13], we know that

$$ (3.1) \qquad \mathrm{dVol}_{g_S, n_S, x}^{\mathrm{WP}}(Y) = \prod_{\lambda \in \Lambda_{\mathrm{in}}} \mathrm{d}y_\lambda \mathrm{d}\alpha_\lambda. $$

To obtain coordinates on $\mathcal{T}_{g_S, n_S}^*$, we simply multiply this by the Lebesgue measure for the length parameters $x = (x_1, \dots, x_{n_S})$, in other words the collection of $y_\lambda$ for $\lambda \in \Lambda_{BC}$. Since $\Lambda = \Lambda_{\mathrm{in}} \cup \Lambda_{BC}$, we obtain the expression

$$ (3.2) \qquad \mathrm{dVol}_{g_S, n_S}^{\mathrm{WP}}(x, Y) = \prod_{\lambda \in \Lambda} \mathrm{d}y_\lambda \prod_{\lambda \in \Lambda_{\mathrm{in}}} \mathrm{d}\alpha_\lambda. $$

**3.4.2. Main statement.** We shall prove the following.

**Theorem 3.11.** Let $c$ be a generalized eight. Then, the map

$$ (3.3) \qquad \begin{cases} \mathcal{T}_{g_S, n_S}^* & \to \mathbb{R}^r_{>0} \times \mathbb{R}^\Theta \\ (x, Y) & \mapsto (\vec{L}, \vec{\theta}) \end{cases} $$

is a $C^1$-diffeomorphism onto its image $\mathcal{D}$, and

$$ (3.4) \qquad 2^{n_S} \prod_{j=1}^{n_S} \sinh\left(\frac{x_j}{2}\right) \mathrm{dVol}_{g_S, n_S}^{\mathrm{WP}}(x, Y) = 2^{2|\Lambda^\beta|} \prod_{\lambda \in \Lambda^\beta} \sinh^2\left(\frac{y_\lambda}{2}\right) \prod_{i=1}^r \sinh(L_i) \mathrm{d}^r \vec{L} \mathrm{d}^{2r} \vec{\theta} $$

where $y_\lambda = \ell_Y(\beta_\lambda)$, $\mathrm{d}^r \vec{L} = \prod_{i=1}^r \mathrm{d}L_i$ and $\mathrm{d}^{2r} \vec{\theta} = \prod_{i=1}^r \mathrm{d}\theta_i^+ \mathrm{d}\theta_i^-$.

We insist on the fact that this formula holds for any coordinate system $(\vec{L}, \vec{\theta})$ obtained by opening the intersections of $c$, meaning that intersection can be opened in the first or second way. This yields different new sets of coordinates. We postpone to §3.5 the description of the image $\mathcal{D}$ of the change of variable.
**Remark 3.12.** Earlier in the paper we assumed that **S** was connected, but since the formula is multiplicative, it will also hold for disconnected **S**.

**3.4.3. Dirac variant.** The integral that we want to evaluate, (1.12), actually relates to the measures

$$ \prod_{j=1}^{k} x_{ij} \delta(x_{ij} - x_{i'j}) \mathrm{dVol}_{g_S, n_S}^{\mathrm{WP}}(\mathbf{x}, Y), $$

where $m = \bigsqcup_{j=1}^k \{i_j, i'_j\}$ is a perfect matching of $\partial S \setminus V$ for a subset $V$ of $\partial S = \{1, \dots, n_S\}$. Note that, by our conventions, $\partial S$ is identified with $\Lambda_{BC}$, so that $V$ can be viewed as a subset of $\Lambda_{BC}$. Theorem 3.11 allows us to write these measures in the new coordinates.

**Corollary 3.13.** For any set $V \subseteq \partial S$ and any perfect matching $m = \bigsqcup_{j=1}^k \{i_j, i'_j\}$ of $\partial S \setminus V$,

$$ (3.5) \qquad \prod_{j=1}^{k} x_{ij} \delta(x_{ij} - x_{i'j}) \mathrm{dVol}_{g_S, n_S}^{\mathrm{WP}}(\mathbf{x}, Y) \\ = 2^{2|\Lambda^{\beta}| - n_S} \frac{\prod_{\lambda \in V^{\beta}} \sinh\left(\frac{y_{\lambda}}{2}\right) \prod_{\lambda \in \Lambda_{\mathrm{in}}^{\beta}} \sinh^2\left(\frac{y_{\lambda}}{2}\right)}{\prod_{\lambda \in V^{\Gamma}} \sinh\left(\frac{y_{\lambda}}{2}\right)} T_{V,m}(\vec{L}, \vec{\theta}) \prod_{i=1}^{r} \sinh(L_i) \mathrm{d}^r \vec{L} \mathrm{d}^{2r} \vec{\theta} $$

where $y_{\lambda} = \ell_{Y}(\Gamma_{\lambda})$ for $\lambda \in \Lambda$, $V^{\beta} = V \cap \Lambda_{BC}^{\beta}$, $V^{\Gamma} = V \cap \Lambda_{BC}^{\Gamma}$, and $T_{V,m}(\vec{L}, \vec{\theta})$ is the expression of the distribution

$$ \frac{\prod_{\lambda \in \Lambda_{BC}^{\beta} \setminus V^{\beta}} \sinh\left(\frac{y_{\lambda}}{2}\right)}{\prod_{\lambda \in \Lambda_{BC}^{\Gamma} \setminus V^{\Gamma}} \sinh\left(\frac{y_{\lambda}}{2}\right)} \prod_{j=1}^{k} x_{ij} \delta(x_{ij} - x_{i'j}) $$

in the new coordinates.

*Proof.* We simply divide the previous expression by $\prod_{j=1}^{n_S} \sinh\left(\frac{x_j}{2}\right) = \prod_{\lambda \in \Lambda_{BC}} \sinh\left(\frac{y_{\lambda}}{2}\right)$, and simplify the factors $\prod_{\lambda \in \Lambda_{BC}^{\beta}} \sinh\left(\frac{y_{\lambda}}{2}\right)$ which appear simultaneously at the numerator and denominator of the resulting fraction. $\square$

We shall never need to develop the full explicit expression of $T_{V,m}(\vec{L}, \vec{\theta})$.

Note that, in the expression above, the lengths $y_{\lambda}$ for $\lambda \in \Lambda_{BC} \cup \Lambda^{\beta}$ should all be expressed in terms of the variables $(\vec{L}, \vec{\theta})$. Obtaining explicit expressions will be the goal of §4. The expression of $y_{\lambda}$ for $\lambda \in \Lambda^{\beta}$ is just a sum of the variables $\theta_q$, but the expressions for $y_{\lambda}$ when $\lambda \in \Lambda^{\Gamma}$ will require more work.

Whilst the lengths $y_{\lambda}$ for $\lambda \in \Lambda_{\mathrm{in}}^{\Gamma}$ do not appear explicitly in the expression (3.5), they are implicitly present in the domain of definition $\mathcal{D}$ of the change of variable, as we will see in Proposition 3.21.

**3.4.4.** *Proof of Theorem 3.11.* The proof goes by induction on the number of intersections $r$.

*The case $r = 1$.* The case $r = 1$ corresponds to three possible situations, represented in Figure 10: **c** is either a figure-eight, opened the first or second way, or **c** = $(c_1, c_2)$ where $c_1$ and $c_2$ are two curves intersecting once, filling a once-holed torus.
<figure><img src="image_12.png" /><figcaption>FIGURE 10. The three cases where $r = 1$.</figcaption></figure>

The first case, of a figure eight opened the first way, is represented in Figure 10a. The set of boundary components of the pair of pants decomposition is $\Lambda = \partial S = \{1, 2, 3\}$, and $\Lambda^\beta = \{1, 2\}$ so that $x_1 = y_1 = \theta_1^+$ and $x_2 = y_2 = \theta_1^-$. Noting that by cutting the pair of pants along its orthogeodesic $B_1$ we obtain a convex right-angled hexagon, the classic trigonometric formula [5, Theorem 2.4.1 (i)] yields

$$ \cosh(L_1) = \frac{\cosh\left(\frac{x_1}{2}\right) \cosh\left(\frac{x_2}{2}\right) + \cosh\left(\frac{x_3}{2}\right)}{\sinh\left(\frac{x_1}{2}\right) \sinh\left(\frac{x_2}{2}\right)}, $$

hence $\sinh\left(\frac{x_3}{2}\right) dx_3 = 2\sinh\left(\frac{x_1}{2}\right) \sinh\left(\frac{x_2}{2}\right) \sinh(L_1) dL_1$, or equivalently

$$ \begin{aligned} & 2^3 \sinh\left(\frac{x_1}{2}\right) \sinh\left(\frac{x_2}{2}\right) \sinh\left(\frac{x_3}{2}\right) dx_1 dx_2 dx_3 \\ & = 2^4 \sinh^2\left(\frac{y_1}{2}\right) \sinh^2\left(\frac{y_2}{2}\right) \sinh(L_1) dL_1 d\theta_1^- d\theta_1^+ \end{aligned} $$

as announced.

The second case, the figure-eight opened the second way, is represented in Figure 10a. In this situation, $\Lambda = \{1, 2, 3\}$ once again, but now $\Lambda^\beta = \{3\}$, and in particular $y_\lambda = x_3 = \theta_1^+ + \theta_1^-$. Cutting the two halves of the pair of pants in right-angled pentagons using orthogeodesics, we obtain

$$ (3.6) \qquad \left\{ \begin{aligned} \cosh\left(\frac{x_1}{2}\right) &= \sinh\left(\frac{\theta_1^+}{2}\right) \sinh\left(\frac{L_1}{2}\right) \\ \cosh\left(\frac{x_2}{2}\right) &= \sinh\left(\frac{x_3 - \theta_1^+}{2}\right) \sinh\left(\frac{L_1}{2}\right). \end{aligned} \right. $$

We then compute the Jacobian of the change of variable $(x_1, x_2) \mapsto (\theta_1^+, L_1)$, where $x_3$ is fixed, by calculating a $2 \times 2$ determinant. We obtain

$$ (3.7) \qquad 2 \sinh\left(\frac{x_1}{2}\right) \sinh\left(\frac{x_2}{2}\right) dx_1 dx_2 = \sinh\left(\frac{x_3}{2}\right) \sinh(L_1) d\theta_1^+ dL_1 $$

which leads to the claim.

In the last case, represented in Figure 10c, the surface $S$ is a once-holed torus, $\beta$ is a curve on $S$, and $\Lambda$ has two components, $\beta$ and $\partial S$. The length of $\beta$ is $y_1 = \theta_1^+ + \theta_1^-$, and we notice that the parameter $\theta_1^+$ can be chosen to be a twist parameter. The Weil-Petersson measure for a once-holed torus of boundary length $x_1$ can then be written

$$ d\mathrm{Vol}_{1,1,x_1}^{\mathrm{WP}}(Y) = dy_1 d\theta_1^+ = d\theta_1^- d\theta_1^+. $$
We can conclude by observing that $\cosh(\frac{x_1}{2}) = \cosh^2(\frac{y_1}{2}) - \sinh^2(\frac{y_1}{2}) \cosh(L_1)$, as in the pair of pants case above, and hence

$$ \sinh\left(\frac{x_1}{2}\right) dx_1 = 2 \sinh^2\left(\frac{y_1}{2}\right) \sinh(L_1) dL_1. $$

□

Let us now perform the induction.

*Proof of the induction of Theorem 3.11.* Let $r \ge 2$, and $c$ be a generalized eight with $r$ self-intersections filling a surface $S$, with a choice of opening for each intersection point. Let $c'_0$ denote the multi-loop obtained by removing the last intersection point from $c$ (as described in Remark 3.5). Then, the intersection $c'$ of $c'_0$ with $S_{r-1}$ is a generalized eight with $r-1$ intersections, filling the surface $S_{r-1}$. We assume that the proposition is true for the surface $S_{r-1}$ and the multi-loop $c'$ (with intersections opened as they are for $c$), and want to prove it for $S = S_r$ by attaching the pair of pants $P_r$.

If $S_r$ is a disjoint union of once-holed tori, then the property is a direct consequence of the result for $r=1$, thanks to its multiplicativity. We hence assume that it is not the case. We therefore have two cases to consider, Case (a) and Case (b).

Let $\vec{L}'$ and $\vec{\theta}' = (\theta'_1^{\pm}, \dots, \theta'_{r-1}^{\pm})$ be the parameters associated to the surface $S_{r-1}$ and to $c'$, as constructed in §3.1. We observe that $\vec{L}' = (L_1, \dots, L_{r-1})$ are the lengths of $\overline{B}_1, \dots, \overline{B}_{r-1}$, and in particular $\vec{L}'$ is the same as $\vec{L}$ with the last component removed.

The parameters $(\theta'_q)_{q \in \{1, \dots, r-1\} \times \{\pm\}}$, are defined as the lengths of simple portions $\overline{\vec{I}}'_q$ contained in $\beta$ and delimited by the endpoints of the bars $\overline{B}_1, \dots, \overline{B}_{r-1}$. We can therefore easily relate $\vec{\theta}'$ and $\vec{\theta}$: among the $2(r-1)$ segments $\overline{\vec{I}}'_q$, two of them contain the endpoints of $\overline{B}_r$. All the other $\overline{\vec{I}}'_q$ are also intervals $\overline{\vec{I}}_q$. As a consequence, for $k \in \{1, \dots, r-1\}$, $\epsilon = \pm$, we have $\theta_k^\epsilon = \theta_k^{\epsilon'}$ for all except two values of $(k, \epsilon)$, namely $(n_+, \epsilon_+)$, $(n_-, \epsilon_-)$, for which we have the simple relations

$$ (3.8) \qquad \theta_{n_+}^{\epsilon_+} = \theta_{n_+}^{\epsilon_+} + \theta_r^+ \quad \text{and} \quad \theta_{n_-}^{\epsilon_-} = \theta_{n_-}^{\epsilon_-} + \theta_r^- $$

We now proceed differently depending on the case to consider, (a) or (b).

*Case (a).* In this case, $S_r$ is obtained from $S_{r-1}$ by gluing $P_r$ to two of its boundary components. Let $\lambda_\pm \in \Lambda$ denote the indices of the two boundary components of $S_{r-1}$ on which we glue the pair of pants $P_r$, which satisfy step($\lambda_\pm$) < $r$. By Remark 3.10, these two boundary components of $S_{r-1}$ are replaced by the $n_S$-th boundary component of $S = S_r$ when $P_r$ is attached to $S_{r-1}$.

Because the endpoints of the orthogeodesic $\overline{B}_r^+$ are on the multi-loop $\beta$, there exist $i_\pm$ such that the bar $\overline{B}_r^+$ goes from the component $\beta_{i_-}$ of $\beta$ to the component $\beta_{i_+}$ of $\beta$. It is homotopic (with endpoints gliding along $\beta$) to a succession of 5 orthogonal segments of respective lengths $t_r^-, \alpha_r^-, T, \alpha_r^+, t_r^+$, as shown on Figure 11. More precisely, we can lift the four geodesics $\beta_{i_-}, \Gamma_{\lambda_-}, \Gamma_{\lambda_+}$ and $\beta_{i_+}$ to the hyperbolic plane, and orient them so that they are aligned and the bar $\overline{B}_r^+$ arrives on the right of $\beta_{i_+}$. Then, we define:

* $H^\pm$ to be the orthogeodesic between $\beta_{i_\pm}$ and $\Gamma_{\lambda_\pm}$, and $H$ between $\Gamma_{\lambda_-}$ and $\Gamma_{\lambda_+}$;

* $t_r^\pm$ and $T$ to be the respective (positive) lengths of $H^\pm$ and $H$;
• the quantities $\alpha_r^+$ and $\alpha_r^-$ to denote the algebraic distance between the endpoints of $H^+$ and $H$ along $\Gamma_{\lambda_+}$ and between the endpoints of $H$ and $H^-$ along $\Gamma_{\lambda_-}$ respectively.

It is important in our discussion to remark that $H^\pm$ are contained in $S_{r-1}$, so that their lengths depend only on $Y_{S_{r-1}}$, the restriction of the metric $Y$ to $S_{r-1}$.

<figure><img src="image_13.png" /><figcaption>FIGURE 11. The geometric construction in Case (a). The black dots symbolize the right-angles between the different segments.</figcaption></figure>

In order to express the Weil-Petersson measure for $S_r$, and relate it to the Weil-Petersson measure for $S_{r-1}$, we need to define twist parameters along the glued components $\Gamma_{\lambda_{\pm}}$. We choose them to be the algebraic lengths $\alpha_r^{\pm}$. As a consequence, the expression of the Weil-Petersson measure in Fenchel-Nielsen coordinates recalled in §3.4 implies the factorization

$$ (3.9) \qquad \mathrm{dVol}_{g_S, n_S, \mathbf{x}}^{\mathrm{WP}}(Y) = d\alpha_r^+ d\alpha_r^- dy_{\lambda_+} dy_{\lambda_-} \mathrm{dVol}_{g_S-1, n_S+1, \mathbf{x}'}^{\mathrm{WP}}(Y_{S_{r-1}}), $$

where $\mathbf{x} = (x_1, \dots, x_{n_S})$ and $\mathbf{x}' = (x_1, \dots, x_{n_S-1}, y_{\lambda_+}, y_{\lambda_-})$ correspond to the respective length-vectors of $S_r$ and $S_{r-1}$.

