*Proof.* We prove the result by induction on the total number $n$ of problematic bars of $\mathbf{D}^{(2)}$. The case $n = 0$ is obvious as we can take $\mathbf{D}^{\text{opt}} = \mathbf{D}^{(2)}$.

Assume that $n \ge 1$, and that the expected property is known until rank $n-1$. By hypothesis, the diagram $\mathbf{D}^{(2)}$ contains a problematic bar $B_q$. Because the bar is problematic regardless of its orientation, we assume w.l.o.g. that $\text{Sign}(q) = +$. By construction of $\mathbf{D}^{(2)}$, the bar $B_q$ goes from one component $\beta'_\lambda$ to itself and is homotopically trivial with gliding endpoints. Denote by $D$ the topological disc bordered by $B_q$ and $\beta'_\lambda$. One of the simple portions adjacent to $B_q$ is shielded. To fix ideas, we assume it is $\mathcal{I}_q$.

A representative $\mathbf{c}^{\text{op}}$ of the homotopy class of $\mathbf{c}$ may be obtained by applying the reconstitution procedure, and allows to write the homotopy class of $\mathbf{c}$ as an alternating concatenation of bars and simple portions. Denote $i_1 = \text{Comp}(q)$ and $i_2 = \text{Comp}(\iota q)$ so that the bars $B_q$ and $B_{\iota q}$ belong to the components $\mathbf{c}_{i_1}^{\text{op}}$ and $\mathbf{c}_{i_2}^{\text{op}}$ of $\mathbf{c}^{\text{op}}$ respectively.

Recall that $B_{\iota q}$ is the same bar as $B_q$ with either the same or the reverse orientation. We treat the case where it has reverse orientation, the other case being similar modulo changes of notation. The local situation near the bar $B_q$ is shown on Figure 42.

<figure><img src="image_55.png" /><figcaption>FIGURE 42. The local situation near the bar $B_q$.</figcaption></figure>

We introduce a new surface $S_\circ \subseteq S$ by taking a regular neighbourhood $\mathcal{N}$ of $\mathbf{D}^{(2)}$ in $S$ and gluing to $\mathcal{N}$ all connected components of $S \setminus \mathbf{D}^{(2)}$ that are contractible, except the connected component contained in the disc $D$ and touching the bar $B_q$.

In the diagram $\mathbf{D}^{(2)}$, now viewed as a diagram on $S_\circ$, the bar $B_q$ is non-problematic. We can therefore apply the induction hypothesis at rank $n-1$ to the multi-loop $\mathbf{c}$ on $S_\circ$. We obtain a good extended diagram $\mathbf{D}_\circ^{\text{opt}}$ on $S_\circ$, containing all non-problematic bars of $(S_\circ, \mathbf{D}^{(2)})$ (and in particular $B_q$), and such that $\mathbf{c}$ is homotopic on $S_\circ$ to a multi-loop stemming from $\mathbf{D}_\circ^{\text{opt}}$.

In $S_\circ$ the diagram $\mathbf{D}_\circ^{\text{opt}}$ is good, and in $S$ it has exactly one problematic bar, namely $B_q$. We shall define the desired diagram $\mathbf{D}^{\text{opt}}$ by replacing $B_q$ by a new non-problematic bar $B_{\text{new}}$.

Applying the reconstitution procedure to $\mathbf{D}_\circ^{\text{opt}}$, for each $1 \le i \le n$, we obtain a representative $\mathbf{c}_i^\circ$ of the homotopy class of $\mathbf{c}_i$ as a concatenation of bars and simple portions of $\mathbf{D}_\circ^{\text{opt}}$. The construction will consist in replacing the two components $\mathbf{c}_{i_1}^\circ$ and $\mathbf{c}_{i_2}^\circ$, containing $B_q$ and $B_{\iota q}$ respectively, by new representatives $\mathbf{c}_{i_1}^{\text{opt}}$ and $\mathbf{c}_{i_2}^{\text{opt}}$. The other components $(\mathbf{c}_i^\circ)_{i \ne i_1, i_2}$ will stay untouched.
Lifting the situation to the hyperbolic plane, for $l \in \{1, 2\}$, we can write the lift $\tilde{c}_{i_l}^l$ of $c_{i_l}^l$ as an infinite concatenation $\cdot_{k \in \mathbb{Z}} \tilde{p}_k^l$ where, for each $k$, $\tilde{p}_k^l$ is the concatenation of lifts $\tilde{B}_k^l$ and $\tilde{\mathcal{I}}_k^l$ of a bar and a simple portion of $\mathbf{D}_\bullet^{\text{opt}}$. We pick the indices so that $\tilde{p}_0^1$ and $\tilde{p}_0^2$ contain a lift of $B_q$ and $B_{tq}$ respectively.

Denote by $\tilde{\beta}_k^l$ the complete geodesic on $\mathbb{H}^2$ containing $\tilde{\mathcal{I}}_k^l$. We have $\tilde{\beta}_0^1 = \tilde{\beta}_0^2$ and we call $m > 0$ the smallest integer such that $\tilde{\beta}_m^1 \neq \tilde{\beta}_m^2$. By the minimality condition for $\mathbf{D}_\bullet^{\text{opt}}$, all $k \in \{0, \dots, m-1\}$ are crossing indices. We define the bar $\tilde{B}_{\text{new}}$ on $\mathbb{H}^2$ as the segment going from $t(\tilde{B}_m^1)$ to $t(\tilde{B}_m^2)$, and let $B_{\text{new}}$ be its projection to $\mathbf{S}$. See Figure 43. The desired diagram $\mathbf{D}^{\text{opt}}$ is defined from $\mathbf{D}_\bullet^{\text{opt}}$ by removing $B_q$ and adding the non-problematic bar $B_{\text{new}}$.

<figure><img src="image_56.png" /><figcaption>FIGURE 43. The new bar $B_{\text{new}}$ replacing $B_q$.</figcaption></figure>

<figure><img src="image_57.png" /></figure>

**Remark 11.22.** At each induction step the number of unshielded simple portions of $\mathbf{D}^{\text{opt}}$ in $\mathbf{S}$ stays the same as for the original $\mathbf{D}^{(2)}$, hence it is smaller that $2\chi(\mathbf{S})$.