We can now, as in the case $r=1$, use the classic trigonometric formula for convex right-angled hexagons [5, Theorem 2.4.1 (i)], which yields

$$ (3.10) \qquad \cosh(T) = \frac{\cosh\left(\frac{y_{\lambda_+}}{2}\right) \cosh\left(\frac{y_{\lambda_-}}{2}\right) + \cosh\left(\frac{x_{n_S}}{2}\right)}{\sinh\left(\frac{y_{\lambda_+}}{2}\right) \sinh\left(\frac{y_{\lambda_-}}{2}\right)}, $$

which implies that the map $x_{n_S} \mapsto T$ is a $C^1$-diffeomorphism onto its image, and

$$ (3.11) \qquad \sinh\left(\frac{x_{n_S}}{2}\right) dx_{n_S} = 2 \sinh\left(\frac{y_{\lambda_+}}{2}\right) \sinh\left(\frac{y_{\lambda_-}}{2}\right) \sinh(T) dT. $$

It follows that $2^{n_S} \prod_{j=1}^{n_S} \sinh\left(\frac{x_j}{2}\right) \mathrm{dVol}_{g_S, n_S}(\mathbf{x}, Y)$ is equal to

$$ \sinh(T) \mathrm{d}T \, d\alpha_r^+ \, d\alpha_r^- \, 2^{n_S+1} \prod_{j=1}^{n_S-1} \sinh\left(\frac{x_j}{2}\right) \sinh\left(\frac{y_{\lambda_+}}{2}\right) \sinh\left(\frac{y_{\lambda_-}}{2}\right) \mathrm{dVol}_{g_S-1, n_S+1}(\mathbf{x}', Y_{S_{r-1}}) $$
which, using the induction hypothesis, implies that

$$ (3.12) \qquad 2^{ns} \prod_{j=1}^{n_S} \sinh\left(\frac{x_j}{2}\right) \mathrm{dVol}_{g_S, n_S}(\mathbf{x}, Y) \\ = 2^{2|\Lambda^\beta|} \sinh(T) \, \mathrm{d}T \, \mathrm{d}\alpha_r^+ \mathrm{d}\alpha_r^- \prod_{\lambda \in \Lambda^\beta} \sinh^2\left(\frac{y_\lambda}{2}\right) \prod_{i=1}^{r-1} \sinh(L_i) \, \mathrm{d}L_i \prod_{k=1}^{r-1} \mathrm{d}\theta_k^{\prime+} \mathrm{d}\theta_k^{\prime-} $$

where we use the observation that the length-vector $\vec{L}'$ is equal to $(L_1, \dots, L_{r-1})$.

Let us now denote by $\epsilon_r^+$ and $\epsilon_r^-$ the algebraic distance from the endpoint of $H^+$ to the endpoint of $\bar{B}_r$ along $\beta_{i_+}$, and from the endpoint of $\bar{B}_r$ to the endpoint of $H^-$ along $\beta_{i_-}$ respectively, as represented in Figure 11. Let us temporarily admit the following formula (which will be checked in §3.4.5):

**Lemma 3.14.** Given a fixed metric $Y_{S_{r-1}}$, the map $(\alpha_r^-, \alpha_r^+, T) \mapsto (\epsilon_r^-, \epsilon_r^+, L_r)$ is a $C^1$-diffeomorphism onto its image, and

$$ (3.13) \qquad \sinh(T) \, \mathrm{d}T \, \mathrm{d}\alpha_r^- \, \mathrm{d}\alpha_r^+ = \sinh(L_r) \, \mathrm{d}L_r \, \mathrm{d}\epsilon_r^- \, \mathrm{d}\epsilon_r^+. $$

We can now remark that

$$ (3.14) \qquad \theta_r^+ + \epsilon_r^+ = \mathrm{fn}(Y_{S_{r-1}}) \quad \text{and} \quad \theta_r^- - \epsilon_r^- = \mathrm{fn}(Y_{S_{r-1}}). $$

Indeed, for instance, the first quantity is the signed distance (measured along $\beta_{i_+}$) between the extremity of $H^+$ lying in $\beta_{i_+}$, and one of the bars $\bar{B}_{m_+}$ with $m_+ < r$. See Figure 12.

<figure><img src="image_14.png" /><figcaption>FIGURE 12. The relations between the algebraic lengths along $\beta_{i_+}$.</figcaption></figure>

The relation (3.14) implies that in (3.13), we can change variables $(\epsilon_r^-, \epsilon_r^+) \mapsto (\theta_r^-, \theta_r^+)$ and replace $d\epsilon_r^- d\epsilon_r^+$ by $d\theta_r^- d\theta_r^+$. Our expression (3.12) then becomes

$$ 2^r \sinh(L_r) \, \mathrm{d}L_r \, \mathrm{d}\theta_r^- \, \mathrm{d}\theta_r^+ \prod_{\lambda \in \Lambda^\beta} \sinh^2\left(\frac{y_\lambda}{2}\right) \prod_{i=1}^{r-1} \sinh(L_i) \, \mathrm{d}L_i \prod_{k=1}^{r-1} \mathrm{d}\theta_k^{\prime+} \mathrm{d}\theta_k^{\prime-}. $$

Now recall the relations between $\theta_k^\epsilon$ and $\theta_k^{\prime\epsilon}$. For $k \le r-1$ and $\epsilon = \pm$, those parameters are equal, except from two sets of indices, for which these values differ by a translation by (3.8). Hence, our final expression for the measure is the announced one. $\square$
**Remark 3.15.** Part of this discussion is irrelevant when $\beta_{i_-} = \Gamma_{\lambda_-}$, because in this case, $H^-$ and $\alpha_r^-$ are ill-defined. The previous discussion has to be modified as follows.

* If $\beta_{i_-}$ is a boundary curve, then there is no twist parameter attached to it and there is no bar $\bar{B}_k$ arriving at $\beta_{i_-}$ on the other side of $\bar{B}_r$: we have $\theta_r^- = x_{i_-} = y_{i_-}$.

* If $\beta_{i_-}$ is not a boundary curve, then we can take directly $\theta_r^-$ as twist parameter, instead of the ill-defined $\alpha_r^-$.

In all cases, the final outcome of the calculation is the same, with less intermediate steps. Similar remarks can be made when $\beta_{i_+} = \Gamma_{\lambda_+}$. We skip details specific to these cases, similar and simpler than the one we treated.

*Case (b).* In this case, we attach $P_r$ to one boundary component $\Gamma_{\lambda}$ of $S_{r-1}$ (with step($\lambda$) < $r$), and get the two new boundary components of $S_r$ labelled $n_S$ and $n_S - 1$.

<figure><img src="image_15.png" /><figcaption>FIGURE 13. The geometric construction in Case (b).</figcaption></figure>

The situation is quite similar to Case (a) with a small difference, namely the fact that $\lambda_+ = \lambda_-$; we therefore denote them $\lambda$. There exist $i_{\pm}$ such that the orthogeodesic $\bar{B}_r^+$ goes from $\beta_{i_-}$ to $\beta_{i_+}$. It is homotopic to a succession of 5 orthogonal segments of respective lengths $t_r^-$, $\alpha_r^-$, $T$, $\alpha_r^+$, $t_r^+$, as shown on Figure 13. Note that, when lifting Figure 11 and 13 in the universal cover, we obtain the same picture, the only difference being that the lifts of $\Gamma_{\lambda_{\pm}}$ now are two different lifts of the same periodic geodesic $\Gamma_{\lambda}$. We therefore define the orthogeodesics $H^{\pm}$ and $H$, as well as the lengths $t_r^{\pm}$, $\alpha_r^{\pm}$ and $T$ as in Case (a).

We choose the twist parameter to be $\alpha_r^+$. As a consequence, the expression of the Weil-Petersson measure in Fenchel-Nielsen coordinates implies the factorization

$$ (3.15) \qquad d\mathrm{Vol}_{g_S, n_S, x}^{\mathrm{WP}}(Y) = d\alpha_r^+ dy_\lambda d\mathrm{Vol}_{g_S, n_S-1, x'}^{\mathrm{WP}}(Y_{S_{r-1}}) $$

where $x' = (x_1, \dots, x_{n_S-2}, y_\lambda)$. The rest of the proof goes along the same lines as in Case (a), now using the formulae (3.6) and the determinant calculation (3.7).
☐

Finally, to conclude, in all cases, the fact that the map $(\mathbf{x}, Y) \in \mathcal{T}_{g_S, n_S}^* \mapsto (\vec{L}, \vec{\theta})$ is a $C^1$-diffeomorphism onto its image can also obtained by induction, since at each step we checked that the maps were $C^1$-diffeomorphisms onto their images. ☐

3.4.5. *Proof of Lemma 3.14.* To finish this section, there remains to prove Lemma 3.14. We also start investigating the range of definition of $\vec{L}$ and $\vec{\theta}$.

Let us consider a family of $M$ aligned geodesics $\gamma_1, \dots, \gamma_M$ on the hyperbolic plane, such that, for all $i$, $\gamma_{i+1}$ is on the left of $\gamma_i$ (for now, we will only take $M=3$ and $4$, but more general situations will arise in §7). For indices $i \neq j$, let $H_{ij}$ denote the orthogeodesic between $\gamma_i$ and $\gamma_j$ and $z_{ij}$ the endpoint of $H_{ij}$ lying on $\gamma_j$. Let us first prove one intermediate lemma in a three-geodesic configuration, illustrated in Figure 14.

**Lemma 3.16.** Let $M=3$. We denote as $L, t, T$ the following orthogeodesic lengths

$$L = \ell(H_{13}) \quad t = \ell(H_{23}) \quad T = \ell(H_{12})$$

and $\epsilon, \alpha$ the algebraic distances

$$\epsilon = \vec{d}(z_{23}, z_{13}) \qquad \alpha = \vec{d}(z_{32}, z_{12}).$$

Then, for fixed $t > 0$, the map

$$\psi_t: \begin{cases} \mathbb{R} \times \mathbb{R}_{>0} & \to \mathbb{R} \times \mathbb{R}_{>0} \\ (\alpha, T) & \mapsto (\epsilon, L) \end{cases}$$

is a $C^1$-diffeomorphism onto the set of real numbers $(\epsilon, L)$ such that

$$(3.16) \qquad \begin{cases} \cosh(t) \cosh(L) - \cosh(\epsilon) \sinh(t) \sinh(L) > 1 \\ L > t, \end{cases}$$

and its Jacobian can be expressed as

$$(3.17) \qquad \sinh(T) \, dT \, d\alpha = \sinh(L) \, dL \, d\epsilon.$$

<figure><img src="image_16.png" /><figcaption>FIGURE 14. The three geodesics and the lengths we will study. The dots mark the points $z_{ij}$, where the geodesics meet perpendicularly.</figcaption></figure>
*Proof.* The fact that the image of {$α ∈ ℝ, T > 0$} is included in the set of $(ε, L)$ satisfying the conditions (3.16) can be read from standard formulas for right-angled hexagons in hyperbolic geometry, see e.g. [5, Ch. 2 §4]:

$$ (3.18) \qquad \cosh(T) = \cosh(t) \cosh(L) - \cosh(\epsilon) \sinh(t) \sinh(L) $$

$$ (3.19) \qquad \cosh(L) = \cosh(t) \cosh(T) + \cosh(\alpha) \sinh(t) \sinh(T). $$

Conversely, if (3.16) holds for parameters $(L, ε)$, then there exists $T ≠ 0$ such that (3.18) holds. This defines $T$ up to the sign, and we choose the positive solution. Another classical hyperbolic trigonometry formula states that, for this geometric situation to hold, we must have

$$ (3.20) \qquad \frac{\sinh(\epsilon)}{\sinh(T)} = \frac{\sinh(\alpha)}{\sinh(L)}. $$

One can then define $α$ so that the relation (3.20) holds. Note that $α$ has the same sign as $ε$. By using $\cosh^2(L) = 1 + \sinh^2(L)$ and the two relations (3.18) and (3.20), one then deduces that

$$ (\cosh(L) - \cosh(t) \cosh(T))^2 = (\cosh(\alpha) \sinh(t) \sinh(T))^2 $$

or, in other words,

$$ \cosh(L) = \cosh(t) \cosh(T) \pm \cosh(\alpha) \sinh(t) \sinh(T). $$

Because we assumed that $L > t$, we actually obtain that (3.19) holds, which means that $L$ is indeed the length of the orthogeodesic $H_{13}$ on Figure 14.

The determinant calculation (3.17) follows from (3.19) and (3.20), but it is worth writing some details. Start writing the jacobian of the change of variable as the determinant of the matrix of partial derivatives:

$$ \cosh(\epsilon) \frac{dL}{dT} \frac{d\epsilon}{d\alpha} = \frac{\partial L}{\partial T} \frac{\partial(\sinh(\epsilon))}{\partial \alpha} - \frac{\partial L}{\partial \alpha} \frac{\partial(\sinh(\epsilon))}{\partial T}. $$

Because of the relation (3.20), we can rewrite this as

$$ \cosh(\epsilon) \frac{dL}{dT} \frac{d\epsilon}{d\alpha} = \frac{\partial L}{\partial T} \frac{\cosh(\alpha) \sinh(T)}{\sinh(L)} - \frac{\partial L}{\partial \alpha} \frac{\sinh(\alpha) \cosh(T)}{\sinh(L)}. $$

An elementary calculation of $\partial L/\partial T$ and $\partial L/\partial \alpha$ from (3.19) yields that

$$ \cosh(\epsilon) \frac{dL}{dT} \frac{d\epsilon}{d\alpha} = \frac{\sinh(T)}{\sinh^2(L)} (\sinh(t) \cosh(T) + \cosh(t) \sinh(T) \cosh(\alpha)). $$

Using another classical trigonometric formula, we recognize the quantity in the bracket to be $\cosh(\epsilon) \sinh(L)$, and hence this simplifies to

$$ \frac{dL}{dT} \frac{d\epsilon}{d\alpha} = \frac{\sinh(T)}{\sinh(L)} $$

which is the desired relation.

□

**Remark 3.17.** We actually have $L ≥ t+T$, so for any given $a > 0$, the set {$α ∈ ℝ, T > a$} is sent to a subset of the set of the set of values $(ε, L)$ such that

$$ \begin{cases} \cosh(t) \cosh(L) - \cosh(\epsilon) \sinh(t) \sinh(L) \ge \cosh(a) \\ L \ge t + a. \end{cases} $$
**Notation 3.18.** This result provides us with a map $\phi_t : (\epsilon, L) \mapsto \vec{d}(z_{31}, z_{21})$, which corresponds to the algebraic distance along the bottom geodesic.

In order to prove Lemma 3.14, we simply apply Lemma 3.16 twice in a four-geodesic configuration, as done below. See also Figure 15.

<figure><img src="image_17.png" /><figcaption>FIGURE 15. The four-geodesic configuration we consider. We have represented the quantities $\tilde{T}^+$ and $\theta^+$ appearing in the statement.</figcaption></figure>

**Lemma 3.19.** Let $M = 4$. We denote as $L$, $t^+$ and $T$ the orthogeodesic lengths

$$L = \ell(H_{14}) \qquad t^+ = \ell(H_{34}) \qquad t^- = \ell(H_{12}) \qquad T = \ell(H_{23})$$

and $\epsilon^\pm$, $\alpha^\pm$ the algebraic lengths

$$\epsilon^+ = \vec{d}(z_{34}, z_{14}) \qquad \alpha^+ = \vec{d}(z_{43}, z_{23}) \qquad \alpha^- = \vec{d}(z_{32}, z_{12}) \qquad \epsilon^- = \vec{d}(z_{41}, z_{21}).$$

For fixed $t^-, t^+ > 0$, the map

$$\begin{cases} \mathbb{R}^2 \times \mathbb{R}_{>0} & \to \mathbb{R}^2 \times \mathbb{R}_{>0} \\ (\alpha^-, \alpha^+, T) & \mapsto (\epsilon^-, \epsilon^+, L) \end{cases}$$

is a $\mathbb{C}^1$-diffeomorphism onto the set of real numbers $(\epsilon^-, \epsilon^+, L)$ such that

$$(3.21) \qquad \begin{cases} \cosh(t^+) \cosh(L) - \cosh(\epsilon^+) \sinh(t^+) \sinh(L) > 1 \\ \cosh(t^-) \cosh(\tilde{T}^+) - \cosh(\epsilon^- - \theta^+) \sinh(t^-) \sinh(\tilde{T}^+) > 1 \\ L > t^+ \quad \text{and} \quad \tilde{T}^+ > t^-, \end{cases}$$

where $\tilde{T}^+$ is the second component of $\psi_{t^+}^{-1}(\epsilon^+, L)$ and $\theta^+ := \phi_{t^+}(\epsilon^+, L)$. Its Jacobian can be expressed as

$$(3.22) \qquad \sinh(T) \, dT \, d\alpha^- \, d\alpha^+ = \sinh(L) \, dL \, d\epsilon^- \, d\epsilon^+.$$

*Proof.* We simply apply the previous result twice, once to the three geodesics $\gamma_1, \gamma_3, \gamma_4$, and another time to $\gamma_3, \gamma_2, \gamma_1$ with reverse orientation. $\square$
**Remark 3.20.** We always have $L \ge t^+ + T + t^-$ and $\tilde{T}^+ \ge t^- + T$. Hence, for any $a > 0$, the set $\{\alpha^-, \alpha^+ \in \mathbb{R}, T > a\}$ corresponds to a subset of the set of real numbers $(\epsilon^-, \epsilon^+, L)$ such that

$$ \begin{cases} \cosh(t^+) \cosh(L) - \cosh(\epsilon^+) \sinh(t^+) \sinh(L) \ge \cosh(a) \\ \cosh(t^-) \cosh(\tilde{T}^+) - \cosh(\epsilon^- - \theta^+) \sinh(t^-) \sinh(\tilde{T}^+) \ge \cosh(a) \\ L > t^+ + a \quad \text{and} \quad \tilde{T}^+ > t^- + a. \end{cases} $$

**3.5. Domain of definition of the new coordinates.** In this section, we discuss the range of definition $\mathcal{D}$ of the coordinates $(\vec{L}, \vec{\theta})$, which will be the domain of the integral (1.12) in the new coordinates. We prove the following statement, which describes the boundary of $\mathcal{D}$.

**Proposition 3.21.** The boundary of $\mathcal{D}$ is contained in $\bigcup_{\lambda \in \Lambda} \{y_\lambda = 0\}$.

*Proof.* We once again proceed by induction, closely following the proof of Theorem 3.11 but making the domain of definition of the variables more explicit.

The result is trivial for $r = 1$. Let us fix a metric $Y_{S_{r-1}} \in \mathcal{T}^*_{g_{S_{r-1}}, n_{S_{r-1}}}$ and describe the domain of definition of the three new coordinates $(\theta_r^-, \theta_r^+, L_r)$ defined when attaching the pair of pants $P_r$. We note that there exists a unique hyperbolic metric associated to any set of Fenchel-Nielsen coordinates. Hence, the constraints on the new variables are:

* $(\epsilon_r^+, \epsilon_r^-, L_r)$ need to belong to the image of the diffeomorphism in Lemma 3.19 – we call this the *local constraint*;

* the new boundary components of the pair of pants need to be of positive length, i.e. $x_{n_S} > 0$ in Case (a) and $x_{n_S}, x_{n_S-1} > 0$ in Case (b) – we call this the *global constraint*.

We prove Proposition 3.21 by proving that the global constraints are stronger than the local constraints. Indeed, in Case (a), because of (3.10), the global constraint is equivalent to

$$ \cosh(T) > \frac{\cosh\left(\frac{y_{\lambda_+}}{2}\right) \cosh\left(\frac{y_{\lambda_-}}{2}\right) + 1}{\sinh\left(\frac{y_{\lambda_+}}{2}\right) \sinh\left(\frac{y_{\lambda_-}}{2}\right)}. $$

Note that the quantity

$$ a := \operatorname{argcosh} \left( \frac{\cosh\left(\frac{y_{\lambda_+}}{2}\right) \cosh\left(\frac{y_{\lambda_-}}{2}\right) + 1}{\sinh\left(\frac{y_{\lambda_+}}{2}\right) \sinh\left(\frac{y_{\lambda_-}}{2}\right)} \right) > 0, $$

only depends $y_{\lambda_+}$ and $y_{\lambda_-}$, i.e. on the metric $Y_{S_{r-1}}$ in $\mathcal{T}^*_{g_{S_{r-1}}, n_{S_{r-1}}}$ that we are considering to be fixed. Hence, the global constraint implies the $T > a$, where $a$ is fixed. By Remark 3.20, this implies strictly stronger conditions than the original local constraints, which is what we wanted to check.

In Case (b), we have

$$ \cosh(T) = \frac{\cosh^2\left(\frac{\alpha}{2}\right) + \cosh(x_{n_S})}{\sinh^2\left(\frac{\alpha}{2}\right)} \quad \text{and} \quad \cosh(T) = \frac{\cosh^2\left(\frac{y_{\lambda-\alpha}}{2}\right) + \cosh(x_{n_S-1})}{\sinh^2\left(\frac{y_{\lambda-\alpha}}{2}\right)} $$
where $\alpha$ is the length of the portion of $\beta_\lambda$ delimited by the orthogeodesic $H$, on the side of $x_{n_S}$. Thus the new global constraints are equivalent to

$$ \cosh(T) > \frac{\cosh^2(\frac{\alpha}{2}) + 1}{\sinh^2(\frac{\alpha}{2})} = 1 + \frac{2}{\sinh^2(\frac{\alpha}{2})} \quad \text{and} \quad \cosh(T) > 1 + \frac{2}{\sinh^2(\frac{y_\lambda - \alpha}{2})}. $$

Define $a := \arg\cosh(1 + 2/\sinh^2(\frac{y_\lambda}{2}))$. Since $0 < \alpha < y_\lambda$, the two global constraints imply that $T > a > 0$, which allows to conclude as in Case (a). □

**3.6. A priori bounds on Fenchel-Nielsen parameters.** We shall need some rough a priori bounds on the parameters $y_\lambda = \ell_Y(\Gamma_\lambda)$ for $\lambda \in \Lambda$ and $\theta_q$ for $q \in \Theta$, that grow at most linearly with respect to $\ell_Y(\mathbf{c})$.

**Proposition 3.22.** On the level set $\{Y \in \mathcal{T}_{g_S,n_S}^* : \ell_Y(\mathbf{c}) = \ell\}$, we have

$$ (3.23) \qquad \forall \lambda \in \Lambda, \quad 0 \le y_\lambda \le \ell $$

$$ (3.24) \qquad \forall q \in \Theta, \quad |\theta_q| \le 2\ell $$

and the twist parameters $(\alpha_\lambda)_{\lambda \in \Lambda_{in}}$ constructed in the proof of Theorem 3.11 satisfy

$$ (3.25) \qquad \forall \lambda \in \Lambda_{in}, \quad |\alpha_\lambda| \le 3\ell. $$

The proof relies on elementary hyperbolic trigonometry, and in particular the following lemma.

**Lemma 3.23.** Let $Y$ be a compact hyperbolic surface with two simple closed geodesics $\gamma_1, \gamma_2$. We assume that $\gamma_1$ and $\gamma_2$ are either disjoint or equal. Let $H$ be a simple orthogeodesic segment going from $\gamma_1$ to $\gamma_2$, not intersecting $\gamma_1$ and $\gamma_2$ in its interior. Then, $\sinh(\ell(\gamma_1)/2) \sinh(\ell(H)) \ge 1$.

*Proof.* If $\gamma_1 = \gamma_2$, this is a direct consequence of the collar lemma [5, Theorem 4.1.1]. If the two geodesics are disjoint, then the surface filled by $\gamma_1, \gamma_2$ and $H$ is a pair of pants. Cutting this pair of pants along its orthogeodesics, we obtain a convex right-angled hexagon with consecutive sides of length $\ell(\gamma_1)/2$, $\ell(H)$ and $\ell(\gamma_2)/2$. We can further cut this hexagon and obtain a right-angled pentagon with consecutive side-lengths $\ell(\gamma_1)/2$ and $\ell(H)$, which implies the claim by [5, Lemma 2.3.5]. □

We use this lemma to prove the following.

**Lemma 3.24.** Let $Y$ be a compact hyperbolic surface with:

* three simple oriented closed geodesics $\gamma_1, \gamma_2, \gamma_3$;

* an orthogeodesic $H_{12}$ leaving from the right of $\gamma_1$ and arriving at the left of $\gamma_2$;

* an orthogeodesic $H_{23}$ leaving from the right of $\gamma_2$ and arriving at the left of $\gamma_3$.

Let $J \subset \gamma_2$ be the segment joining $t(H_{12})$ to $o(H_{23})$ and $H$ be the orthogeodesic from $\gamma_1$ to $\gamma_3$, homotopic with gliding endpoints to $H_{12} \cdot J \cdot H_{23}$. Then, if $\gamma$ denotes the geodesic representative of the homotopy class $\gamma_1 \cdot H \cdot \gamma_3 \cdot H^{-1}$, we have

$$ (3.26) \qquad \cosh\left(\frac{\ell(\gamma)}{2}\right) \ge \cosh(\ell(J)). $$

Of course, the orientations of $\gamma_1, \gamma_2$ and $\gamma_3$ are irrelevant in the length estimate, but serve to describe the isotopy class of the picture $\gamma_1 \cup \gamma_2 \cup \gamma_2 \cup H_{12} \cup H_{23}$.
Proof. The geodesic $\gamma$ is a figure eight, going around the two closed geodesics $\gamma_1$ and $\gamma_3$, connected by the orthogeodesic $H$. It then follows that

$$ \cosh\left(\frac{\ell(\gamma)}{2}\right) = \cosh\left(\frac{\ell(\gamma_1)}{2}\right) \cosh\left(\frac{\ell(\gamma_3)}{2}\right) + \cosh(H) \sinh\left(\frac{\ell(\gamma_1)}{2}\right) \sinh\left(\frac{\ell(\gamma_3)}{2}\right) \\ \geq \cosh(H) \sinh\left(\frac{\ell(\gamma_1)}{2}\right) \sinh\left(\frac{\ell(\gamma_3)}{2}\right). $$

Expressing the length of $H$ in terms of the lengths of $J$ and $H_{12}$, $H_{23}$, we obtain

$$ \cosh(H) = \cosh(\ell(H_{12})) \cosh(\ell(H_{23})) + \cosh(\ell(J)) \sinh(\ell(H_{12})) \sinh(\ell(H_{23})) \\ \geq \cosh(\ell(J)) \sinh(\ell(H_{12})) \sinh(\ell(H_{23})). $$

By Lemma 3.23, $\sinh(\ell(\gamma_1)/2) \sinh(\ell(H_{12})) \geq 1$ and $\sinh(\ell(\gamma_3)/2) \sinh(\ell(H_{23})) \geq 1$, which leads to our claim. $\square$

We are now ready to proceed to the proofs of the bounds.

*Proof of Proposition 3.22.* The upper bound on $y_\lambda$ is a direct consequence of Remark 3.7, which states that $\Gamma_\lambda$ is freely homotopic to a path $\Gamma_{\lambda, \text{init}}$ on $c$, and hence

$$ y_{\lambda} = \ell_{Y}(\Gamma_{\lambda}) \leq \ell(\Gamma_{\lambda, \text{init}}) \leq \ell(c) = \ell. $$

Let us now prove the bound on $\theta_q$ for an index $q \in \Theta$.

* If $q$ is a U-turn parameter, then we simply observe that $0 \leq \theta_q \leq \ell(\beta_{\lambda(q)}) \leq \ell$.

* If $q$ is a crossing parameter, we apply Lemma 3.24 to the simple closed geodesics $\gamma_1 = \beta_{\lambda(\sigma^{-1}q)}$, $\gamma_2 = \beta_{\lambda(q)}$ and $\gamma_3 = \beta_{\lambda(\sigma q)}$, with the orthogeodesics $H_{12} = \bar{B}_q$ and $H_{23} = \bar{B}_{\sigma q}$, so that $J = \bar{I}_q$. We check that the figure-eight $\gamma$ produced by the lemma is freely homotopic to a loop $\gamma_{\text{init}}$ drawn on $c$, going along $\beta_{\lambda(\sigma^{-1}q), \text{init}}$, $\beta_{\lambda(\sigma q), \text{init}}$ and twice $\mathcal{I}_{q, \text{init}}$. We conclude that

$$ \cosh(\theta_q) \leq \cosh(\ell(\gamma)) \leq \cosh(2\ell). $$

The bounds on the twist parameters are obtained the same way, constructing for each case (Case (a) and (b)) a figure-eight $\gamma$ such that $|\alpha_\lambda| \leq \ell(\gamma) \leq 3\ell$ (the factor of 3 being there for Case (b), due to repeated portions of $c$ used to describe the homotopy class of $\gamma$). $\square$

3.7. **Polynomial bounds on volumes and the case of double-filling loops.** One can straightforwardly deduce from the apriori bound on Fenchel-Nielsen parameters, Proposition 3.22, a rough polynomial upper bound on the volume of the set of hyperbolic surfaces $Y \in \mathcal{T}^*_{g_S, n_S}$ such that $\ell_Y(c) \leq \ell$. A similar statement was obtained when $S$ is a once-holed torus in [2, eq. (8.2)].

**Corollary 3.25.** For any family of distinct indices $(i_j, i'_j)_{1 \leq j \leq k}$ in $\partial S$, any $\ell \geq 0$,

$$ (3.27) \qquad \int_{\ell_Y(c)} \leq \ell \prod_{j=1}^{k} \delta(x_{i_j} - x_{i'_j}) \mathrm{dVol}^{\mathrm{WP}}_{g_S, n_S}(\mathbf{x}, Y) \leq (3\ell)^{3\chi(S)}. $$

*Proof.* We simply write the expression of $\mathrm{dVol}^{\mathrm{WP}}_{g_S, n_S}(\mathbf{x}, Y)$ in Fenchel-Nielsen coordinates as in (3.1), and use the bounds given by Proposition 3.22 on the lengths and twist angles. $\square$
**Remark 3.26.** We have assumed at the beginning of §3 that the loop **c** is a generalized eight, and therefore, as of now, we have only proven Corollary 3.25 for generalized eights. For the sake of the following application, it will be useful to observe that this result can be extended to any multi-loop **c**. This proof relies on definitions from Section 11 and can be omitted at first read; it is only presented here for completeness and to allow to state the interesting Lemma 3.27 below.

*Proof of Corollary 3.25 for general loops.* Let **c**' be a skeleton of **c**, and **S**' be the surface filled by **c**'. If **S**' = **S**, then by Proposition 11.13, $\ell_Y(\mathbf{c}') \le \ell_Y(\mathbf{c})$, from which we deduce the claim. Otherwise, **S** is obtained from **S**' by gluing a finite number of cylinders and adding cylindrical bars. We add the corresponding twist parameters to our coordinate system and remove the $\theta_q$ that have become redundant. Corollary 3.25 persists because we can choose those twist parameters so that they still satisfy (3.25), by the same proof. □

As in [2, §8.1], we can use polynomial volume bounds to prove the following special case of our main result, Theorem 1.22, for double-filling loops (defined in Definition 1.20).

**Lemma 3.27.** If **c** is double-filling, then $\mathcal{J} \in \mathcal{R}_w^N$ and

$$ \|\mathcal{J}\|_{\mathcal{R}_w^N} \le \mathrm{fn}(g_S, n_S, K, N) \prod_{j \in V} \|\tilde{g}_j\|_{\mathcal{F}^{K_j, N_j}}. $$

*Proof of Lemma 3.27.* Let us write

$$ \int_0^\ell |\mathcal{J}(s)| \, ds \le \int_{\ell_Y(\mathbf{c}) \le \ell} \prod_{j \in V} |f_j(x_j)| \prod_{j=1}^k x_{i_j} \delta(x_{i_j} - x_{i'_j}) \, d\mathrm{Vol}_{g_S, n_S}^{\mathrm{WP}}(\mathbf{x}, Y). $$

By hypothesis on the functions $f_j$ and using equation (1.7) on $\tilde{g}_j$, we have for each $j \in V$,

$$ |f_j(x_j)| \le \mathrm{fn}(K, N) \|\tilde{g}_j\|_{\mathcal{F}^{K_j, N_j}} (1 + x_j)^{K_j - 1} e^{x_j/2} $$

and for each $1 \le j \le k$, $x_{i_j} \le e^{x_{i_j}/2}$. By Proposition 1.29, if $\ell_Y(\mathbf{c}) \le \ell$ and **c** is double-filling, then $\sum_{j=1}^{n_S} x_j \le \ell$ and hence

$$ \prod_{j \in V} |f_j(x_j)| \prod_{j=1}^k x_{i_j} \le \mathrm{fn}(K, N) \prod_{j \in V} \|\tilde{g}_j\|_{\mathcal{F}^{K_j, N_j}} (1 + \ell)^{\sum_{j \in V} (K_j - 1)} e^{\ell/2}. $$

With Corollary 3.25, using the hypothesis $K_j \le N_j$ and the definition $N = \sum_{j \in V} N_j + 3\chi(\mathbf{S}) + 1$, this implies that

$$ \int_0^\ell |\mathcal{J}(s)| \, ds \le \mathrm{fn}(g_S, n_S, K, N) \prod_{j \in V} \|\tilde{g}_j\|_{\mathcal{F}^{K_j, N_j}} (1 + \ell)^{N-1} e^{\ell/2}, $$

which allows us to conclude and to obtain the claimed bound. □
**Remark 3.28.** Had we not assumed that $\mathbf{c}$ was double-filling, Proposition 1.29 would only have implied that

$$ (3.28) \qquad \int_0^\ell |\mathcal{J}(s)| \, ds \le \operatorname{fn}(g_S, n_S, K, N) \prod_{j \in V} \|\tilde{g}_j\|_{\mathcal{F}^{K_j, N_j}} (1+\ell)^{N-1} e^\ell, $$

i.e. a similar upper bound but without the factor $1/2$ in the exponential, which is not enough to conclude to Theorem 1.22.

### 4. EXPRESSION OF LENGTH FUNCTIONS IN THE NEW COORDINATE SYSTEM

The aim of this section is to give explicit expressions in the coordinates $(\vec{L}, \vec{\theta})$ for the length $\ell_Y(\mathbf{c})$, but also for all the lengths $y_\lambda$ of the multi-curve $(\Gamma_\lambda)_{\lambda \in \Lambda}$ defined in §3.2.

**4.1. Trajectories in $T^1S$ and multiplication in $SL(2, \mathbb{R})$.** It is a classical fact that $T_{g_S,n_S}^*$ may be seen as the space of discrete faithful representations of the fundamental group $\pi_1(S)$ into $PSL(2, \mathbb{R})$ (modulo conjugacy). Moreover, if $Y \in T_{g_S,n_S}^*$ corresponds to the representation $\rho: \pi_1(S) \to PSL(2, \mathbb{R})$, we can express all lengths thanks to the trace of $\rho$:

$$ (4.1) \qquad 2 \cosh \left( \frac{\ell_Y(\gamma)}{2} \right) = |\operatorname{Tr} \rho(\gamma)| $$

(where $\rho(\gamma) \in PSL(2, \mathbb{R})$ is the image under $\rho$ of the homotopy class of $\gamma$). Our method to express lengths in the new coordinates consists in providing an explicit expression for $\rho(\gamma)$ in (4.1), by writing a representative of the homotopy class of $\gamma$ as a succession of orthogonal segments.

We recall that the unit tangent bundle $T^1\mathbb{H}^2$ itself can be identified with $PSL(2, \mathbb{R})$. We can then identify different geometric actions on $T^1\mathbb{H}^2$ with the right-multiplication by a specific matrix: $a^t$ is the action of the geodesic flow at time $t$, $w^t$ the conjugate of the geodesic flow by a rotation of angle $\pi/2$, and $k^\theta$ the rotation of angle $\theta$ by $k^\theta$, where

$$ (4.2) \qquad a^t = \begin{pmatrix} e^{\frac{t}{2}} & 0 \\ 0 & e^{-\frac{t}{2}} \end{pmatrix}, \quad w^t = \begin{pmatrix} \cosh(\frac{t}{2}) & \sinh(\frac{t}{2}) \\ \sinh(\frac{t}{2}) & \cosh(\frac{t}{2}) \end{pmatrix}, \quad k^\theta = \begin{pmatrix} \cos(\frac{\theta}{2}) & \sin(\frac{\theta}{2}) \\ -\sin(\frac{\theta}{2}) & \cos(\frac{\theta}{2}) \end{pmatrix}. $$

These moves are represented on Figure 16.