11.4. **Sketch of proof for general topologies.** We now explain how to prove Theorem 1.22 for general multi-loops $\mathbf{c}$.

11.4.1. *Coordinates on Teichmüller spaces.* Since the diagram $\mathbf{D}'$ represents a generalized eight $\mathbf{c}'$ in $\mathbf{S}'$, we can use it to define coordinates $(\tilde{L}, \tilde{\theta}) \in \mathbb{R}_{\ge 0}^{r'} \times \mathbb{R}^{\Theta'}$ on $\mathcal{T}_{g_{\mathbf{S}'}, n_{\mathbf{S}'}}^*$.

The surface $\mathbf{S}$ is obtained by gluing cylinders to $\mathbf{S}'$. This gives rise to coordinates $(\tilde{L}, \tilde{\theta}, \tilde{\alpha})$ on $\mathcal{T}_{g_{\mathbf{S}}, n_{\mathbf{S}}}^*$, obtained by adding some twist coordinates $\tilde{\alpha}$ along the boundary of these cylinders, and getting rid of an equivalent number of parameters $\theta_q$ that have become redundant (corresponding to components of $\beta'$ homotopic in $\mathbf{S}$). Note that $\mathcal{T}_{g_{\mathbf{S}'}, n_{\mathbf{S}'}}^*$ and $\mathcal{T}_{g_{\mathbf{S}}, n_{\mathbf{S}}}^*$ have the same dimension, that is, $3\chi(\mathbf{S})$.
Recall that certain simple portions of $c'$ are declared to be a priori neutral. The corresponding parameters $\theta_q$ are a priori neutral coordinates. A priori neutral parameters, as well as all the twist parameters $\vec{\alpha}$, will be neutral parameters throughout this discussion.

**Remark 11.23.** If $c$ is double-filling, all parameters $(\theta_q)_{q \in \Theta}$ are a priori neutral.

11.4.2. *Expression of $\ell_Y(c)$.* Denote by $D^{\text{opt}} = (\beta^{\text{opt}}, B^{\text{opt}})$ the good extended diagram constructed in Theorem 11.20. The reconstitution procedure, applied to the diagram $D^{\text{opt}}$, leads to a representative of the homotopy class of $c$ in the form of an alternance of bars and segments similar to (2.7).

Since none of the bars of $D^{\text{opt}}$ is homotopically trivial with gliding endpoints, we can replace the writing (2.7) by a geodesic representative, as described in §4.2.1. Indeed, we pick the representative of the metric $Y \in T_{g_S, n_S}^*$ such that $\beta'$ is a multi-geodesic, and glide the bars $B_q^{\text{opt}}$ to their orthogeodesic representatives $\bar{B}_q$, and then take a representative of $c$ of the form (4.3). In the special case of multiple components $\beta'_i$ in the same homotopy class, their geodesic representatives are collapsed to the same curve, and the cylindrical bars $\bar{B}_q$ running between them are reduced to a point. Theorem 4.6 then provides an expression for the length of each component $c_i$ with $1 \le i \le n$, with the bars and segments corresponding to those of $D^{\text{opt}}$.

11.4.3. *Rest of the proof.* Once the expression for $\ell_Y(c)$ is obtained, the proof is similar to the one for generalized eights:

(1) we get rid of negative crossing parameters using the erasing procedure from §7;

(2) we write the integral in the new coordinates, with an additional set of a priori neutral coordinates defined in §11.4.1 (note that each non-neutral coordinate $\theta$ coincides with the length of an unshielded simple portion of $D^{\text{opt}}$, up to addition of a function of neutral variables);

(3) we apply our results about generalized convolutions in the active variables $\Theta''_{ac} \subseteq \Theta$;

(4) we conclude with a comparison estimate, which is easier to prove now that some lengths $\ell(\Gamma_\lambda)$ (corresponding to contractible components) vanish.

**Remark 11.24.** The choice of $K$ in Theorem 1.22 is still given by equation (9.6), but now the the a-priori-neutral parameters have been removed from the new set of active parameter $\Theta''_{ac}$. Hence,

$$ (11.2) \qquad K = \sum_{q \in \Theta''_{ac}} (K_q + 1) \le \sum_{j \in V'} (K_j - 1) + \rho[\mathbf{T}] $$

where the sum $V' \subseteq V$ is the set of $\lambda \in \Lambda_{BC}^{\beta} \cap V$ such that $\beta_{\lambda}$ contains at least an unshielded simple portion of $c$, as mentioned in Remark 1.24. In particular, we can take $K = 0$ if $\rho[\mathbf{T}] = 0$.

## 12. VARIANT OF THE MAIN THEOREM FOR LC-SURFACES

Our core motivation for proving Theorem 1.22 is the spectral gap problem, i.e. proving that $\lambda_1$ is close to $1/4$ for typical hyperbolic surfaces of high genus. We have explained the link between these questions, and presented the arguments leading to our spectral gap result (Theorem 1.6), in [4]. In the proof, we need to restrict the Weil-Petersson
measure to the set of tangle-free surfaces. The presence of an indicator function in the restricted integral causes problems, that are overcome by the introduction of a so-called Moebius function, constructed in [3]. The T-averages $\langle F \rangle_g^T$ of a test function F, defined in (1.1), are then replaced by expressions of the form

$$ (12.1) \qquad \langle \mu \otimes F \rangle_{g,Q}^{\mathcal{T}} := \mathbb{E}_g^{\text{WP}} \left[ \sum_{\substack{\zeta \in \mathcal{G}(X) \\ \tau \in \mathcal{S}(X) \\ (\zeta, \tau) \sim \mathcal{T}}} \mu(\tau) F(\ell(\zeta)) \right] $$

where:

* $\mu$ is the Moebius function constructed in [3], defined on the “extended moduli space” of metric spaces made of disjoint unions of circles and hyperbolic surfaces with boundary;

* $\zeta$ is a geodesic loop of length $\ell(\zeta)$;

* $\tau$ is a subset of X with various connected components, which may be either surfaces with geodesic boundary, or non-contractible simple closed geodesics;

* $\mathcal{T}$ is now a local topological type of pairs $(\zeta, \tau)$.

In the following, a test function F is a measurable function $F: \mathbb{R}_{\ge 0} \to \mathbb{R}$ which is bounded and compactly supported.