<figure><img src="image_18.png" /><figcaption>FIGURE 16. The moves corresponding to the matrices $a^t$, $w^t$ and $k^\theta$.</figcaption></figure>

It is important to note that, while $a^t$ always has non-negative entries, this is true of $w^t$ only if $t \ge 0$. In what follows, a lot of effort is devoted to using $w^t$ only for $t \ge 0$.
4.2. **Expression of $\ell_Y(c)$ in the new coordinates.** Let us now focus on finding an expression for $\ell_Y(c)$ in terms of $(\vec{L}, \vec{\theta})$. We shall do so component by component, i.e. provide an expression for the $i$th component $c_i$ of $c$ for $i \in \{1, \dots, n\}$. Oftenwise, we will focus on one component $c_i$ at a time, in which case we will drop the mention of $i$ to lighten notations. This shall always be mentioned, to avoid any confusion.

4.2.1. Piecewise geodesic representative of $c_i$. First, we recall that we saw in (2.7) that the homotopy class of $c_i^{\text{op}}$ can be written, for any index $q \in \Theta^i$, as

$$c_i^{\text{op}} = p(q) \cdot p(\sigma q) \cdot \dots \cdot p(\sigma^{n_i-1} q)$$

where, for any $q'$, $p(q')$ is the concatenation of the bar $B_{q'}$ and the simple portion $\mathcal{I}_{q'}$.

Consider a point $Y \in \mathcal{T}_{g_S, n_S}^*$. As suggested in Remark 2.8, we can pick our representative so that $\beta$ is a multi-geodesic. We then consider the representative of the homotopy class of $c_i$ given by a cyclic succession of orthogonal geodesics segments:

$$(4.3) \qquad \bar{p}(q) \cdot \bar{p}(\sigma(q)) \cdot \dots \cdot \bar{p}(\sigma^{n_i-1}q)$$

for a $q \in \Theta^i$, where $\bar{p}(q) := \bar{B}_q \cdot \bar{\mathcal{I}}_q$. Each bar $\bar{B}_q$ is now an orthogeodesic from $\beta_{\lambda(\sigma^{-1}q)}$ to $\beta_{\lambda(q)}$.

For the rest of §4.2, it will be convenient to use the relabelling of $\Theta$ introduced in §2.5.2, which allows us to identify $\Theta^i$ with the cyclic set $\mathbb{Z}_{n_i} = \mathbb{Z}/n_i\mathbb{Z}$. Through this identification, going through the elements $k \in \mathbb{Z}_{n_i}$ corresponds to going through the elements $\sigma^k q$ in (4.3). We will therefore write $\bar{B}(i, k) = \bar{B}(\sigma^k q)$, $\bar{\mathcal{I}}(i, k) = \bar{\mathcal{I}}(\sigma^k q)$, $\theta(i, k) = \theta(\sigma^k q)$ etc. Following Notation 3.3, we will also write $L(i, k) = L(\sigma^k q)$ and $\epsilon L(i, k) = \epsilon L(\sigma^k q)$. As a consequence, the consecutive lengths of all the segments encountered in (4.3) are (up to cyclic permutation)

$$L(i, 1), \theta(i, 1), \dots, L(i, n_i), \theta(i, n_i).$$

The sign of $\epsilon L(i, k)$ tells us in which direction the bar $\bar{B}(i, k)$ is taken at the step $k$.

4.2.2. *Lift of c to the universal cover.* We now consider a lift of the closed path (4.3) to the universal cover of $S$, identified with a subset of the hyperbolic plane. For each index $q \in \Theta^i$, let $\tilde{\beta}_q \subset \mathbb{H}^2$ be a lift of the geodesic $\beta_{\lambda(q)}$ such that, if $\tilde{B}_{\sigma^{-1}q,q}$ denotes the oriented orthogeodesic going from $\tilde{\beta}_{\sigma^{-1}q}$ to $\tilde{\beta}_q$, then $\tilde{B}_{\sigma^{-1}q,q}$ projects down to $\bar{B}_q$ on the surface.

Whilst there was a priori no natural orientation for the multi-curve $\beta$ on $S$, we can now orient the lifts $\tilde{\beta}_q$ so that the bar $\tilde{B}_{\sigma^{-1}q,q}$ arrives on the right of $\tilde{\beta}_q$ for each $q \in \Theta^i$ with $\text{Sign}(q) = +$ and on the left otherwise. We then make the following crucial observation.

**Lemma 4.1.** *The geodesics $(\tilde{\beta}_q)_{q \in \Theta^i}$ are disjoint and, for any $q \in \Theta^i$, $\tilde{\beta}_q$ and $\tilde{\beta}_{\sigma q}$ are aligned. $\tilde{\beta}_{\sigma q}$ is on the left of $\tilde{\beta}_q$ if and only if $\text{Sign}(q) = +$.*

We denote by $\tilde{\mathcal{I}}_q \subset \tilde{\beta}_q$ the segment between the endpoint of $\tilde{B}_{\sigma^{-1}q,q}$ and the origin of $\tilde{B}_{q,\sigma q}$. Then the segment $\tilde{\mathcal{I}}_q$ projects down to the segment $\bar{\mathcal{I}}_q$ on the surface. If we call $\tilde{p}(q) = \tilde{B}_{\sigma^{-1}q,q} \cdot \tilde{\mathcal{I}}(q)$, then the infinite concatenation $\cdot_{q \in \Theta^i} \tilde{p}(q)$ of orthogonal geodesics segments in the hyperbolic plane projects down to (4.3) on the surface. We shall denote by $\tilde{c}_i$ the infinite geodesic homotopic to $\cdot_{q \in \Theta^i} \tilde{p}(q)$ in the hyperbolic plane.
**Notation 4.2.** The notation $\tilde{B}_{mn}$, for the moment only defined for $n = \sigma m$, may be extended to arbitrary $m, n \in \Theta^i$. We call $\tilde{B}_{mn} \subset \mathbb{H}^2$ the oriented orthogeodesic segment from $\tilde{\beta}_m$ to $\tilde{\beta}_n$. We denote by $L_{mn} > 0$ its length and $z_{mn} \in \tilde{\beta}_n$ its terminus.

**4.2.3. Expression of the length of $c_i$ as a trace.** Starting from (4.3) we now find a formula expressing each $\ell_Y(c_i)$ as the trace of a product of matrices.

**Lemma 4.3.** For $i \in \{1, \dots, n\}$, the length of the component $c_i$ is given by the formula

$$ (4.4) \qquad \cosh\left(\frac{\ell_Y(c_i)}{2}\right) = \frac{1}{2} \operatorname{Tr}(a_1^i b_1^i \dots a_{n_i}^i b_{n_i}^i) $$

where the matrices $a_i^i, b_i^i$ are defined by

$$ (4.5) \qquad a_k^i = a^{\ell L(i,k)} = \begin{pmatrix} e^{\frac{\ell L(i,k)}{2}} & 0 \\ 0 & e^{-\frac{\ell L(i,k)}{2}} \end{pmatrix} \qquad b_k^i = w^{\theta(i,k)} = \begin{pmatrix} \cosh\left(\frac{\theta(i,k)}{2}\right) & \sinh\left(\frac{\theta(i,k)}{2}\right) \\ \sinh\left(\frac{\theta(i,k)}{2}\right) & \cosh\left(\frac{\theta(i,k)}{2}\right) \end{pmatrix}. $$

*Proof.* For the proof, let us fix a value of $i$. As explained at the beginning of this section, the elements of $\Theta^i$ are labelled $(i, k)$ with $k \in \mathbb{Z}_{n_i}$. To lighten notations, we shall omit the mention of $i$, for instance writing $n_i = n$, $L_k = L(i, k)$, $\theta(i, k) = \theta_k$, etc.

Let $v$ denote the unit vector tangent to the bar $\bar{B}_1 = \bar{B}(i, 1)$ based at its origin. We consider a lift $\tilde{v}$ of $v$ in $T^1\mathbb{H}^2$ identified with PSL(2, $\mathbb{R}$). The decomposition (4.3) into a succession of orthogonal geodesic segments precisely tells us that $\tilde{v}a_1b_1\dots a_nb_n = \rho(c)\tilde{v}$. This implies that

$$ (4.6) \qquad \rho(c) = \tilde{v}a_1b_1\dots a_nb_n\tilde{v}^{-1}. $$

In light of (4.1), we obtain $\cosh(\ell_Y(c)/2) = \frac{1}{2}|\operatorname{Tr}(a_1b_1\dots a_nb_n)|$, which is our claim up to the presence of the absolute value.

To prove the proposition, we need to argue that we can remove the absolute value. In order to do so, we realise the matrix $a_1b_1\dots a_nb_n$ as the endpoint of a continuous path

$$ \tilde{a}_1 \diamond \tilde{b}_1 \diamond \dots \diamond \tilde{a}_n \diamond \tilde{b}_n $$

in SL(2, $\mathbb{R}$) starting at the identity matrix. More precisely, for any $k$, let us define the paths

$$ \tilde{a}_k : [0, L_k] \to \mathrm{SL}(2, \mathbb{R}) $$

$$ t \mapsto a^{\mathrm{Sign}(k)t} $$

$$ \tilde{b}_k : [0, \theta_k] \to \mathrm{SL}(2, \mathbb{R}) $$

$$ t \mapsto w^t, $$

which respectively correspond to the geodesic motion along the bar $\bar{B}_k$ (in positive or negative time depending on $\mathrm{Sign}(k)$), and the parallel transport of a vector orthogonal to $\bar{I}_k$. Note that the endpoint of $\tilde{a}_k$ and $\tilde{b}_k$ are the matrices $a_k$ and $b_k$. We then define the path $\tilde{a}_1 \diamond \tilde{b}_1 \diamond \dots \diamond \tilde{a}_n \diamond \tilde{b}_n$ as the concatenation of the successive paths with matching endpoints $t \mapsto \tilde{a}_1(t)$, $t \mapsto a_1\tilde{b}_1(t)$, $t \mapsto a_1b_1\tilde{a}_2(t)$, $t \mapsto a_1b_1a_2\tilde{b}_2(t)$, ..., and $t \mapsto a_1b_1\dots a_nb_n\tilde{b}_n(t)$.

Thanks to our prescription of orientation rules, the path $c_i^{\mathrm{op}}$ traced in $\mathbf{S}$ may be lifted to a path $c_i^{\mathrm{nl}}$ traced in the unit tangent bundle $T^1\mathbf{S}$, projecting down to $c_i^{\mathrm{op}}$, satisfying the following: $c_i^{\mathrm{nl}}$ is made of unit vectors that point to the left of each oriented segment
$\bar{\mathcal{I}}_k$, and are tangent to the bars $\bar{B}_k$. We call this a *nice lift* of $\mathbf{c}_i^{\text{op}}$. To $\mathbf{c}_i^{\text{nl}}$, we may apply the following statement.

**Lemma 4.4.** Suppose we have a piecewise smooth closed path $(\gamma, v)$ drawn in $T^1\mathbf{S}$, where the path $t \mapsto \gamma(t) \in \mathbf{S}$ is non-contractible, in minimal position, and $v(t) \in T_{\gamma(t)}^1\mathbf{S}$ is always on the left of $\gamma$. Then there is a homotopy from $(\gamma, v)$ to $(\gamma_g, v_n)$, where $\gamma_g$ is the closed geodesic homotopic to $\gamma$ and $v_n(t)$ is the left-normal vector to $\gamma_g(t)$.

We can then conclude that the path $c_i^{\text{nl}}$ in $T^1\mathbf{S}$ can be continuously deformed to

$$[0, \ell_Y(\mathbf{c}_i)] \to T^1\mathbf{S} \\ t \mapsto v_0 w^t$$

where $v_0$ is normal to the geodesic representative of $\mathbf{c}_i$ and lies on its left. As a consequence, the path $v\tilde{a}_1 \diamond \tilde{b}_1 \diamond \dots \diamond \tilde{a}_n \diamond \tilde{b}_n$ can also be continuously deformed to $t \mapsto v_0 w^t$. Each path in this continuous family leads to the same value of $\text{Tr} \rho(\mathbf{c}_i)$. Since the trace of $w^{\ell(\mathbf{c}_i)}$ is positive, we must have $\text{Tr}(a_1 b_1 \dots a_n b_n) > 0$ as well, proving the proposition. $\square$

4.2.4. The formula for the length of $\mathbf{c}_i$. Expanding (4.4), we can deduce the following expression for the length of $\mathbf{c}_i$. We shall use a shorthand notation for hyperbolic trigonometric functions.

**Notation 4.5.** We denote $\text{hyp}_+ := \cosh$, $\text{hyp}_- := \sinh$. For $\delta = (\delta_1, \dots, \delta_n) \in \{\pm 1\}^n$ and $t = (t_1, \dots, t_n) \in \mathbb{R}^n$, we set

$$(4.7) \qquad \text{hyp}_{\delta}(t) := \prod_{i=1}^{n} \text{hyp}_{\delta_i}(t_i).$$

We prove the following.

**Theorem 4.6.** For any $Y \in \mathcal{T}_{g_S, n_S}^*$ of coordinates $(\vec{L}, \vec{\theta})$, any $i \in \{1, \dots, n\}$,

$$\cosh\left(\frac{\ell_Y(\mathbf{c}_i)}{2}\right) = \sum_{\substack{\delta \in \{\pm 1\}^{n_i} \\ \delta_1 \dots \delta_{n_i} = +1}} \text{hyp}_\delta\left(\frac{\theta}{2}\right) \cosh\left(\frac{\rho^\delta \cdot \epsilon L}{2}\right)$$

where $\rho_k^\delta = \prod_{1 \le j < k} \delta_j$ (for $k = 1$ this empty product is equal to 1), $\epsilon L$ is the vector $(\epsilon L(i, 1), \dots, \epsilon L(i, n_i))$, and $\cdot$ denotes the scalar product, so that $\rho^\delta \cdot \epsilon L = \sum_{k=1}^{n_i} \epsilon_k L_k \rho_k^\delta$.

Note that this result holds for any choice of openings of the intersections of $\mathbf{c}$, and provides different expressions for the length of $\mathbf{c}$ in different coordinate systems depending on that choice.

*Proof.* As in the previous proof, we momentarily drop all references to the index $i$. By equation (4.4), the desired quantity can be expressed as $\frac{1}{2}\text{Tr}(A_1 \dots A_n)$ where for any $k$, $A_k = a_k b_k$, and $a_k, b_k$ are defined in (4.5). Calculating the product, we find

$$A_k = a_k b_k = \begin{pmatrix} \cosh\left(\frac{\theta_k}{2}\right) e^{\epsilon_k L_k / 2} & \sinh\left(\frac{\theta_k}{2}\right) e^{\epsilon_k L_k / 2} \\ \sinh\left(\frac{\theta_k}{2}\right) e^{-\epsilon_k L_k / 2} & \cosh\left(\frac{\theta_k}{2}\right) e^{-\epsilon_k L_k / 2} \end{pmatrix} \\ = \text{hyp}_+(\frac{\theta_k}{2}) E_{+1}(\frac{\epsilon_k L_k}{2}) + \text{hyp}_-(\frac{\theta_k}{2}) E_{-1}(\frac{\epsilon_k L_k}{2})$$
with the matrices $E_{\pm 1}$ defined by

$$ (4.8) \qquad E_{+1}(s) = \begin{pmatrix} e^s & 0 \\ 0 & e^{-s} \end{pmatrix} \qquad E_{-1}(s) = \begin{pmatrix} 0 & e^s \\ e^{-s} & 0 \end{pmatrix}. $$

We need to understand products of these matrices. Using the fact that

$$ \begin{align*} E_{+1}(s)E_{+1}(s') &= E_{+1}(s+s') & E_{+1}(s)E_{-1}(s') &= E_{-1}(s+s') \\ E_{-1}(s)E_{+1}(s') &= E_{-1}(s-s') & E_{-1}(s)E_{-1}(s') &= E_{+1}(s-s') \end{align*} $$

one proves by a straightforward induction that for any $s \in \mathbb{R}^n$ and $\delta \in \{\pm 1\}^n$,

$$ E_{\delta_1}(s_1) \dots E_{\delta_n}(s_n) = E_{\delta_1 \dots \delta_n}(\rho^{\delta} \cdot s) $$

where $\rho_k^\delta$ is defined in the statement of the Lemma. Therefore, expanding the product of the matrices $A_k$, we obtain

$$ (4.9) \qquad \cosh\left(\frac{\ell_Y(\mathbf{c}_i)}{2}\right) = \frac{1}{2} \sum_{\delta \in \{\pm 1\}^n} \operatorname{hyp}_\delta\left(\frac{\theta}{2}\right) \operatorname{Tr}\left(E_{\delta_1 \dots \delta_n}\left(\frac{\rho^\delta \cdot \epsilon L}{2}\right)\right) $$

which allows us to conclude, since $\operatorname{Tr}(E_{+1}(s)) = 2 \cosh(s)$ and $\operatorname{Tr}(E_{-1}(s)) = 0$. $\square$

<figure><img src="image_19.png" /></figure>

Motivated by Proposition 4.6, we introduce the following function $M_n$.