The precise definition of $\mu$ and of the “extended moduli space” is given in [3, §2] and partially recalled below. We also recall the new notion of local topological type adapted to pairs $(\zeta, \tau)$, and finally explain how to extend Theorem 1.22 to the average (12.1). This provides the required technical generalization to complete the proof of Theorem 1.6.

**12.1. c-surfaces.** Let us recall the notion of c-surface, developed in more detail in [3, §2]. In a nutshell, a c-surface is the disjoint union of a surface and of simple curves.

Let $\mathbb{N}_0 = \mathbb{Z}_{\ge 0}$ be the set of non-negative integers. We order the pairs $(g, n) \in \mathbb{N}_0^2$ by the order relation: $(g, n) \le (g', n')$ if $2g+n-2 < 2g'+n'-2$, or if $2g+n-2 = 2g'+n'-2$ and $g \le g'$. For $j \ge 1$, denote by $\mathfrak{S}^{(j)}$ the set of $2j$-tuples $(\mathbf{g}, \mathbf{n}) = ((g_1, n_1), \dots, (g_j, n_j)) \in (\mathbb{N}_0^2)^j$, satisfying $(g_i, n_i) = (0, 2)$ or $2g_i - 2 + n_i > 0$ for $1 \le i \le j$, and $(g_i, n_i) \le (g_{i+1}, n_{i+1})$ for $1 \le i < j$.

**Definition 12.1.** A c-surface of signature $(\mathbf{g}, \mathbf{n}) \in \mathfrak{S}^{(j)}$ is a non-empty topological space S with j connected components $S = \bigsqcup_{i=1}^j \tau_i$ labelled from 1 to j, where:

* if $(g_i, n_i) = (0, 2)$, then $\tau_i$ is a 1-dimensional oriented manifold diffeomorphic to a circle;

* if $2g_i + n_i - 2 > 0$, then $\tau_i$ is a compact 2-dimensional orientable manifold with (or without) boundary, of signature $(g_i, n_i)$, with boundary components numbered by $\{1, \dots, n_i\}$.

**Notation 12.2.** The c-surface S can be decomposed as $S = (c, \sigma)$, where $c = (c_1, \dots, c_{j_1})$ and $\sigma = (\sigma_1, \dots, \sigma_{j_2})$ are, respectively, collections of 1-dimensional oriented manifolds and 2-dimensional oriented compact manifolds with boundary ($j = j_1 + j_2$ is the total number of connected components). We shall denote by $c(S) = j_1$ the number of 1d
components, and $\chi(\mathbf{S})$ the total absolute Euler characteristic, that is

$$ \chi(\mathbf{S}) = \sum_{i=1}^{j_2} \chi(\sigma_i) \quad \text{where} \quad \chi(\sigma_i) = 2g_{j_1+i} + n_{j_1+i} - 2. $$

We say that $\mathbf{S}$ is purely 2-dimensional if $c(\mathbf{S}) = 0$, and purely 1-dimensional if $\chi(\mathbf{S}) = 0$.

**Notation 12.3 (Boundary of a c-surface).** For $\mathbf{S} = (\tau_1, \dots, \tau_j) = (c_1, \dots, c_{j_1}, \sigma_1, \dots, \sigma_{j_2})$, if $\partial\sigma_i$ is the boundary of the surface $\sigma_i$, we let $\partial\mathbf{S}$ be the (usual) boundary of $\mathbf{S}$

$$ \partial\mathbf{S} = (\bigsqcup_{i=1}^{j_1} c_i) \sqcup (\bigsqcup_{i=1}^{j_2} \partial\sigma_i). $$

The total number of boundary components of $\sigma$ is defined as

$$ n_{\sigma} := \sum_{i=1}^{j_2} n_{j_1+i}. $$

**Definition 12.4 (Sub-c-surface).** If $\mathbf{S}$ is a c-surface, we call *sub-c-surface* of $\mathbf{S}$ a subset $\mathbf{S}' \subseteq \mathbf{S}$ which (for the induced topology and differential structure) is a c-surface, and such that $\partial\mathbf{S}'$ is a multi-curve in $\mathbf{S}$, i.e. its components are non-contractible and homotopically distinct.

**Notation 12.5 (Geodesic representative).** Let $\mathbf{S}$ be a c-surface of signature $(\mathbf{g}, \mathbf{n})$ and $\mathbf{S}' = (c', \sigma')$ be a sub-c-surface of $\mathbf{S}$. Assume $\mathbf{S}$ is endowed with a metric $Z \in \mathcal{T}_{\mathbf{g},\mathbf{n}}^*$, the Teichmüller space of Riemannian metrics on $\mathbf{S}$ that are hyperbolic on the 2-dimensional components. Then $\mathbf{S}'$ is isotopic to a geodesic sub-c-surface $\mathbf{S}'_Z = (c'_Z, \sigma'_Z)$: we mean by this a sub-c-surface with geodesic boundary, that we call the *geodesic representative* of $\mathbf{S}'$ in $Z$.

We denote by $\mathcal{S}(Z)$ the set of geodesic sub-c-surfaces of $\mathbf{S}$ endowed with the metric $Z$.

In [3, §2], we also defined the extended moduli space $\mathcal{M}$, which is – roughly speaking – the moduli space of all possible Riemannian structures on all possible c-surfaces, with hyperbolic metrics on the 2d-components. In the context of Definition 12.5, $\mathbf{S}'_Z$ may be seen as a representative of an element of $\mathcal{M}$.

The following notation will also be useful.