**Notation 4.7.** For an integer $n \ge 1$, $x = (x_1, \dots, x_n)$ and $t = (t_1, \dots, t_n) \in \mathbb{R}^n$, we define

$$ (4.10) \qquad M_n(x,t) = 2 \operatorname{argcosh} \sum_{\substack{\delta \in \{\pm 1\}^n \\ \delta_1 \dots \delta_n = +1}} \operatorname{hyp}_\delta \left(\frac{t}{2}\right) \cosh\left(\frac{\rho^\delta \cdot x}{2}\right). $$

Then, Theorem 4.6 tells us that

$$ (4.11) \qquad \ell_Y(\mathbf{c}_i) = M_{n_i} (\epsilon L(i, 1), \dots, \epsilon L(i, n_i), \theta(i, 1), \dots, \theta(i, n_i)) $$

where we recall that $\epsilon L(i, k)$ is short-hand for $\epsilon(i, k)L(i, k)$. Adding the lengths of all components of $\mathbf{c}$, we obtain

$$ (4.12) \qquad \ell_Y(\mathbf{c}) = \sum_{i=1}^{n} M_{n_i} (\epsilon L(i, 1), \dots, \epsilon L(i, n_i), \theta(i, 1), \dots, \theta(i, n_i)). $$

### 4.3. Parameters $\tau$. 
In the following section, we shall apply similar techniques to find the expression of the lengths $(\ell_Y(\Gamma_\lambda))_{\lambda \in \Lambda}$ of the pair of pants decomposition constructed in §3.2. As in the case of $\mathbf{c}$, we can realise the path $\Gamma_\lambda$ by going along a succession of bars $B_q$ and simple portions $\mathcal{I}_q$, now with different orientation rules (this was already observed in Remark 3.6). This motivates the following definitions.

**For the rest of this section (and hence until the end of §10), we shall assume that all intersections are opened the first way.** As a consequence, the multi-curve $\beta$ is oriented. This convention will lighten notations for the expressions of $\ell_Y(\Gamma_\lambda)$; however, the methods we develop can be adapted to other situations.
4.3.1. *Definition*. Let $q, q' \in \Theta$ be two indices such that $\lambda(q) = \lambda(q')$ and $\text{Sign}(q) = \text{Sign}(q') =: \epsilon$, i.e. both bars $B_q$ and $B_{q'}$ arrive on the same component of $\beta$ and on the same side. We denote as $\mathcal{K}_{qq'}$ the oriented segment on $\beta$ going from the terminus point $t(B_q)$ to the terminus point $t(B_{q'})$, following the orientation of $\beta$ if $\epsilon = +$, and its reverse orientation otherwise.

Once the bars $B_q, B_{q'}$ have glided to the orthogeodesics $\bar{B}_q, \bar{B}_{q'}$, the segment $\mathcal{K}_{qq'}$ glides to a new segment $\bar{\mathcal{K}}_{qq'}$, and we denote by $\tau_{qq'}$ its length. Thanks to the Useful Remark (Remark 2.11), $\tau_{qq'}$ is always strictly positive.

<figure><img src="image_20.png" /><figcaption>FIGURE 17. The segment $\mathcal{K}_{qq'}$.</figcaption></figure>

In the special case when a segment $\mathcal{K}_{qq'}$ does not contain any other point $t(B_{q''})$ in its interior, with $B_{q''}$ also arriving on the same side as $B_q$ and $B_{q'}$, then we simply denote by $\mathcal{K}_q$ this segment. This notation (where $q'$ has been removed) is justified by the remark that there is exactly one $q'$ with this property. We then denote $\tau_q := \tau_{qq'}$ (or, when convenient, $\tau(q)$).

4.3.2. *Relation between $\tau$ and $\theta$.* Each parameters $\tau_{qq'}$ is a sum of a certain number of parameters $\theta_{q''}$: there exists a (uniquely defined) set of indices $\Theta_{\tau}(qq') \subseteq \Theta$ such that

$$ (4.13) \qquad \tau_{qq'} = \sum_{q'' \in \Theta_{\tau}(qq')} \theta_{q''}, $$

reflecting a partition of $\mathcal{K}_{qq'}$ into $\mathcal{K}_{qq'} = \bigsqcup_{q'' \in \Theta_{\tau}(qq')} \mathcal{I}_{q''}$.

In the case of $\tau_q$, we adopt as slightly simpler notation, writing that there exists a set of indices $\Theta_{\tau}(q) \subseteq \Theta$ such that

$$ (4.14) \qquad \tau_q = \sum_{q' \in \Theta_{\tau}(q)} \theta_{q'}. $$

For a set $W \subseteq \Theta$, we let $\Theta_{\tau}(W) := \bigcup_{q \in W} \Theta_{\tau}(q)$. Note that this union is not necessarily disjoint.

**4.4. Expression of the boundary lengths in the new coordinates.** We are now ready to find the expression of the length of the curves $\Gamma_{\lambda}$ for $\lambda \in \Lambda$. We recall that we assumed, for the sake of simplicity, that the intersections have been opened the first way (see Remark 4.9 for the general case).
4.4.1. Case where $\Gamma_\lambda$ is a component of $\beta$. First, we note that in the special case where the curve $\Gamma_\lambda$ is a component of the multi-curve $\beta$, i.e. when $\lambda \in \Lambda^\beta$, then we can simply write $y_\lambda = \ell_Y(\beta_\lambda)$, which has the simple expression in terms of the parameters $\vec{\theta}$:

$$ (4.15) \qquad y_{\lambda} = \sum_{q:\lambda(q)=\lambda} \theta_{q} = \sum_{q \in \Theta_{t}(\lambda)} \theta_{q}, $$

where we recall the definition of the set $\Theta_t(\lambda)$ from Notation 2.16.

4.4.2. Polygonal curve associated to $\Gamma_\lambda$. Let us now fix $\lambda \in \Lambda^\Gamma = \Lambda \setminus \Lambda^\beta$ (i.e. $\Gamma_\lambda$ is not a component of $\beta$). The length of $\Gamma_\lambda$ does not depend on its orientation; we orient it so that, if $j := \text{step}(\lambda)$ is the step of the construction of the pair of pants decomposition at which $\Gamma_\lambda$ appears, then $\mathbf{P}_j$ lies on the left of $\Gamma_\lambda$.

<figure><img src="image_21.png" /><figcaption>FIGURE 18. Following two (non-necessarily distinct) boundary components $\Gamma_\lambda$ and $\Gamma_{\lambda'}$ of $\mathbf{S}$ near a bar $B_q$ with $\text{Sign}(q) = +$.</figcaption></figure>

Let us first assume that $\lambda \in \Lambda_{BC}$, i.e. $\lambda$ is a boundary component of $\mathbf{S}$. Examining the local picture at each intersection and following the boundary of the regular neighbourhood of the loop $\mathbf{c}$ (as represented in Figure 18), we obtain that we can write the boundary component $\Gamma_\lambda$ as an alternating succession of bars $B_q$ and segments $\mathcal{K}_q$ defined in §4.3, i.e. as

$$ (4.16) \qquad \bar{p}_1^\lambda \cdots \bar{p}_m^\lambda $$

for some integer $m(\lambda)$, where each piece $\bar{p}_k^\lambda$ is a concatenation $\bar{B}_q \cdot \bar{\mathcal{K}}_q$ for some index $q = q_k^\lambda$ in $\Theta$.

**Remark 4.8.** We also notice that $\text{Sign} q_k^\lambda = (-1)^k$, and in particular the integer $m(\lambda)$ is even, but this will not play any role in the discussion.

More generally, if $\lambda \in \Lambda_{in}$, then we realise $\Gamma_\lambda$ as a boundary component of the surface $\mathbf{S}_j$ explored at the step $j$ of the construction of the pair of pants decomposition. We then obtain that $\Gamma_\lambda$ can, too, be written under the form (4.16), but now with bars $\bar{p}_k^\lambda = \bar{B}_q \cdot \bar{\mathcal{K}}_{qq'}$ for some $q = q_k^\lambda$ and $q' = q_k'^\lambda \in \Theta$. See Figure 19 for an example.

This expression for the homotopy class of $\Gamma_\lambda$ is the equivalent of (4.3) for the curve $\mathbf{c}_i$ and will, as before, be our starting point to compute the length of $\Gamma_\lambda$. The lengths
<figure><img src="image_22.png" /><figcaption>FIGURE 19. An example of the two new boundary components $\Gamma_{\lambda}$ for a case where $r = 2$, and their description in terms of a succession of bars and segments $\mathcal{K}$. We use the shorthand notation $\tau_{ij}^{\epsilon} := \tau_{(i,\epsilon),(j,\epsilon)}$.</figcaption></figure>

appearing in this decomposition are the lengths of the bars $\overline{B}_q$, which are the same lengths $L_q$ as before, now together with the lengths $\tau_{qq'}$ of the segments $\overline{\mathcal{K}}_{qq'}$.

The important observation here is that, as we go along the curve $\Gamma_{\lambda}$, we follow a succession of geodesic segments, only making right turns (as seen on Figure 18, and in the orientation convention of $\tau_{qq'}$ in §4.3). Such a path will be called a polygonal curve in Definition 8.1.

**Remark 4.9.** If we open the intersections both ways, we can still obtain a description of $\Gamma_{\lambda}$ as a polygonal curve (a succession of orthogonal geodesic segments making only right turns), but the notations for the orientations of the bars and the parameters $\tau$ need to be adapted. In particular, we no longer have alternate signs for the bars, and $m(\lambda)$ is not necessarily even (see for instance a figure-eight opened the second way).

4.4.3. *Re-labelling.* Let us now relabel the segments and lengths appearing in the decomposition (4.16) following the order in which they occur in $\Gamma_{\lambda}$. We shall therefore denote $\overline{B}_{k}^{\lambda} := \overline{B}_{q_{k}^{\lambda}}$ and $\mathcal{K}_{k}^{\lambda} := \mathcal{K}_{q_{k}^{\lambda}q_{k}^{\lambda}}$, and call $L_{k}^{\lambda}$, $\tau_{k}^{\lambda}$ their respective lengths. Note that both these lengths are always positive quantities, by definition for $L_{k}^{\lambda}$, and in virtue of the Useful Remark for $\tau_{k}^{\lambda}$. We denote as $\vec{L}^{\lambda}$ and $\vec{\tau}^{\lambda}$ the vectors $\vec{L}^{\lambda} = (L_{1}^{\lambda}, \dots, L_{m(\lambda)}^{\lambda})$ and $\vec{\tau}^{\lambda} = (\tau_{1}^{\lambda}, \dots, \tau_{m(\lambda)}^{\lambda})$.

4.4.4. *Expression of the length.* We now follow quite closely the discussion of §4.1.

**Lemma 4.10.** For any $Y \in \mathcal{T}_{g_S, n_S}^*$, any $\lambda \in \Lambda^{\Gamma}$, we have

$$ (4.17) \qquad \cosh\left(\frac{\ell_Y(\Gamma_\lambda)}{2}\right) = \frac{1}{2} \operatorname{Tr}\left(a_1^\lambda b_1^\lambda \dots a_{m(\lambda)}^\lambda b_{m(\lambda)}^\lambda\right) $$

where, for any index $k$, $a_k^\lambda := A(L_k^\lambda)$ and $b_k^\lambda := A(\tau_k^\lambda)$ with

$$ (4.18) \qquad A(s) := w^s k^{-\frac{\pi}{2}} = \frac{\sqrt{2}}{2} \begin{pmatrix} e^{\frac{s}{2}} & -e^{-\frac{s}{2}} \\ e^{\frac{s}{2}} & e^{-\frac{s}{2}} \end{pmatrix}. $$
*Proof.* The proof is the same as the proof of Lemma 4.3. The new path in $T^1S$ that we consider is represented on Figure 20. The key argument to remove the absolute value is

<figure><img src="image_23.png" /><figcaption>FIGURE 20. The path in $T^1S$ followed in the proof of Lemma 4.10.</figcaption></figure>

that this path has the property required by Lemma 4.4, namely that the vectors always point to its left. □

Expanding the trace, we can deduce the following formula giving the lengths of the curves $\Gamma_{\lambda}$.

**Theorem 4.11.** For any $Y \in \mathcal{T}_{g_S,n_S}^*$, any $\lambda \in \Lambda^{\Gamma}$, we have

$$ \cosh\left(\frac{\ell_Y(\Gamma_\lambda)}{2}\right) = \frac{1}{2} \sum_{\alpha \in \{\pm 1\}^{m(\lambda)}} \alpha_1 \alpha_2 \dots \alpha_{m(\lambda)} \operatorname{hyp}_{-\nabla\alpha}\left(\frac{L}{2}\right) \exp\left(\frac{\alpha \cdot \tau}{2}\right) $$

where $(\nabla\alpha)_k := \alpha_k/\alpha_{k-1} = \alpha_k\alpha_{k-1}$ with the convention $\alpha_0 = \alpha_{m(\lambda)}$.

**Remark 4.12.** For $m=1$ this reduces to

$$ \cosh\left(\frac{\ell_Y(\Gamma)}{2}\right) = \sinh\left(\frac{L_1}{2}\right) \sinh\left(\frac{\tau_1}{2}\right). $$

*Proof.* To simplify notations, we shall remove the dependency on $\lambda$ from the name of the matrices and lengths. By equation (4.17), the announced quantity can be expressed as

$$ \frac{1}{2} \operatorname{Tr}(A(L_1)A(\tau_1) \dots A(L_m)A(\tau_m)) $$

where $A(s)$ is the matrix introduced in (4.18). Calculating the product, we find

$$ A(L)A(\tau) = \begin{pmatrix} e^{\frac{\tau}{2}} \sinh\left(\frac{L}{2}\right) & -e^{-\frac{\tau}{2}} \cosh\left(\frac{L}{2}\right) \\ e^{\frac{\tau}{2}} \cosh\left(\frac{L}{2}\right) & -e^{-\frac{\tau}{2}} \sinh\left(\frac{L}{2}\right) \end{pmatrix} = \cosh\left(\frac{L}{2}\right) e_{-1}\left(\frac{\tau}{2}\right) + \sinh\left(\frac{L}{2}\right) e_{+1}\left(\frac{\tau}{2}\right) $$

with the matrices $e_{\pm 1}$ defined by

$$ e_{+1}(s) = \begin{pmatrix} e^s & 0 \\ 0 & -e^{-s} \end{pmatrix} \qquad e_{-1}(s) = \begin{pmatrix} 0 & -e^{-s} \\ e^s & 0 \end{pmatrix}. $$
We can use once again the matrices $E_{-1}$, $E_{+1}$ introduced in equation (4.8), and note the relations

$$ 
\begin{aligned}
e_{+1}(s)e_{+1}(s') &= E_{+1}(s+s') & e_{+1}(s)e_{-1}(s') &= -E_{-1}(s-s') \\
e_{-1}(s)e_{+1}(s') &= E_{-1}(-s-s') & e_{-1}(s)e_{-1}(s') &= -E_{+1}(s'-s)
\end{aligned}
$$

and

$$ 
\begin{aligned}
E_{+1}(s)e_{+1}(s') &= E_{-1}(s)e_{-1}(s') = e_{+1}(s+s') \\
E_{+1}(s)e_{-1}(s') &= E_{-1}(s)e_{+1}(s') = e_{-1}(s'-s).
\end{aligned}
$$

A straightforward induction then shows that for any $s \in \mathbb{R}^{2k}$ and $\delta \in \{\pm 1\}^{2k}$,

$$ e_{\delta_1}(s_1)e_{\delta_2}(s_2)\dots e_{\delta_{2k}}(s_{2k}) = \delta_2\delta_4\dots\delta_{2k} E_{\pi(\delta)}(\alpha^\delta \cdot s), $$

where $\pi(\delta) = \prod_k \delta_k$ and $\alpha_k^\delta := \prod_{j \le k} \delta_j$, and that for any $s \in \mathbb{R}^{2k+1}$ and $\delta \in \{\pm 1\}^{2k+1}$,

$$ e_{\delta_1}(s_1)e_{\delta_2}(s_2)\dots e_{\delta_{2k+1}}(s_{2k+1}) = \delta_2\delta_4\dots\delta_{2k} e_{\pi(\delta)}(\pi(\delta)\alpha^\delta \cdot s). $$

Therefore, in the case $m = 2k$,

$$ \cosh\left(\frac{\ell(\Gamma)}{2}\right) = \frac{1}{2} \sum_{\delta \in \{\pm 1\}^{2k}} \delta_2 \delta_4 \dots \delta_m \operatorname{hyp}_{-\delta}\left(\frac{L}{2}\right) \operatorname{Tr}\left(E_{\pi(\delta)}\left(\frac{\alpha^\delta \cdot \tau}{2}\right)\right) $$

which implies that

$$ \cosh\left(\frac{\ell(\Gamma)}{2}\right) = \sum_{\substack{\delta \in \{\pm 1\}^m \\ \pi(\delta) = +1}} \delta_2 \delta_4 \dots \delta_m \operatorname{hyp}_{-\delta}\left(\frac{L}{2}\right) \cosh\left(\frac{\alpha^\delta \cdot \tau}{2}\right) $$

since $\operatorname{Tr}(E_{+1}(s)) = 2\cosh(s)$ and $\operatorname{Tr}(E_{-1}(s)) = 0$. By expanding $\cosh(\frac{\alpha^\delta \cdot \tau}{2})$ into exponential functions, and letting $\alpha_{k-1}\alpha_k = \delta_k = \alpha_k/\alpha_{k-1}$, we obtain the claimed formula.