**Notation 12.6.** For $\mathbf{S}$ as in Notation 12.3, if now $\tilde{\partial}c_i := c_i \times \{\pm\}$ (interpreted as the two boundary curves of a cylinder containing $c_i$), we let

$$ \tilde{\partial}\mathbf{S} = (\bigsqcup_{i=1}^{j_1} \tilde{\partial}c_i) \sqcup (\bigsqcup_{i=1}^{j_2} \partial\sigma_i). $$

The components of $\tilde{\partial}\mathbf{S}$ are labelled by the set

$$ (\{1, \dots, j_1\} \times \{\pm\}) \sqcup \{(i, l), 1 \le i \le j_2, 1 \le l \le n_i\}. $$

When needed, we can put a total lexicographic order on this set. We let $n_{\mathbf{S}} := 2j + n_{\sigma}$. Notice that, listing all the boundary components in the natural order, we can also identify

$$ \tilde{\partial}\mathbf{S} = \tilde{\partial}c \sqcup \{1, \dots, n_{\sigma}\} = (\{1, \dots, j_1\} \times \{\pm\}) \sqcup \{1, \dots, n_{\sigma}\}. $$
**12.2. lc-surfaces and their local topological types.** We enrich the notion of local topological type, to a notion designed to sort out expressions such as (12.1). The topological object to consider is now $(\mathbf{S}, \zeta, \tau)$, where $\zeta$ is a loop and $\tau$ is a c-surface, which together fill $\mathbf{S}$. Such a pair will be called a lc-surface. We define these objects in [3, Appendix A].

**Notation 12.7.** To any $(\mathbf{g}, \mathbf{n}) \in \mathfrak{S}^{(j)}$, $j \ge 0$, we associate a *fixed smooth oriented c-surface* $\mathbf{S}$ of signature $(\mathbf{g}, \mathbf{n})$. The data of $(\mathbf{g}, \mathbf{n})$, or equivalently of $\mathbf{S}$, is called a *c-filling type*.

**Definition 12.8.** A local lc-surface is a triplet $(\mathbf{S}, \zeta, \tau)$, where $\mathbf{S}$ is a c-filling type, $\zeta$ is a loop in $\mathbf{S}$, $\tau$ is a sub-c-surface of $\mathbf{S}$, and the pair $(\zeta, \tau)$ fills $\mathbf{S}$.

Two local lc-surfaces $(\mathbf{S}, \zeta, \tau)$ and $(\mathbf{S}', \zeta', \tau')$ are said to be locally equivalent if $\mathbf{S} = \mathbf{S}'$ (i.e. $(\mathbf{g}\mathbf{s}, \mathbf{n}\mathbf{s}) = (\mathbf{g}\mathbf{s}', \mathbf{n}\mathbf{s}')$), and there exists a positive homeomorphism $\psi : \mathbf{S} \to \mathbf{S}$, possibly permuting the boundary components and connected components of $\mathbf{S}$, such that $\psi \circ \zeta$ is freely homotopic to $\zeta'$ and $\psi(\tau) = \tau'$. We ask that the homotopy between $\psi \circ \zeta$ and $\zeta'$ preserves the orientation, and that $\psi$ respects the numbering of boundary components of $\tau$ and $\tau'$.

This defines an equivalence relation $\sim_{\text{loc}}$ on local lc-surfaces. Equivalence classes for this relation are denoted as $[\mathbf{S}, \zeta, \tau]_{\text{loc}}$ and called *local topological types* of lc-surfaces.

**Notation 12.9.** If $(\mathbf{S}, \zeta, \tau)$ is a local lc-surface, we will denote by $\mathbf{S}_{\partial} \subseteq \mathbf{S}$ the c-surface filled by the multi-loop $(\zeta, \partial\tau)$, and $\mathbf{S}_{\text{in}}$ its complement in $\mathbf{S}$.

Necessarily, the 1d-components of $\tau$ which lie in the 2d-components of $\mathbf{S}$ must intersect $\zeta$, otherwise $\tau \cup \zeta$ would not fill $\mathbf{S}$. As a consequence, $\mathbf{S}_{\partial}$ contains the 1d part of $\tau$, and $\mathbf{S}_{\text{in}}$ is contained in the 2d-part of $\tau$.

**Definition 12.10.** Let $\mathcal{T} = [\mathbf{S}, \zeta, \tau]_{\text{loc}}$ be a local topological type. For a pair $(\gamma, \tau_0)$ formed by a loop $\gamma$ and a sub-c-surface $\tau_0$ on a hyperbolic surface $X$, we denote as $S(\gamma, \tau_0)$ the surface filled by $(\gamma, \tau_0)$ in $X$ (defined in details in [2, Definition 4.1]). Then, $(\gamma, \tau_0)$ is said to belong to the local topological type $[\mathbf{S}, \zeta, \tau]_{\text{loc}}$ if there exists a positive homeomorphism $\phi : S(\gamma, \tau_0) \to \mathbf{S}$ such that the loops $\phi(\gamma)$ and $\zeta$ are freely homotopic in $\mathbf{S}$, and such that $\phi(\tau_0) = \tau$. In that case, we write $(\gamma, \tau_0) \sim \mathcal{T}$.

**Notation 12.11.** For any $j \ge 1$, there is a natural bijection from the set of c-surfaces with $c(\mathbf{S}) = 0$ to the set of c-surfaces with $c(\mathbf{S}) = j$, which consists in adding $j$ 1-dimensional connected components: we denote it by $\rho_j$.

Similarly, if $\mathcal{T} = [\mathbf{S}, \zeta, \tau]_{\text{loc}}$ is a local topological type of lc-surface such that $c(\mathbf{S}) = 0$, we denote by $\rho_j \mathcal{T}$ the local topological type obtained by adding $j$ 1-dimensional connected components to the local topological type $\mathcal{T}$. If $\mathcal{T}$ fills $\mathbf{S}$, then $\rho_j \mathcal{T}$ fills $\rho_j \mathbf{S}$.

**12.3. Volume functions for local topological types of lc-surfaces.** We now study expressions such as (12.1) and state analogues of Theorem 1.4. As explained in [4], to prove our spectral gap result, Theorem 1.6, we can restrict attention to subsurfaces of $X$ that disconnect the hyperbolic surface $X$ into a bounded number of components. This is formalised as follows.

For an integer $Q$, we denote as $\text{MC}_X(Q)$ the set of multi-curves which separate $X$ into at most $Q$ connected components, following [2, Notation A.2]. Let us define $S_Q(X) \subset$
$S(X)$ as the set of geodesic sub-c-surfaces of $X$, with a 1d-part belonging to $MC_X(Q)$. For any test function $F$, any bounded measurable function $\nu$ on the extended moduli space $M$, we let

$$ (12.2) \qquad \langle \nu \otimes F \rangle_{g,Q}^{\mathfrak{T}} := \mathbb{E}_g^{\text{WP}} \left[ \sum_{\substack{\gamma \in \mathcal{G}(X) \\ \tau \in \mathcal{S}_Q(X) \\ (\gamma, \tau) \sim \mathfrak{T}}} \nu(\tau) F(\ell(\gamma)) \right]. $$

**Notation 12.12.** In the following, the function $\nu$ on $M$ will be assumed to satisfy the following properties for some positive real numbers $\kappa, R$ with $\kappa < R$.

* If $\tau = (c_1, \dots, c_{j_1}, \sigma_1, \dots, \sigma_{j_2})$ is such that $\nu(\tau) \neq 0$, then
  - for $1 \le i \le j_1$, $\ell(c_i) \le \kappa$, where $\ell(c_i)$ is the length of $c_i$ for the metric $\tau$;
  - for $1 \le i \le j_2$, $\ell(\partial\sigma_i) \le 9R\chi(\sigma_i)$, where $\ell(\partial\sigma_i)$ is the total boundary length of $\sigma_i$.

* There exists a bound of the form

$$ (12.3) \qquad |\nu(\tau)| \le f(c(\tau), \chi(\tau)). $$

We assume that $f(j, \chi) \le f(j', \chi')$ if $j \ge j'$ and $\chi \le \chi'$.

**Remark 12.13.** Notice that the above conditions imply that, for any $\tau$ in the support of $\nu$,

$$ (12.4) \qquad \ell(\partial\tau) \le \kappa c(\tau) + 9R\chi(\tau). $$

If we further assume that $\tau$ is a sub-c-surface of an ambient c-surface $S$ with $c(S) = 0$, then $\ell(\partial\tau) \le 12R\chi(S)$ since the maximal number of components for a multi-curve in $S$ is $3\chi(S)$.

**Example 12.14.** The hypotheses above are satisfied for $\nu = \mu_{\kappa,R}$, the Moebius function constructed in [3], with the bound

$$ (12.5) \qquad f(j, \chi) := \frac{U_1(\chi)}{j!} e^{RU_2(\chi)} $$

where $U_1$ and $U_2$ are increasing sequences (for which we do not need an explicit expression).

The following generalizes Proposition 1.2, with exactly the same proof:

**Proposition 12.15.** Let $\mathfrak{T} = [S, \zeta, \tau]_{\text{loc}}$ be a local topological type of lc-surface with $c(S) = 0$. For any $j \ge 0$, there exists a continuous functions $V_{g,Q}^{\nu, \mathfrak{T}, j}$ such that, for any test function $F$,

$$ \langle \nu \otimes F \rangle_{g,Q}^{\rho_j \mathfrak{T}} = \int_0^{+\infty} F(\ell) V_{g,Q}^{\nu, \mathfrak{T}, j}(\ell) d\ell. $$

We are now ready to state the Friedman-Ramanujan property that we will prove.

**Theorem 12.16.** Let $\mathfrak{T} = [S, \zeta, \tau]_{\text{loc}}$ be a local topological type of lc-surface such that $c(S) = 0$. For any integers $Q, N \ge 0$, any large enough $g$, any $j \le \log g$, there exists a
Friedman-Ramanujan function $h_{g,Q,N}^{\nu, \bar{\tau}, j}$ such that, for any test function $F$, any $\eta > 0$,

$$ \left| \langle \nu \otimes F \rangle_{g,Q}^{\rho_j \bar{\tau}} - \frac{1}{g \chi(\mathbf{S})} \int_0^{+\infty} F(\ell) h_{g,Q,N}^{\nu, \bar{\tau}, j}(\ell) \, d\ell \right| \\ \le \mathrm{fn}(Q, N, \chi(\mathbf{S}), \eta) M^j \mathfrak{f}(j, \chi(\tau)) e^{13R\chi(\mathbf{S})} \frac{\|F(\ell) e^{(1+\eta)\ell}\|_{\infty}}{g^{N+1}} $$

where $M = \mathrm{fn}(Q, N, n_S, \kappa) = 2^{2(Q+nS)} \tilde{a}_N^2 (e^\kappa - 1)$ with the constant $\tilde{a}_N$ from [2, Theorem B.1]. More precisely, there exists $\bar{K} = \bar{K}(N, \chi(\mathbf{S}))$ and $N = N(N, \chi(\mathbf{S}))$ such that $h_{g,Q,N}^{\nu, \bar{\tau}, j} \in \mathcal{F}_w^{\bar{K},N}$, and

$$ (12.6) \qquad \|h_{g,Q,N}^{\nu, \bar{\tau}, j}\|_{\mathcal{F}^{\bar{K},N}}^w \le \mathrm{fn}(Q, N, \chi(\mathbf{S})) \tilde{a}_N^{2j} \mathfrak{f}(j, \chi(\tau)) e^{6R\chi(\mathbf{S})}. $$

It is essential in this statement that the exponents and constants are independent from the number of 1d components $j$ and the specific topology of $(\zeta, \tau)$ amongst pairs filling $\mathbf{S}$.

**12.4. Writing down the volume functions for lc-local types.** Let us now sketch the proof of Theorem 12.16, which is very similar to the proof of Theorem 1.22.

As before, we start by writing down an explicit expression for the volume function $V_{g,Q}^{\nu, \bar{\tau}, j}$. We then use the asymptotic expansion from [1, Theorem 1.1] to obtain an approximate expression for this volume function, up to errors decaying like $1/g^{N+1}$. The Friedman-Ramanujan property follows from a variant of Theorem 1.22. The main new difficulty here is to control the dependency of the estimates on the integer $j$, the number of 1d-components of our lc-local-type (as we will need to understand this dependency well for the spectral gap result).

**12.4.1. Realizations of a lc-local type.** In order to write down the volume functions $V_{g,Q}^{\nu, \bar{\tau}, j}$ with a formula similar to (1.8), we need an enumeration of the possible embeddings of the c-filling type $\mathbf{S}$ into a surface $S_g$ of genus $g$. We therefore extend from [2, §4.5] the notion of realization of a filling type into $S_g$.

For the sake of convenience, we treat separately purely 2d and 1d c-filling types, before merging the two definitions. Recall from Notation 12.3 that we can put an order on the labels of the connected components of $\partial \mathbf{S}$.