In the case $m = 2k + 1$, we conclude similarly, now using

$$ \cosh\left(\frac{\ell(\Gamma)}{2}\right) = \sum_{\substack{\delta \in \{\pm 1\}^m \\ \pi(\delta) = +1}} \delta_2 \delta_4 \dots \delta_{m-1} \operatorname{hyp}_{-\delta}\left(\frac{L}{2}\right) \sinh\left(\frac{\alpha^\delta \cdot \tau}{2}\right). $$

□

4.4.5. Reformulation and final outcome. Motivated by Theorem 4.11, we now introduce the following notation.

**Notation 4.13.** For an integer $m \ge 1$ and $x = (x_1, \dots, x_m)$, $t = (t_1, \dots, t_m)$, we define

$$ (4.19) \qquad Q_m(x,t) = 2 \operatorname{argcosh} \frac{1}{2} \sum_{\alpha \in \{\pm 1\}^m} \alpha_1 \alpha_2 \dots \alpha_m \operatorname{hyp}_{-\nabla \alpha} \left(\frac{x}{2}\right) \exp\left(\frac{\alpha \cdot t}{2}\right) $$

and

$$ (4.20) \quad F_m(x,t) = \frac{\cosh\left(\frac{1}{2}Q_m(x,t)\right)}{\exp\left(\frac{1}{2}\sum_{n=1}^m t_n\right)} = \frac{1}{2} \sum_{\alpha \in \{0,1\}^m} (-1)^{\alpha_1 + \dots + \alpha_m} \operatorname{hyp}_{-\partial\alpha}\left(\frac{x}{2}\right) \exp(-\alpha \cdot t) $$

with $(\partial\alpha)_n := 1$ if $\alpha_{n-1} = \alpha_n$ and $-1$ if $\alpha_{n-1} \neq \alpha_n$.
**Remark 4.14.** Note that we go from (4.19) to (4.20) by the change of indices $(\alpha_n) \rightsquigarrow (\frac{\alpha_n+1}{2})$.

We have proved that

$$ (4.21) \qquad \ell_Y(\Gamma_\lambda) = Q_{m(\lambda)}(\vec{L}^\lambda, \vec{\tau}^\lambda). $$

We observe that the relation between the functions $Q_m$ and $F_m$ can be rewritten

$$ (4.22) \qquad U(Q_m(x,t)) = \sum_{n=1}^{m} t_n + 2 \log F_m(x,t), $$

for the function

$$ (4.23) \qquad U(x) = 2 \log \cosh\left(\frac{x}{2}\right) = x - 2 \log 2 + \mathcal{O}(e^{-x}). $$

The final outcome of the calculation then is

$$ (4.24) \qquad \ell_Y(\Gamma_\lambda) = \sum_{n=1}^{m(\lambda)} \tau_n^\lambda - 2 \log 2 + 2 \log F_{m(\lambda)}(\vec{L}^\lambda, \vec{\tau}^\lambda) + \mathcal{O}(e^{-\ell_Y(\Gamma_\lambda)}). $$

4.4.6. *Set of indices associated to* $\lambda$. Recall from (4.14) (the labelling by $q$ being replaced by $(\lambda, k)$) that the lengths $\tau_k^\lambda$ can be expressed by

$$ (4.25) \qquad \tau_k^\lambda = \sum_{q \in \Theta_\tau(\lambda, k)} \theta_q, $$

for some uniquely defined set of indices $\Theta_\tau(\lambda, k) \subseteq \Theta$. Now, we define the following index notation.

**Notation 4.15.** For $\lambda \in \Lambda^\Gamma$, let

$$ (4.26) \qquad \Theta_\ell(\lambda) := \bigcup_{k=1}^{m(\lambda)} \Theta_\tau(\lambda, k) \subseteq \Theta. $$

We extend the notation $\Theta_\ell(\lambda)$ to the indices $\lambda \in \Lambda^\beta$ by letting $\Theta_\ell(\lambda) := \Theta_t(\lambda)$ the set of indices introduced in Notation 2.16. If $W$ is a subset of $\Lambda$ we denote

$$ (4.27) \qquad \Theta_\ell(W) := \bigcup_{\lambda \in W} \Theta_\ell(\lambda). $$

For any $\lambda \in \Lambda$, $\Theta_\ell(\lambda)$ denotes the set of indices $q \in \Theta$ necessary to express the length $\ell_Y(\Gamma_\lambda)$.

**Remark 4.16.** Keep in mind that the sets $\Theta_\ell(\lambda)$ are not disjoint: in fact, each $q \in \Theta$ may belong to two sets $\Theta_\ell(\lambda)$ with $\lambda \in \Lambda_{BC}$, expressing the fact that each simple portion $I_q$ has a boundary component of $S$ on its left and one on its right. As a consequence, one has an inclusion $(\Theta_\ell(W))^c \subseteq \Theta_\ell(W^c)$ (where $V^c$ denotes the complement of the set $V$), which in general is strict.
5. EXPRESSION OF THE VOLUME FUNCTIONS AS PSEUDO-CONVOLUTIONS OF FR
FUNCTIONS

Let us now, in this brief section, come back to our initial objective, i.e. proving
Theorem 1.22, and reformulate it using the new notions and formulae presented in §2
to 4. More precisely, we shall explain how to compute the volume functions defined in
(1.12) by

$$
\mathcal{J} : \ell \mapsto \int_{\ell_Y(\mathbf{c})=\ell} \prod_{j \in V} f_j(x_j) \prod_{j=1}^{k} x_{i_j} \delta(x_{i_j} - x_{i'_j}) \frac{\mathrm{d}\mathrm{Vol}_{g_S, n_S}^{\mathrm{WP}}(\mathbf{x}, Y)}{\mathrm{d}\ell}
$$

as pseudo-convolutions of functions in the class $\mathcal{F}$ – a notion that will be formalized
in §6.

We recall that, in the integral above, V is a subset of ∂S. By our conventions intro-
duced in §3.3, we identify ∂S with ΛBC, and hence V with a subset of ΛBC ⊆ Λ. We
write Vβ := V ∩ Λβ and VΓ := V ∩ ΛΓ, so that Vβ are the components of V that are
components of the multi-curve β, and VΓ those that are not.

5.1. **Change of variables.** Let us change variables using our Jacobian computation, and more precisely its Dirac variant, Corollary 3.13. We obtain that:

$$
\mathcal{J}(\ell) = 2^{2|\Lambda^\beta|-n_S} \int_{\ell_Y(\mathbf{c})=\ell} \mathbb{1}_{\mathcal{D}}(\vec{L}, \vec{\theta}) \prod_{\lambda \in V} f_\lambda(y_\lambda) \frac{\prod_{\lambda \in V^\beta} \sinh\left(\frac{y_\lambda}{2}\right) \prod_{\lambda \in \Lambda_{\text{in}}^\beta} \sinh^2\left(\frac{y_\lambda}{2}\right)}{\prod_{\lambda \in V^\Gamma} \sinh\left(\frac{y_\lambda}{2}\right)} \\
T_{V,m}(\vec{L}, \vec{\theta}) \prod_{i=1}^r \sinh(L_i) \frac{d^r \vec{L} \, d^{2r} \vec{\theta}}{d\ell}
$$

where $\mathcal{D}$ is the domain of variation of the new variables $(\vec{L}, \vec{\theta})$, studied in §3.5, and $T_{V,m}$ is an explicit distribution which we shall leave as such. The integral $\mathcal{J}(\ell)$ can be rewritten under the simpler form

$$
(5.1) \qquad 2^{2|\Lambda^\beta|-n_S} \int_{\ell_Y(\mathbf{c})=\ell} \mathbb{1}_{\mathcal{D}}(\vec{L}, \vec{\theta}) \prod_{\lambda \in V_+^\beta} g_\lambda(y_\lambda) \prod_{\lambda \in V^\Gamma} u_\lambda(y_\lambda) T_{V,m}(\vec{L}, \vec{\theta}) \prod_{i=1}^r \sinh(L_i) \frac{d^r \vec{L} \, d^{2r} \vec{\theta}}{d\ell},
$$

by setting $V_+^\beta := V^\beta \cup \Lambda_{\text{in}}^\beta$ and, for $\lambda \in \Lambda$,

$$
g_{\lambda}(y) := f_{\lambda}(y) \sinh\left(\frac{y}{2}\right) \quad \text{and} \quad u_{\lambda}(y) := \frac{f_{\lambda}(y)}{\sinh\left(\frac{y}{2}\right)}
$$

where $f_\lambda$ is already defined for $\lambda \in V \subseteq \Lambda_{BC}$, and is extended to $\lambda \in \Lambda$ by letting

$$
\begin{cases}
f_{\lambda}(y) := 1 & \text{for } \lambda \in \Lambda_{\text{BC}} \setminus V \\
f_{\lambda}(y) := \sinh\left(\frac{y}{2}\right) & \text{for } \lambda \in \Lambda_{\text{in}}.
\end{cases}
$$

5.2. Expansion of the product in the integral. Let us now focus on the factor

$$
(5.2) \qquad \prod_{\lambda \in V_+^\beta} g_\lambda(y_\lambda) \prod_{V^\Gamma} u_\lambda(y_\lambda)
$$

appearing in the integral $\mathcal{J}$.
5.2.1. The functions $g_\lambda$. For technical considerations related to cancellations at zero in the integral, let us first rewrite for each $\lambda \in V_+^\beta$

$$g_{\lambda}(y) := \tilde{g}_{\lambda}(y) \chi_{z}(y)^{2},$$

where $\chi_z$ is the function introduced in Notation 1.21. We notice that we can apply the Friedman-Ramanujan hypothesis to every term $\tilde{g}_\lambda$ appearing in the product above. Indeed, for $\lambda \in V^\beta$, by the hypothesis on the functions $(f_\lambda)_{\lambda \in V}$ in Notation 1.21, we have that $\tilde{g}_\lambda \in \mathcal{F}^{K_\lambda, N_\lambda}$ and, if $p_\lambda$ denotes its principal term,

$$(5.3) \qquad |\tilde{g}_{\lambda}(y) - p_{\lambda}(y) e^{y}| \le \|\tilde{g}_{\lambda}\|_{\mathcal{F}^{K_{\lambda}, N_{\lambda}}} (1 + y)^{N_{\lambda}-1} e^{y/2}$$

where $p_\lambda$ is a polynomial of degree $\le K_\lambda - 1$. For $\lambda \in \Lambda_{\text{in}}^\beta$, we simply notice that $\tilde{g}_\lambda(y) = \sinh^2(y/2)/\chi_z(y)^2$ which belongs to $\mathcal{F}^{K_\lambda, N_\lambda}$ for the integers $(K_\lambda, N_\lambda) := (1, 1)$, thanks to the fact that $\sinh^2(y/2)$ has a zero of order 2 at 0. Its principal term is $p_\lambda(y)e^y$ with $p_\lambda(y) := 1/4$. We can write the Friedman-Ramanujan hypothesis explicitly for these terms and obtain

$$(5.4) \qquad |\tilde{g}_{\lambda}(y) - p_{\lambda}(y) e^{y}| \le \|\tilde{g}_{\lambda}\|_{\mathcal{F}^{K_{\lambda}, N_{\lambda}}} e^{y/2}$$

where $\|\tilde{g}_\lambda\|_{\mathcal{F}^{K_\lambda, N_\lambda}}$ is a fixed universal constant.

We pick a set of indices $W^\beta \subseteq V_+^\beta$ for which we take the remainder term in the approximations (5.3) and (5.4). This allows us to expand the product as a sum of terms of the form

$$\prod_{\lambda \in V_+^\beta} \chi_z(y_\lambda)^2 \prod_{\lambda \in V_+^\beta \setminus W^\beta} p_\lambda(y_\lambda) e^{y_\lambda} \prod_{\lambda \in W^\beta} \|\tilde{g}_\lambda\|_{\mathcal{F}^{K_\lambda, N_\lambda}} r_\lambda(y_\lambda)$$

where for all $\lambda \in W^\beta$, $|r_\lambda(y_\lambda)| \le (1 + y_\lambda)^{N_\lambda - 1} e^{y_\lambda/2}$.

We then use the fact that, for $\lambda \in \Lambda^\beta$, $y_\lambda$ is a linear combination of the components of the vector $\vec{\theta}$, given by the simple expression (4.15):

$$(5.5) \qquad y_{\lambda} = \sum_{q \in \Theta_{\ell}(\lambda)} \theta_{q}.$$

Note that the sets $\Theta_\ell(\lambda)$ for $\lambda \in V_+^\beta \setminus W^\beta$ are disjoint and their union is $\Theta_\ell(V_+^\beta \setminus W^\beta)$ (note that this is not a priori the case for the sets $\Theta_\ell(\lambda)$ for $\lambda \in \Lambda$, and we here use the fact that $\lambda \in \Lambda^\beta$). We obtain that we can further expand the product as a sum of contribution of the form

$$(5.6) \qquad \mathrm{fn}((\tilde{g}_{\lambda})_{\lambda \in V^{\beta}}) \prod_{\lambda \in V_{+}^{\beta}} \chi_{z}(y_{\lambda})^{2} \prod_{\lambda \in V_{+}^{\beta} \setminus W^{\beta}} q \in \Theta_{\ell}(\lambda) \prod_{q \in \Theta_{\ell}(V_{+}^{\beta} \setminus W^{\beta})} \theta_{q}^{K_{\lambda}} \prod_{\lambda \in W^{\beta}} e^{\theta_{q}} \prod_{\lambda \in W^{\beta}} r_{\lambda}(y_{\lambda})$$

where for $\lambda \in V_+^\beta \setminus W^\beta$, $\sum_{q \in \Theta_\ell(\lambda)} K_q^\lambda \le K_\lambda - 1$. In particular, for $\lambda \in \Lambda_{\text{in}}^\beta$, $K_q^\lambda = 0$. We also note that the constant in front can be bounded by $\prod_{\lambda \in V^\beta} \|\tilde{g}_\lambda\|_{\mathcal{F}^{K_\lambda, N_\lambda}}$.

5.2.2. The functions $u_\lambda$. Similarly, applying the Friedman-Ramanujan hypothesis to the functions $\tilde{g}_\lambda(y) = f_\lambda(y) \sinh(y/2)/\chi_z(y)^2 \in \mathcal{F}^{K_\lambda, N_\lambda}$ for $\lambda \in V^\Gamma$ allows us to replace the functions $u_\lambda$ by an approximation, this time by the polynomial function $p_\lambda$ coming from
the principal term of $\tilde{g}_{\lambda}$, up to an exponentially decaying error. More precisely, using the relation

$$u_{\lambda}(y) = \frac{4\chi_{z}(y)^{2}}{(1 - e^{-y})^{2}}e^{-y}\tilde{g}_{\lambda}(y)$$

together with the hypothesis (1.11) on the function $\chi_{z}$, we see that

$$|u_{\lambda}(y) - p_{\lambda}(y)| \leq 32 \|\tilde{g}_{\lambda}\|_{\mathcal{F}^{K_{\lambda}, N_{\lambda}}} (1 + y)^{N_{\lambda}-1} e^{-y/2}.$$

We now replace each term in the product by this expansion, and obtain it can be rewritten as a sum of terms of the form

$$\prod_{\lambda \in V^{\Gamma} \setminus W_{0}^{\Gamma}} p_{\lambda}(y_{\lambda}) \prod_{\lambda \in W_{0}^{\Gamma}} 32 \|\tilde{g}_{\lambda}\|_{\mathcal{F}^{K_{\lambda}, N_{\lambda}}} r_{\lambda}^{0}(y_{\lambda})$$

where $W_{0}^{\Gamma} \subseteq V^{\Gamma}$ and $|r_{\lambda}^{0}(y)| \leq (1 + y)^{N_{\lambda}-1}e^{-y/2}$ for all $\lambda \in W_{0}^{\Gamma}$.

We now express $p_{\lambda}(y_{\lambda})$ as a linear combination of monomials, and use the expression for $y_{\lambda}$ in terms of $(\vec{L}, \vec{\theta})$ obtained in §4.4. Note that, for these terms which are not components of $\beta$, the expression is more complex, and not a simple linear combination. More precisely, by (4.24),

$$(5.7) \qquad y_{\lambda} = \sum_{n=1}^{m(\lambda)} \tau_{n}^{\lambda} - 2 \log 2 + 2 \log F_{m(\lambda)}(\vec{L}^{\lambda}, \vec{\tau}^{\lambda}) + \mathcal{O}(e^{-y_{\lambda}})$$

where $\vec{L}^{\lambda} = (L_{1}^{\lambda}, \dots, L_{m(\lambda)}^{\lambda})$ and $\vec{\tau}^{\lambda} = (\tau_{1}^{\lambda}, \dots, \tau_{m(\lambda)}^{\lambda})$ are the lengths of the bars and segments $\mathcal{K}$ appearing in the description of $\Gamma_{\lambda}$ as a polygonal curve, and $F_{m(\lambda)}$ is the function defined in (4.20). We can further express the parameters $\tau$ appearing in the sum in (5.7) in terms of $\vec{\theta}$ using (4.25) which states that

$$\tau_{n}^{\lambda} = \sum_{q \in \Theta_{\tau}(\lambda, n)} \theta_{q}.$$

The set of indices involved in writing $\vec{\tau}^{\lambda}$ is $\Theta_{\ell}(\lambda) = \bigcup_{n=1}^{m(\lambda)} \Theta_{\tau}(\lambda, n)$. Altogether, we obtain that the product of the $u_{\lambda}$ terms can be expressed as a sum of terms of the form

$$(5.8) \qquad \mathrm{fn}((\tilde{g}_{\lambda})_{\lambda \in V^{\Gamma}}) \prod_{\lambda \in V^{\Gamma} \setminus W^{\Gamma}} \left[ (\log F_{m(\lambda)}(\vec{L}^{\lambda}, \vec{\tau}^{\lambda}))^{d_{\lambda}} \prod_{q \in \Theta_{\ell}(\lambda)} \theta_{q}^{K_{q}^{\lambda}} \right] \prod_{\lambda \in W^{\Gamma}} r_{\lambda}(\vec{L}, \vec{\theta}^{\lambda})$$

where:

* $W^{\Gamma} \subseteq V^{\Gamma}$ is the set of indices with exponential decay (it contains the set $W_{0}^{\Gamma}$ above, but can contain more elements due to the exponentially decaying term in (5.7));

* for $\lambda \in W^{\Gamma}$, $\vec{\theta}^{\lambda} = (\theta_{q})_{q \in \Theta_{\ell}(\lambda)}$ is the set of $\theta$ parameters required to express $y_{\lambda}$ and

$$(5.9) \qquad |r_{\lambda}(\vec{L}, \vec{\theta}^{\lambda})| \leq (1 + y_{\lambda})^{N_{\lambda}-1} e^{-y_{\lambda}/2};$$

* for $\lambda \in V^{\Gamma} \setminus W^{\Gamma}$, $\sum_{q \in \Theta_{\ell}(\lambda)} K_{q}^{\lambda} + d_{\lambda} \leq K_{\lambda} - 1$;

* the constant factor is bounded by $\prod_{\lambda \in V^{\Gamma}} \|\tilde{g}_{\lambda}\|_{\mathcal{F}^{K_{\lambda}, N_{\lambda}}}$.
5.2.3. Conclusion and substitution in the integral. Taking equations (5.1), (5.6) and (5.8), we obtain that the integral $\mathcal{J}(\ell)$ can be rewritten as a fixed sum of terms of the form

$$ (5.10) \qquad \int_{\ell_Y(\mathbf{c})=\ell} \mathbb{1}_{\mathcal{D}}(\vec{L}, \vec{\theta}) \prod_{\lambda \in V_+^\beta} \chi_z(y_\lambda)^2 \prod_{q \in \Theta_\ell(V_+ \setminus W)} \theta_q^{K_q} \prod_{q \in \Theta_\ell(V_+^\beta \setminus W^\beta)} e^{\theta_q} \\ \qquad \prod_{\lambda \in V^\Gamma \setminus W^\Gamma} (\log F_{m(\lambda)}(\vec{L}^\lambda, \vec{\tau}^\lambda))^{d_\lambda} \prod_{\lambda \in W} r_\lambda(\vec{L}, \vec{\theta}^\lambda) T_{V,m}(\vec{L}, \vec{\theta}) \prod_{i=1}^r \sinh(L_i) \frac{d^r \vec{L} \, d^{2r} \vec{\theta}}{d\ell}, $$

where:

* $V_+ := V \cup \Lambda_{in}^\beta$ and $W := W^\beta \cup W^\Gamma \subseteq V_+$;

* $K_q := \sum_{\lambda \in V_+ \setminus W, \Theta_\ell(\lambda) \not\equiv q} K_q^\lambda$ (with $K_q^\lambda$ defined to be 0 if non already defined);

* $|r_\lambda(\vec{L}, \vec{\theta}^\lambda)| \le (1 + y_\lambda)^{N_\lambda - 1} e^{\alpha_\lambda y_\lambda / 2}$ with $\alpha_\lambda = 1$ if $\lambda \in W^\beta$ and $-1$ if $\lambda \in W^\Gamma$

* the constant factor is bounded by $\text{fn}(r) \prod_{\lambda \in V} \|\tilde{g}_\lambda\|_{\mathcal{F}^{K_\lambda, N_\lambda}}$.