**Definition 12.17.** Let $\mathbf{S} = (\sigma_1, \dots, \sigma_j)$ be a purely 2-dimensional c-filling type, and let $g \ge 2$. We call realization of $\mathbf{S}$ in a connected surface $S_g$ of genus $g$ any pair $\mathfrak{R} = (\vec{I}, \vec{g})$, where:

* $\vec{I}$ is a partition of $\partial \mathbf{S}$ into $q \ge 1$ non-empty sets $I_1, \dots, I_q$, numbered such that $i \mapsto \min I_i$ is an increasing function;

* we ask furthermore the partition to be transitive in the sense given below;

* $\vec{g} = (g_1, \dots, g_q)$ is a vector of non-negative integers;

* if for any $1 \le i \le q$, we denote $n_i := \#I_i$ and $\chi_i := 2g_i - 2 + n_i$, then $\chi_i > 0$ or $(g_i, n_i) = (0, 2)$, and we have:

$$ (12.7) \qquad \chi(\mathbf{S}) + \sum_{i=1}^{q} \chi_i = 2g - 2. $$
The set of realizations of $\mathbf{S}$ in $S_g$ is denoted as $R_g(\mathbf{S})$.

Recall that $\partial\mathbf{S}$ comes naturally partitioned into $\bigsqcup_{i=1}^j \partial\sigma_i$. The partition $\vec{I}$ being transitive means that there does not exist a strict non-empty subset of $\partial\mathbf{S}$ that can be realized simultaneously as a union of sets $(\partial\sigma_i)_{1\le i\le j}$ and as a union of sets $(I_i)_{1\le i\le q}$.

Realizations are meant to enumerate all possible embeddings of a local type in a closed surface of genus $g$ (i.e. absolute Euler characteristic $2g-2$), modulo the action of the mapping class group ([2, §4.5]). The transitivity assumption guarantees that the resulting surface is connected.

For a purely 1d c-filling type, i.e. a multi-curve, we have a similar definition for realizations of a multicurve in a connected surface $S_{g,n}$ of signature $(g, n)$.

**Definition 12.18.** Let $c = (c_1, \dots, c_j)$ be a purely 1-dimensional c-filling type. Let $(g, n) \in \mathbb{N}_0^2$ be such that $2g-2+n>0$ or $(g, n) = (0, 2)$. We call realization of $c$ in a connected surface $S_{g,n}$ of signature $(g, n)$ any pair $\mathfrak{R} = (\vec{J}, \vec{g})$, where:

* $\vec{J}$ is a partition of $\{1, \dots, n\} \sqcup \tilde{\partial}c$ into $\mathfrak{q} \ge 1$ non-empty sets $J_1, \dots, J_{\mathfrak{q}}$, numbered so that $i \mapsto \min J_i$ is an increasing function;

* we ask furthermore the partition to be transitive in the sense given below;

* $\vec{g} = (g_1, \dots, g_{\mathfrak{q}})$ is a vector of non-negative integers;

* if for any $1 \le i \le \mathfrak{q}$, we denote $n_i := \#J_i$ and $\chi_i := 2g_i - 2 + n_i$, then $\chi_i > 0$ or $(g_i, n_i) = (0, 2)$, and in the latter case, $J_i$ must contain exactly one element of $\{1, \dots, n\}$ and one element of $\tilde{\partial}c$;

* we have:

$$ (12.8) \qquad \sum_{i=1}^{\mathfrak{q}} \chi_i = 2g - 2 + n. $$

The set of realizations of $(c_1, \dots, c_j)$ in $S_{g,n}$ is denoted as $R_{g,n}^1(j)$.

We recall that here $\tilde{\partial}c = \{1, \dots, j\} \times \{\pm\}$ is the set of boundary curves of the cylinders containing the multi-curve $c$. Note that the superscript 1 refers to the fact that we consider realizations of 1d curves. If the curves, instead of being numbered by $\{1, \dots, j\}$, are numbered by an abstract set $B$ of cardinality $j$, we write $R_{g,n}^1(B)$. Similarly, if the boundary components of $S_{g,n}$ are indexed by a set $I$ of cardinality $n$, we denote $S_{g,I}$ and $R_{g,I}^1(B)$.

Realizations are meant to enumerate (modulo the action of the mapping class group) all possible embeddings of a multi-curve with $j$ components into a surface of signature $(g, n)$. In the fourth point of the definition, we impose that two curves cannot be in the same homotopy class, but we authorize the case where one of the curves is homotopic to a boundary component. In particular, if $(g, n) = (0, 2)$ we must have $j \le 1$.

Because we are only interested in connected surfaces $S_{g,n}$, we again impose a transitivity condition, defined as follows. Remark that $\{1, \dots, n\} \sqcup \tilde{\partial}c$ comes naturally partitioned into

$$ (\bigsqcup_{k=1}^n \{k\}) \sqcup (\bigsqcup_{i=1}^j (\{i\} \times \{\pm\})) . $$
The partition $\vec{J}$ being transitive means that there does not exist a strict non-empty subset of $\{1, \dots, n\} \sqcup \tilde{\partial}c$ that can be realized simultaneously as a union of sets $\{k\}$ and $\{i\} \times \{\pm\}$ as above, and as a union of sets $J_i$ with $1 \le i \le q$.

Remark that the definition makes sense for $j = 0$, in this case we must have $q = 1$.

The definitions above can be extended for realizations of an arbitrary c-filling type $S = (c, \sigma)$ (having 1d and 2d components $c$ and $\sigma$) into a connected surface $S_g$. For the purpose of controlling the dependency of volume functions on $j$, we opt for the following formulation in two steps: we first realize the 2d part $\sigma$ into $S_g$, and in a second step we realize $c$ in $S_g \setminus \sigma$.

**Definition 12.19.** Let $S = (c, \sigma)$ be a general c-filling type, separated into its 1d and 2d components $c = (c_1, \dots, c_j)$ and $\sigma$. Let $g \ge 2$. We call realization of $S$ in a connected surface $S_g$ of genus $g$ the data of $\mathfrak{R} = (\mathfrak{R}_\sigma, \pi, (\mathfrak{R}_l)_{1 \le l \le q})$, where

* $\mathfrak{R}_\sigma = (\vec{I}, \vec{g}) = (I_1, \dots, I_q, g_1, \dots, g_q)$ is a realization of $\sigma$ in $S_g$ with $q$ components;

* $\pi$ is a partition of $\{1, \dots, j\}$ into (possibly empty) sets $\pi_1, \dots, \pi_q$;

* for each $1 \le l \le q$, $\mathfrak{R}_l = (\vec{J}_l, \vec{g}_l) \in R_{g_l, I_l}^1(\pi_l)$ is a realization of $(c_i)_{i \in \pi_l}$ in a connected surface $S_l = S_{g_l, n_l}$ of signature $(g_l, n_l)$, where $n_l = \#I_l$.

The set of realizations of $S$ in $S_g$ is denoted as $R_g(S)$.

For such a realization, the integer $q$ corresponds to the number of connected components of $S_g \setminus \sigma$. If those components are called $S_1, \dots, S_q$, then $S_l$ has $n_l$ boundary components indexed by $I_l$. The set $\pi_l$ corresponds to the curves in $(c_1, \dots, c_j)$ that lie inside the component $S_l$.

For $1 \le l \le q$, we denote $\vec{J}_l = (J_l^1, \dots, J_l^{q_l})$, i.e. we let $q_l$ be the number of connected components of the surface $S_l \setminus (\cup_{i \in \pi_l} c_i)$ (some of those components may be peripheral annuli). Then $(J_l^i)_{1 \le i \le q_l}$ forms a partition of $I_l \sqcup (\pi_l \times \{\pm\})$.

The collection $(J_l^i)_{1 \le l \le q, 1 \le i \le q}$ forms a partition of $\tilde{\partial}S = \{1, \dots, n_\sigma\} \sqcup \tilde{\partial}c$. Letting $n_l^i = \#J_l^i$, we have

$$ \sum_{i=1}^{q_l} n_l^i = n_l + 2|\pi_l| \quad \text{and} \quad \sum_{l=1}^{q} \sum_{i=1}^{q_l} n_l^i = n_\sigma + 2j = n_S. $$

**Notation 12.20.** With the notations of Definition 12.19, for $Q \ge 0$ an integer, we denote by $R_{g,Q}(S)$ the set of realizations in $R_g(S)$ for which $c \in \text{MC}_{S_g}(Q)$, that is to say, the 1d part $c$ disconnects $S_g$ into at most $Q$ connected components.

Remark that the number of connected components of $S_g \setminus S$ coincides with $\sum_{l=1}^q q_l$. For a realization in $R_{g,Q}(S)$, by definition, the surface $S_g \setminus c$ has at most $Q$ connected components. Each connected component of $S_g \setminus S$ is either a connected component of $S_g \setminus c$, or shares a boundary with one boundary component of $\sigma$. Hence,

$$ \sum_{l=1}^{q} q_l \le Q + n_\sigma. $$

12.4.2. *Volume function associated to a realization.* For a realization $\mathfrak{R} \in R_g(S)$, we define the volume function $V_{\mathfrak{R}}(z)$, which is a function of $n_S = n_\sigma + 2j$ variables $z =$
$(z_t)_{t \in \tilde{\partial}S}$ defined by

$$V_{\mathfrak{R}}(\mathbf{z}) := \prod_{l=1}^{q} \prod_{i=1}^{q_l} V_{g_l^i, n_l^i}(\mathbf{z}_{J_l^i}) \quad \text{where} \quad \mathbf{z}_{J_l^i} = (z_t, t \in J_l^i),$$

where we recall that for any $(g, n)$ with $2g-2+n > 0$, $V_{g,n}(x_1, \dots, x_n)$ is the total volume of the moduli space of surfaces of signature $(g, n)$ with boundary lengths $x_1, \dots, x_n$, and

$$V_{0,2}(x_1, x_2) = \frac{1}{x_1} \delta(x_1 - x_2)$$

where $\delta$ denotes the Dirac mass at 0.

For $1 \le l \le q$, let

$$J_l^{\text{cyl}} := \bigcup_{\substack{1 \le i \le q_l \\ \chi_l^i = 0}} J_l^i,$$

where $\chi_l^i = 2g_n^i - 2 + n_l^i \ge 0$. We then define $J^{\text{cyl}} := \bigcup_{l=1}^q J_l^{\text{cyl}}$. This corresponds to the set of boundary components of $S$ that are glued to one another by cylinders in the realization.

### 12.4.3. Upper bound on volume functions.
Let us prove a rough upper bound on the volume functions.

**Lemma 12.21.** For a fixed partition $(\vec{J}_l)_{1 \le l \le q}$ of $\tilde{\partial}S$ as above, the sum over all corresponding realizations possessing this datum satisfies, for a universal constant $C$,

$$\frac{1}{V_g} \sum_{\mathfrak{R}} \prod_{t \in \tilde{\partial}S} z_t V_{\mathfrak{R}}(\mathbf{z}) \le \frac{C}{g^{\chi(S)}} \prod_{t \in \tilde{\partial}S \setminus J^{\text{cyl}}} 2 \sinh\left(\frac{z_t}{2}\right) \prod_{\substack{\chi_l^i=0 \\ J_l^i=\{t,t'\}}} z_t \delta(z_t - z_{t'})$$

to be interpreted as an inequality between two positive measures. The number of possible data sets $(\vec{J}_l)_{1 \le l \le q}$ is at most $2^{n_S(Q+n_\sigma)}$.