Note that we have

$$ (5.11) \qquad \sum_{q \in \Theta_\ell(V_+ \setminus W)} K_q + \sum_{\lambda \in V^\Gamma \setminus W^\Gamma} d_\lambda \le \sum_{\lambda \in V} (K_\lambda - 1). $$

5.3. **Active and neutral parameters.** Our final aim is to prove that the integral $\mathcal{J}$ is a Friedman-Ramanujan function following the method sketched in §1.3.8 to prove that a convolution $f_1 * f_2$ is a Friedman-Ramanujan function. We recall that, in this case, the idea is to observe that $\mathcal{L}^{K_1+K_2}(f_1 * f_2) = \mathcal{L}^{K_1} f_1 * \mathcal{L}^{K_2} f_2$. Hence, if we choose the indices $K_1$ and $K_2$ so that $\mathcal{L}^{K_i} f_i$ are elements of $\mathcal{R}$ (which we can do if $f_i \in \mathcal{F}$), then we can easily deduce by an upper bound that $\mathcal{L}^{K_1+K_2}(f_1 * f_2) \in \mathcal{R}$, which in turn implies that $f_1 * f_2 \in \mathcal{F}$.

Now, if a function $f_i$ is already an element of $\mathcal{R}$, we do not need to cancel its principal term; in other words, we do not need to apply the operator $\mathcal{L}$ to $f_i$ in order to establish the Friedman-Ramanujan property for $f_1 * f_2$ (we can take $K_i = 0$). In our language, this means that the variable $x_i$ corresponding to $f_i$ in the convolution $f_1 * f_2(x) = \int_{x_1+x_2=x} f_1(x_1)f_2(x_2) \frac{dx_1}{dx_2}$ is a *neutral parameter*. To the contrary, a variable which appears with a principal term in the integral (and will therefore need to be cancelled by applying the operator $\mathcal{L}$) will be called an *active parameter*.

**Notation 5.1.** We partition the set of $\Theta$ into sets of *active and neutral parameters* $\Theta = \Theta_{ac} \sqcup \Theta_{nt}$ where

$$ (5.12) \qquad \Theta_{ac} := \{q \in \Theta_{\ell}(V_{+}^{\beta}) : q \notin \Theta_{\ell}(W)\}. $$

We denote as $\vec{\theta}_{ac} = (\theta_q)_{q \in \Theta_{ac}}$ and $\vec{\theta}_{nt} = (\theta_q)_{q \in \Theta_{nt}}$ the components of $\vec{\theta} = (\theta_q)_{q \in \Theta}$ in those two sets of indices. We will call *neutral parameters* the components of $\vec{\theta}_{nt}$ together with the vector $\vec{L} = (L_1, \dots, L_r)$, and *active parameters* the components of $\vec{\theta}_{ac}$.

In the following, the convolution argument will concern the variables $\vec{\theta}_{ac}$, while $\vec{L}$ and $\vec{\theta}_{nt}$ are kept fixed. We will not need to apply the operator $\mathcal{L}$ with respect to the neutral variables, because their contribution does not yield a principal term in the product (5.10).
**Remark 5.2.** The reason why we do not need to work on the neutral parameters is that they will appear without a factor 2 in our comparison estimate, Proposition 9.3. We have already commented on the importance of factors of 2 in Proposition 1.29, a simple comparison estimate. We saw in Lemma 3.27 how double-filling loops (for which all parameters are neutral) can easily be proven to be Friedman-Ramanujan remainders, by direct inequalities, without any need to apply the operator $\mathcal{L}$.

We now set aside the neutral parameters in (5.10) and consider them as constants. The focus of the remainder of the paper is thus on integrals of the form

$$ (5.13) \qquad \mathrm{Int}(\ell) = \int_{h(\vec{\theta}_{ac})=\ell} \varphi_0(\vec{\theta}_{ac}) \prod_{q \in \Theta_{ac}} \theta_q^{K_q} e^{\theta_q} \frac{d\vec{\theta}_{ac}}{d\ell} $$

where:

* all neutral variables $\vec{L}, \vec{\theta}_{nt}$ are fixed;

* we have the upper bound

$$ (5.14) \qquad \sum_{q \in \Theta_{ac}} K_q \le \sum_{\substack{\lambda \in V_+^\beta \\ \Theta_\ell(\lambda) \not\subseteq \Theta_\ell(W)}} (K_\lambda - 1) \le \sum_{\lambda \in V_+^\beta \setminus W} (K_\lambda - 1); $$

* the weight function $\varphi_0$ is defined as

$$ (5.15) \qquad \varphi_0(\vec{\theta}_{ac}) = \mathbb{1}_{\mathcal{D}}(\vec{L}, \vec{\theta}) \prod_{\lambda \in V_+^\beta} \chi_z(y_\lambda)^2 \prod_{\lambda \in V^\Gamma \setminus W^\Gamma} (\log F_{m(\lambda)}(\vec{L}^\lambda, \vec{\tau}^\lambda))^{d_\lambda} $$

and, for each $\lambda \in V^\Gamma \setminus W^\Gamma$, $d_\lambda \le K_\lambda - 1$;

* the level function $h$ is the length function of $c$,

$$ (5.16) \qquad h(\vec{\theta}_{ac}) = \ell_Y(c) = \sum_{i=1}^{n} \ell_Y(c_i). $$

It will be useful in the following to keep track of the quantities we have taken out of the integral due to depending only on neutral variables:

$$ (5.17) \qquad \mathrm{fn}((\tilde{g}_\lambda)_{\lambda \in V}, r) \prod_{q \in \Theta_\ell(V_+) \cap \Theta_{nt}} \theta_q^{K_q} \prod_{\lambda \in W} r_\lambda(\vec{L}, \vec{\theta}_{nt}) T_{V,m}(\vec{L}, \vec{\theta}_{nt}) \prod_{i=1}^r \sinh(L_i). $$

Comparing with the forthcoming definition (6.3), this appears as the $(h, \varphi_0)$-convolution of the Friedman-Ramanujan functions $(\theta_q^{K_q} e^{\theta_q})_{q \in \Theta_{ac}}$. The reason why we use the word *convolution* is that the function $h(\vec{\theta}_{ac})$ behaves in many respects like the sum $\sum_{q \in \Theta_{ac}} \theta_q$ (plus a function of the neutral variables), and the function $\varphi_0$ will be shown to have “small” derivatives in some sense, so it can be thought of as close to being constant. The purpose of §6 is to make these notions precise.

## 6. PSEUDO-CONVOLUTIONS AND STABILITY OF THE FR HYPOTHESIS

In this section, we introduce a notion of pseudo-convolution $f_1 \star \dots \star f_n|_\varphi^h$ of functions $(f_i)_{1 \le i \le n}$ with the level function $h$ and the weight $\varphi$, which generalizes the usual convolution $f_1 * \dots * f_n$, and is motivated by the form of (5.13). We then study the way the
operator $\mathcal{L} = \text{id} - \mathcal{P}$ acts on pseudo-convolutions in §6.2. This will provide statements of the following kind: if $f_1, \dots, f_n$ are Friedman-Ramanujan functions, and if $(h, \varphi)$ satisfy adequate assumptions, then $f_1 \star \dots \star f_n|_{\varphi}^h$ is also a Friedman-Ramanujan function; see Theorem 6.28. Roughly speaking, the function $h(x_1, \dots, x_n)$ is assumed to be close to $x_1 + \dots + x_n + C$ and $\varphi$ to a constant (in a certain topology defined in §6.3).

**6.1. Definition of the notion of pseudo-convolution.** We make the following definition.

**Definition 6.1.** Take $n \ge 1$. Let $\varphi : \mathbb{R}^n \to \mathbb{R}$ be a locally bounded, measurable function, and $h$ be a real-valued measurable function defined on a set containing the support $\text{supp}(\varphi)$ of $\varphi$. We assume that for every compact subset $K \subset \mathbb{R}$, the set $h^{-1}(K) \cap \text{supp}(\varphi)$ has finite measure.

We call $(h, \varphi)$-convolution of $n$ locally integrable functions $f_1, \dots, f_n : \mathbb{R} \to \mathbb{R}$ the pushforward of the product measure $\varphi(x_1, \dots, x_n) \prod_{i=1}^n f_i(x_i) \, dx_i$ by the map $h$. In other words, it is the distribution $\nu$ on $\mathbb{R}_{\ge 0}$ defined by the fact that, for every continuous function $F : \mathbb{R}_{\ge 0} \to \mathbb{R}$ with compact support,

$$ \int_{\mathbb{R}} F(x)\nu(dx) = \int_{\mathbb{R}^n} F(h(x_1, \dots, x_n)) \varphi(x_1, \dots, x_n) \prod_{i=1}^n f_i(x_i) \, dx_i. $$

In our applications, the functions $h$ and $\varphi$ will always be much more than measurable. More precisely, throughout this article, we suppose the following.

**Notation 6.2 (Assumptions on $(\varphi, h)$).** Take $a > 0$. We assume the following.

* $\varphi : \mathbb{R}^n \to \mathbb{R}$ is a $C^1$ function of support included in $(a, +\infty)^n$.

* $h : [a, +\infty)^n \to \mathbb{R}$ is a $C^1$ function.

* For any index $1 \le i \le n$ and any fixed $\hat{\mathbf{x}}_i = (x_j)_{j \ne i} \in [a, +\infty)^{n-1}$, the function

$$ h_{i}^{\hat{\mathbf{x}}_{i}} : x_{i} \mapsto h(x_{1}, \dots, x_{n}) $$

is a $C^1$-diffeomorphism from $[a, +\infty)$ onto its image; we denote $\ell \mapsto h_i^{-1}(\ell, \hat{\mathbf{x}}_i)$ its inverse.

* The derivative $(\ell, \hat{\mathbf{x}}_i) \mapsto \partial_{\ell} h_i^{-1}(\ell, \hat{\mathbf{x}}_i)$ is bounded on its domain of definition.

The following proposition follows directly from the change of variable formula:

**Proposition 6.3.** Under the hypotheses of Notation 6.2, the $(h, \varphi)$-convolution $\nu$ of any $n$ locally integrable functions $f_1, \dots, f_n$ admits a density with respect to the Lebesgue measure, with the following expression

$$ (6.1) \quad \frac{d\nu}{d\ell}(\ell) = \int \varphi(x_1, \dots, h_i^{-1}(\ell, \hat{\mathbf{x}}_i), \dots, x_n) f_i(h_i^{-1}(\ell, \hat{\mathbf{x}}_i)) \prod_{j \ne i} f_j(x_j) \partial_{\ell} h_i^{-1}(\ell, \hat{\mathbf{x}}_i) \, d\hat{\mathbf{x}}_i $$

where $d\hat{\mathbf{x}}_i := \prod_{j \ne i} dx_j$. In particular this is independent of the choice of the index $i$.

**Notation 6.4.** The expression (6.1) will be denoted $f_1 \star \dots \star f_n|_{\varphi}^h$.

If $x = (x_1, \dots, x_n)$ are related by the implicit relation $h(x) = \ell$, we will denote $\frac{dx_1 \dots dx_n}{d\ell} = \frac{dx}{d\ell}$ any of the equivalent expressions of the measure

$$ (6.2) \qquad \frac{dx_1 \dots dx_n}{d\ell} = \frac{dx}{d\ell} := \partial_{\ell} h_i^{-1}(\ell, \hat{\mathbf{x}}_i) \, d\hat{\mathbf{x}}_i. $$
This corresponds to the disintegration of the Lebesgue measure $dx = dx_1... dx_n$ on the level-sets of $h$. Thus, we can write

$$ (6.3) \qquad f_1 \star \dots \star f_n|_{\varphi}^h(\ell) = \int_{h(x)=\ell} \varphi(x) \prod_{i=1}^n f_i(x_i) \frac{dx}{d\ell}. $$

**Example 6.5.** The usual convolution corresponds to $h(x) = x_1 + \dots + x_n$, and either $\varphi \equiv 1$, or $\varphi(x) = \mathbb{1}_{x_1 \ge 0, \dots, x_n \ge 0}$ (in the case of convolutions of functions defined on $\mathbb{R}_{\ge 0}$).

**Remark 6.6 (Associativity).** The associativity of the usual convolution follows from associativity of the addition. For a general function $h$, the associativity rule may be replaced by formulae such as

$$ f_1 \star \dots \star f_n|_{\varphi}^h(\ell) = \int f_2 \star \dots \star f_n|_{\varphi_{x_1}}^{h_{x_1}}(\ell) f_1(x_1) dx_1 $$

where $\varphi_{x_1}$ (and similarly $h_{x_1}$) is the function of $n-1$ variables defined by freezing the first variable to $x_1$:

$$ \varphi_{x_1}(x_2, x_3, \dots, x_n) := \varphi(x_1, x_2, x_3, \dots, x_n). $$

This simply expresses the fact that we can do the convolution $f_2 \star \dots \star f_n|_{\varphi_{x_1}}^{h_{x_1}}$ with respect to $x_2, \dots, x_n$ (while $x_1$ is kept fixed) and then integrate with respect to $f_1(x_1) dx_1$, to obtain the pseudo-convolution $f_1 \star \dots \star f_n|_{\varphi}^h$.

In the rest of the article, we will need to use this associativity rule for multiple variables and pairs $(h, \varphi)$. In order to avoid cumbersome notations, we will often omit the index $x_1$ in the freezed functions $h_{x_1}$ and $\varphi_{x_1}$ when the context allows to avoid any confusion.

**6.2. Action of the operator $\mathcal{L}_\ell$ on pseudo-convolution.** In §1.3.7 we defined two operators acting on locally integrable functions, $P_x = \int_0^x$ (taking a primitive) and $\mathcal{L}_x = \text{id} - P_x$. Since we are dealing with functions of several variables $(x_1, \dots, x_n)$, the subscript $x_i$ will serve to indicate that the primitive is taken with respect to the variable $x_i$.

The goal of this section is to compare $\mathcal{L}_\ell^{K_1+\dots+K_n}(f_1 \star \dots \star f_n|_{\varphi}^h)$ with $(\mathcal{L}_{x_1}^{K_1} f_1) \star \dots \star (\mathcal{L}_{x_n}^{K_n} f_n)|_{\varphi}^h$. With the usual convolution on functions defined on $\mathbb{R}$, we have the exact identity

$$ (6.4) \qquad \mathcal{L}_\ell^{K_1+\dots+K_n}(f_1 \star \dots \star f_n) = (\mathcal{L}_{x_1}^{K_1} f_1) \star \dots \star (\mathcal{L}_{x_n}^{K_n} f_n). $$

Note that, in the case of functions defined on $\mathbb{R}_{\ge 0}$ only, there are additional boundary terms coming from $x_i = 0$ in this relation (we leave this observation as an exercise to the reader).

For general functions $h$ and $\varphi$, the two expressions $\mathcal{L}_\ell^{K_1+\dots+K_n}(f_1 \star \dots \star f_n|_{\varphi}^h)$ and $(\mathcal{L}_{x_1}^{K_1} f_1) \star \dots \star (\mathcal{L}_{x_n}^{K_n} f_n)|_{\varphi}^h$ will not be equal. We provide in Theorem 6.16 a detailed expression for the error term produced by replacing one by the other.

We saw in §1.3.8 how the identity (6.4) can be exploited to prove the stability of the Friedman-Ramanujan hypothesis by convolution. In §6.4, we will use the results of this section to provide a similar stability result, under certain hypotheses on the level function $h$ and the weight $\varphi$.
### 6.2.1. Basic calculus in one variable.
We gather in the following lemmas the behavior of $\mathcal{P}$ and $\mathcal{L}$ under change of variables and multiplication. The change of variable formula yields:

**Lemma 6.7.** Let $h : (0, +\infty) \to (h_{\inf}, +\infty)$ be a $\mathcal{C}^1$-diffeomorphism with $h_{\inf} \ge 0$, and $h^{-1}$ be its inverse. Then, for any continuous function $G : \mathbb{R}_{\ge 0} \to \mathbb{R}$,

$$ \mathcal{P}_{\ell}(G \circ h^{-1}) = (\mathcal{P}_{x}G) \circ h^{-1} + R_{x}G \circ h^{-1} $$

where $R_x G(x) := \mathcal{P}_x(G(\partial_x h - 1))$.

*Proof.* This is just a way of writing

$$ \int_{h_{\inf}}^{\ell} G(h^{-1}(u)) \, du = \int_{0}^{h^{-1}(\ell)} G(t) \, dt + \int_{0}^{h^{-1}(\ell)} G(t)(h'(t) - 1) \, dt. $$

It is important to note that $\ell \mapsto G(h^{-1}(\ell))$ is only defined on $(h_{\inf}, +\infty)$, so the operator $\mathcal{P}_{\ell}$ should, a priori, be defined as $\int_{h_{\inf}}^{\ell} \cdot$. However, if we extend $G(h^{-1}(\ell))$ to be 0 if $\ell \le h_{\inf}$, the formula remains true taking $\mathcal{P}_{\ell} = \int_{0}^{\ell} \cdot$.

□

Integration by parts can be written in the following form:

**Lemma 6.8.** For any continuous function $G : \mathbb{R}_{\ge 0} \to \mathbb{R}$, any $\mathcal{C}^1$ function $\varphi : \mathbb{R}_{\ge 0} \to \mathbb{R}$,
$$ \mathcal{P}_x(\varphi G) = \varphi \mathcal{P}_x G - \mathcal{P}_x(\partial_x \varphi \mathcal{P}_x G). $$

*Proof.* This just says that

$$ \int_{0}^{x} \varphi(t) G(t) \, dt = \varphi(x) \int_{0}^{x} G(t) \, dt - \int_{0}^{x} \varphi'(t) \left( \int_{0}^{t} G(s) \, ds \right) dt $$

which is indeed true by integration by parts.

□

By combining both previous formulas and iterating them, we obtain:

**Proposition 6.9.** With the notations of Lemma 6.7 and 6.8, for any $K \ge 1$,

$$ \begin{aligned} \mathcal{L}_{\ell}^{K}((\varphi G) \circ h^{-1}) = & (\varphi \mathcal{L}_{x}^{K} G) \circ h^{-1} \\ & - \sum_{t=0}^{K-1} \mathcal{L}_{\ell}^{K-1-t}((\varphi R_x(\mathcal{L}_x^t G)) \circ h^{-1}) \\ & + \sum_{t=0}^{K-1} \mathcal{L}_{\ell}^{K-1-t} \mathcal{P}_{\ell}(\partial_{\ell}(\varphi \circ h^{-1}) \mathcal{P}_{\ell}(\mathcal{L}_x^t G \circ h^{-1})). \end{aligned} $$

*Proof.* For $K=1$, a combination of Lemmas 6.7 and 6.8 (w.r.t. the variable $\ell$) yields

$$ \begin{aligned} \mathcal{L}_{\ell}((\varphi G) \circ h^{-1}) = & (\varphi G) \circ h^{-1} - \mathcal{P}_{\ell}(\varphi \circ h^{-1} \cdot G \circ h^{-1}) \\ \stackrel{6.8}{=} & (\varphi G) \circ h^{-1} - \varphi \circ h^{-1} \cdot \mathcal{P}_{\ell}(G \circ h^{-1}) + \mathcal{P}_{\ell}(\partial_{\ell}(\varphi \circ h^{-1}) \mathcal{P}_{\ell}(G \circ h^{-1})) \\ \stackrel{6.7}{=} & (\varphi \mathcal{L}_x G) \circ h^{-1} - (\varphi R_x G) \circ h^{-1} + \mathcal{P}_{\ell}(\partial_{\ell}(\varphi \circ h^{-1}) \mathcal{P}_{\ell}(G \circ h^{-1})) \end{aligned} $$

which gives the expression of the commutator $[L, T]$ for with $L = \mathcal{L}$ and $TG = (\varphi G) \circ h^{-1}$:

$$ [L, T]G = -(\varphi R_x G) \circ h^{-1} + \mathcal{P}_{\ell}(\partial_{\ell}(\varphi \circ h^{-1}) \mathcal{P}_{\ell}(G \circ h^{-1})). $$
We then use the algebraic identity

$$L^K T = T L^K + \sum_{t=0}^{K-1} L^{K-1-t} [L, T] L^t.$$

to obtain the claimed formula for general K.

<figure><img src="image_24.png" /></figure>

### 6.2.2. Application to the first variable.
Proposition 6.9 allows us to compare the action $\mathcal{L}_{\ell}^{K}(f_1 \star \dots \star f_n|_{\varphi}^{h})$ of $\mathcal{L}^{K}$ on a pseudo-convolution with $(\mathcal{L}_{x_1}^{K} f_1) \star f_2 \dots \star f_n|_{\varphi}^{h}$, which is a first step towards the aim of this subsection. We prove the following.

**Proposition 6.10.** For any family of continuous functions $(f_j)_{1 \le j \le n}$ and any $K \ge 1$, the quantity $\mathcal{L}_{\ell}^{K}(f_1 \star \dots \star f_n|_{\varphi}^{h})(\ell)$ can be expressed as

$$(6.5) \qquad (\mathcal{L}_{x_1}^{K} f_1) \star f_2 \star \dots \star f_n|_{\varphi}^{h} \frac{\partial h}{\partial x_1}(\ell) + \mathcal{L}_{\ell}^{K}(f_1 \star \dots \star f_n|_{\varphi(1-\frac{\partial h}{\partial x_1})}^{h})(\ell)$$

$$(6.6) \qquad - \sum_{t=0}^{K-1} \mathcal{L}_{\ell}^{K-1-t} \left( \int f_2 \star \dots \star f_n \left|_{\varphi \frac{\partial h}{\partial x_1}}^{h_{x_1}} R_{x_1}(\mathcal{L}_{x_1}^t f_1) \right.(\ell) \mathrm{d}x_1 \right)$$

$$(6.7) \qquad + \sum_{t=0}^{K-1} \mathcal{L}_{\ell}^{K-1-t} \mathcal{P}_{\ell} \left( \int f_2 \star \dots \star f_n \left|_{\frac{\partial \varphi}{\partial x_1}}^{h_{x_1}} \mathcal{P}_{x_1}(\mathcal{L}_{x_1}^t f_1 \frac{\partial h}{\partial x_1}) \right.(\ell) \mathrm{d}x_1 \right).$$

**Remark 6.11.** For the usual convolution, we have $h(x_1, \dots, x_n) = x_1 + \dots + x_n$ and $\varphi = 1$, so $\frac{\partial \varphi}{\partial x_1} = 0$ and $\frac{\partial h}{\partial x_1} = 1$ (and hence $R_{x_1} = 0$). We see that only the first term remains:

$$\mathcal{L}_{\ell}^{K}(f_1 \star \dots \star f_n) = (\mathcal{L}_{x_1}^{K} f_1) \star f_2 \star \dots \star f_n.$$

In the case $\varphi(x_1, \dots, x_n) = \mathbb{1}_{x_1 \ge 0, \dots, x_n \ge 0}$, we get a boundary term with a Dirac mass at $x_1 = 0$, it corresponds to the term (6.7).

*Proof.* We start by writing our pseudo-convolution as:

$$(6.8) \qquad f_1 \star \dots \star f_n|_{\varphi}^{h}(\ell) = \int \varphi(h_1^{-1}(\ell, \hat{\mathbf{x}}_1), \hat{\mathbf{x}}_1) f_1(h_1^{-1}(\ell, \hat{\mathbf{x}}_1)) \prod_{i=2}^{n} f_i(x_i) \partial_{\ell} h_1^{-1}(\ell, \hat{\mathbf{x}}_1) \mathrm{d}x_2 \dots \mathrm{d}x_n.$$

We break (6.8) into two terms, writing

$$\partial_{\ell} h_1^{-1}(\ell, \hat{\mathbf{x}}_1) = 1 + \partial_{\ell} h_1^{-1}(\ell, \hat{\mathbf{x}}_1) \left(1 - \frac{\partial h}{\partial x_1}(h_1^{-1}(\ell, \hat{\mathbf{x}}_1))\right).$$

We observe that the second term on the right-hand-side yields exactly the second announced term in (6.5). We are therefore left to apply the operator $\mathcal{L}_{\ell}^{K}$ to the function

$$\int \varphi(h_1^{-1}(\ell, \hat{\mathbf{x}}_1), \hat{\mathbf{x}}_1) f_1(h_1^{-1}(\ell, \hat{\mathbf{x}}_1)) \prod_{i=2}^{n} f_i(x_i) \mathrm{d}x_2 \dots \mathrm{d}x_n$$
which we do by applying Proposition 6.9 with respect to the variable $x_1$ (while $x_2, \dots, x_n$ are frozen parameters) and the function $h = h_1$. We obtain:

$$ (6.9) \qquad \mathcal{L}_{\ell}^{\mathrm{K}}((\varphi f_1) \circ h_1^{-1}) = \varphi \circ h_1^{-1} (\mathcal{L}_{x_1}^{\mathrm{K}} f_1) \circ h_1^{-1} $$

$$ (6.10) \qquad - \sum_{t=0}^{K-1} \mathcal{L}_{\ell}^{K-1-t} (\varphi \circ h_1^{-1} (R_{x_1}(\mathcal{L}_{x_1}^t f_1)) \circ h_1^{-1}) $$

$$ (6.11) \qquad + \sum_{t=0}^{K-1} \mathcal{L}_{\ell}^{K-1-t} \mathcal{P}_{\ell} (\partial_{\ell} (\varphi \circ h_1^{-1}) \mathcal{P}_{\ell} (\mathcal{L}_{x_1}^t f_1 \circ h_1^{-1})). $$

Integrating the terms (6.9) and (6.10) along $x_2, \dots, x_n$ respectively yield the first term of (6.5) and the term (6.6) in the theorem. For the last term (6.11), we use the chain rule

$$ \partial_{\ell} (\varphi \circ h_1^{-1}) = \left( \frac{\partial \varphi}{\partial x_1} \circ h_1^{-1} \right) \partial_{\ell} h_1^{-1}, $$

and the change of variable

$$ \mathcal{P}_{\ell}(\mathcal{L}_{x_1}^t f_1 \circ h_1^{-1}) = \mathcal{P}_{x_1}(\mathcal{L}_{x_1}^t f_1 \frac{\partial h}{\partial x_1}) \circ h_1^{-1} $$

which leads to the claim once again by integration. $\square$

**Notation 6.12.** For $1 \le j \le n$, $x = (x_1, \dots, x_n)$ and $0 \le t \le K-1$, introduce the functions

$$ F_j^K(x) = \mathcal{L}_{x_j}^K f_j(x_j) \frac{\partial h}{\partial x_j}(x) $$

$$ G_j(x) = f_j(x_j) \left(1 - \frac{\partial h}{\partial x_j}(x)\right) $$

$$ H_j^t(x) = -R_{x_j}(\mathcal{L}_{x_j}^t f_j)(x_j) \frac{\partial h}{\partial x_j}(x) $$

$$ I_j^t(x) = \mathcal{P}_{x_j}(\mathcal{L}_{x_j}^t f_j \frac{\partial h}{\partial x_j})(x). $$

We can rephrase Proposition 6.10 as follows:

**Corollary 6.13.** For any family of continuous functions $(f_j)_{1 \le j \le n}$ and any integer $K \ge 1$, $\mathcal{L}_{\ell}^K(f_1 \star \dots \star f_n|_{\varphi}^h)$ is the sum of the following functions, or of the images of these functions under some power $\mathcal{L}_{\ell}^t$ where $0 \le t \le K-1$:

* $\int f_2 \star \dots \star f_n|_{T_1 \varphi}^{h_{x_1}}(\ell) dx_1$ where the operator $T_1$ is the multiplication by the function $F_1^K, G_1$ or $H_1^t$ with $0 \le t \le K-1$; or

* $\mathcal{P}_{\ell}(\int f_2 \star \dots \star f_n|_{T_1 \varphi}^{h_{x_1}}(\ell) dx_1)$ where $T_1 = I_1^t \partial_1$ with $0 \le t \le K-1$, i.e. $T_1$ is the derivative with respect to $x_1$ followed by multiplication by $I_1^t$.

*Proof.* We observe that by the associativity rule, Theorem 6.10 can also be written

$$ \begin{aligned} \mathcal{L}_{\ell}^{\mathrm{K}}(f_1 \star \dots \star f_n|_{\varphi}^{h}) (\ell) = & \int f_2 \star \dots \star f_n|_{F_1^{\mathrm{K}} \varphi}^{h_{x_1}}(\ell) dx_1 + \mathcal{L}_{\ell}^{\mathrm{K}} \left( \int f_2 \star \dots \star f_n|_{G_1 \varphi}^{h_{x_1}}(\ell) dx_1 \right) \\ & + \sum_{t=0}^{K-1} \mathcal{L}_{\ell}^{\mathrm{K}-1-t} \int f_2 \star \dots \star f_n(\ell)|_{H_1^t \varphi}^{h_{x_1}} dx_1 \\ & + \sum_{t=0}^{K-1} \mathcal{L}_{\ell}^{\mathrm{K}-1-t} \mathcal{P}_{\ell} \left( \int f_2 \star \dots \star f_n|_{I_1^t \partial_1 \varphi}^{h_{x_1}}(\ell) dx_1 \right). \end{aligned} $$
The conclusion then follows directly.

☐

6.2.3. The multi-variable result. We shall now successively apply Corollary 6.13 in the variables $x_1, \dots, x_n$. In order to do so, the following notations for multi-indices will be helpful.

**Notation 6.14 (Multi-indices).** If $\alpha = (\alpha_1, \dots, \alpha_n) \in \mathbb{Z}_{\ge 0}^n$ is a multi-index, we use the standard notations $|\alpha| = \sum_{i=1}^n \alpha_i$, $\partial^\alpha = \prod_{i=1}^n \partial_i^{\alpha_i}$ (with $\partial_i = \frac{\partial}{\partial x_i}$) and $\alpha \cdot x = \sum_{i=1}^n \alpha_i x_i$.

Most of the time, all of the coefficients will be $\alpha_i = 0$ or 1, that is to say, we will manipulate differential operators of order at most one in each variable. This motivates the following convention.

**Notation 6.15.** If $\pi$ is a subset of $\{1, \dots, n\}$, we identify $\pi$ with the multi-index $\alpha = \mathbb{1}_\pi$ (the indicator function of $\pi$), and hence denote $\partial^\pi = \partial^{\mathbb{1}_\pi} = \prod_{i \in \pi} \partial_i$. Then, $\mathbb{1}_\pi \cdot x = \sum_{i \in \pi} x_i$.

We now have the following.

**Theorem 6.16.** For any family of continuous functions $(f_j)_{1 \le j \le n}$ and any integers $(K_j)_{1 \le j \le n}$ with $K_j \ge 1$, if $K = \sum_{j=1}^n K_j$, then the function $\mathcal{L}_\ell^K (f_1 \star \dots \star f_n|_\varphi^h)$ can be expressed as the sum of the following functions, or their images under $\mathcal{L}_\ell^t \mathcal{P}_\ell^{t'}$ for $0 \le t \le K-n$ and $0 \le t' \le n$,

$$ (6.12) \qquad \mathrm{Int}_\pi(\ell) := \int_{h(x)=\ell} \partial^{\pi_0} \varphi(x) \prod_{j=1}^n \partial^{\pi_j} \Phi_j(x) \frac{dx}{d\ell} $$

where:

* $\pi = (\pi_j)_{0 \le j \le n}$ is a family of disjoint subsets of $\{1, \dots, n\}$ such that $j \notin \pi_j$ for all $j$;

* if $B := \bigsqcup_{j=0}^n \pi_j$, then for any $1 \le j \le n$,

$$ (6.13) \qquad \Phi_j = \begin{cases} F_j^{K_j}, & G_j \text{ or } H_j^{t''} \text{ with } 0 \le t'' \le K_j - 1 & \text{if } j \notin B \\ I_j^{t''} & \text{with } 0 \le t'' \le K_j - 1 & \text{if } j \in B. \end{cases} $$

**Remark 6.17.** The set $B$ corresponds to the set of “bad” parameters, i.e. parameters for which $\Phi_j = I_j^{t''}$ (we will see in §6.4.2 why these terms are more problematic). It is also the set of variables w.r.t. which a derivative appears in (6.12).

**Remark 6.18.** The number of possible terms of form (6.12) is $\mathrm{fn}(n, K)$.

The proof is an induction on the number of variables to which the operators are applied (Corollary 6.13 corresponding to one variable). More precisely, we prove the following result, and deduce Theorem 6.16 by taking $k = n$.

**Lemma 6.19.** With the notations of Theorem 6.16, for any $k \in \{1, \dots, n\}$, the function $\mathcal{L}_\ell^{K_1+\dots+K_k} (f_1 \star \dots \star f_n|_\varphi^h)$ can be expressed as the sum of the following functions, or their images under $\mathcal{L}_\ell^t \mathcal{P}_\ell^{t'}$ for $0 \le t \le \sum_{j=1}^k (K_j - 1)$ and $0 \le t' \le k$,

$$ (6.14) \qquad \mathrm{Int}_\pi^k(\ell) := \int_{h(x)=\ell} \partial^{\pi_0} \varphi(x) \prod_{j=1}^k \partial^{\pi_j} \Phi_j(x) \prod_{j=k+1}^n f_j(x_j) \frac{dx}{d\ell} $$