*Proof.* The choice of $(\vec{J}_l)_{1 \le l \le q}$ automatically fixes $q$, $\vec{I}$, $\pi$ and $J^{\text{cyl}}$. This reduces to a sum over all possible values of $g_1, \dots, g_q$ and $\vec{g}_l$. This sum is controlled by [12, Lemma 24]:

$$\sum_{\mathfrak{R}} \prod_{t \in \tilde{\partial}S} z_t V_{\mathfrak{R}}(\mathbf{z}) \le \prod_{l=1}^{q} W_{\chi_l} \prod_{t \in \tilde{\partial}S \setminus J^{\text{cyl}}} 2 \sinh\left(\frac{z_t}{2}\right) \prod_{\substack{\chi_l^i=0 \\ J_l^i=\{t,t'\}}} z_t \delta(z_t - z_{t'})$$

where, for $\chi > 0$,

$$W_{\chi} = \begin{cases} V_{\frac{\chi}{2}+1,0} & \text{if } \chi \text{ is even;} \\ V_{\frac{\chi+1}{2},1} & \text{otherwise.} \end{cases}$$

The product $\prod_{l=1}^q W_{\chi_l}$ itself is bounded by $O(V_g/g^{\chi(S)})$. Then, $2^{(n_\sigma+2j)(Q+n_\sigma)}$ is an upper bound on the number of partitions of a set with $n_S = n_\sigma + 2j$ elements into at most $Q+n_\sigma$ subsets. $\square$
**Remark 12.22.** If we want to further impose that $\mathbf{S} = (c, \sigma)$ is realised as a sub-c-surface, we have to impose that $\chi_j > 0$ and $\chi_l^i > 0$ in Definitions 12.18 and 12.19. It is also possible to impose this restriction only on some elements $J_j$ or $J_l^i$ of the partitions; that is to say, forbid certain components to be cylinders. Obviously, imposing restrictions on the set of allowed realizations does not modify the following discussion.

**12.5. Proof of our main results.** We now turn to the proof of the main results of this section, Proposition 12.15 and Theorem 12.16. We omit technical details that are similar to earlier considerations: we focus on the main new difficulty, which is to control the dependency on the integer $j$. This requires to used refined estimates, with explicit dependency in the constants, as developed in [2, Appendix B].

**Remark 12.23.** At some point it seems necessary to assume some upper bound, say $j \ll g^{1/2}$. It is not clear if this is strictly necessary, or only a consequence of our imperfect optimisation of the estimates. In any case, the application to spectral gap results only requires $j \le \log g$.

**12.5.1. Expression of the volume function.** Let $\mathcal{T} = [\mathbf{S}, \zeta, \tau]_{\text{loc}}$ be a local topological type of lc-surface such that $c(\mathbf{S}) = 0$ (thus, $n_\sigma = n_\mathbf{S}$). Recall that $\rho_j \mathcal{T}$ stands for the local topological type obtained by adding $j$ 1-dimensional connected components to $\mathcal{T}$. If $\mathcal{T}$ fills the c-surface $\mathbf{S}$, then $\rho_j \mathcal{T}$ fills the c-surface $\rho_j \mathbf{S}$ and the decomposition of $\rho_j \mathbf{S}$ into 1d and 2d-components can be written as $\rho_j \mathbf{S} = (c_1, \dots, c_j, \mathbf{S})$.

We now straightforwardly apply the method developed in [2, Thm. 5.7 and Prop. 5.22] to establish Proposition 12.15, and see that the volume function $V_{g,Q}^{\nu, \bar{\mathcal{T}}, j}(\ell)$ has the form

$$ (12.9) \qquad \int_{\bar{\alpha} \in \mathbb{R}_{>0}^j} \int_{\mathbf{x} \in \mathbb{R}_{>0}^{n_S}} \int_{\ell_Y(\zeta) = \ell} \int_{Y \in \mathcal{M}_{\mathbf{x}}(\mathbf{S}|\mathcal{S}_{\partial})} \phi(\mathbf{x}, \bar{\alpha}) \nu(\tau_{(\mathbf{x},Y)}, \bar{\alpha}) \frac{\mathrm{d}\mathrm{Vol}_{g_S, n_S}^{\mathrm{WP}}(\mathbf{x}, Y)}{\mathrm{d}\ell} \mathrm{d}\bar{\alpha}, $$

where, for $\bar{\alpha} = (\alpha_1, \dots, \alpha_j) \in \mathbb{R}_{>0}^j$ and $\mathbf{x} \in \mathbb{R}_{>0}^{n_S}$,

$$ \phi(\mathbf{x}, \alpha_1, \dots, \alpha_j) = \frac{1}{\alpha_1 \dots \alpha_j} \Phi_{g,Q}^{\bar{\mathcal{T}},j}(\mathbf{x}, \alpha_1, \alpha_1, \dots, \alpha_j, \alpha_j) $$

and $\Phi_{g,Q}^{\bar{\mathcal{T}},j}$ is defined by

$$ (12.10) \qquad \Phi_{g,Q}^{\bar{\mathcal{T}},j}(\mathbf{z}) := \frac{1}{n(\mathcal{T})} \frac{1}{V_g} \prod_{t \in \partial \mathbf{S}} z_t \sum_{\mathfrak{R} \in R_{g,Q}(\rho_j \mathbf{S})} V_{\mathfrak{R}}(\mathbf{z}). $$

Recall from Notation 12.5 that for $Z = (\mathbf{x}, Y)$, $\tau_Z \in S(Z)$ is the geodesic representative of $\tau$ for the metric $Z$. We should actually restrict the sum to realizations $\mathfrak{R}$ such that $(c_1, \dots, c_j, \tau)$ stands as a sub-c-surface in $S_g$ (see Remark 12.22).

Careful attention needs to be paid to the domain of integration $\mathcal{M}_{\mathbf{x}}(\mathbf{S}|\mathcal{S}_{\partial})$. Recall from Notation 12.9 that $\mathbf{S} = \mathcal{S}_{\partial} \sqcup \mathcal{S}_{\text{in}}$ where $\mathcal{S}_{\partial}$ is filled by $\zeta$ and $\partial\tau$. By hypothesis, the function $Y \mapsto \nu(\tau_{(\mathbf{x},Y)}, \bar{\alpha})$ is invariant by the mapping class group of $\tau$. Thus the pair $(\ell_Y(\zeta), \nu(\tau_{(\mathbf{x},Y)}, \bar{\alpha}))$, initially defined on the Teichmüller space $\mathcal{T}_{\mathbf{x}}(\mathbf{S})$, is invariant by MCG($\mathcal{S}_{\text{in}}$). It follows that the integral takes place on $\mathcal{M}_{\mathbf{x}}(\mathbf{S}|\mathcal{S}_{\partial})$, defined as the quotient of $\mathcal{T}_{\mathbf{x}}(\mathbf{S})$ by MCG($\mathcal{S}_{\text{in}}$) and the Dehn twists along the shared boundary components of
$\mathbf{S}_{\partial}$ and $\mathbf{S}_{\text{in}}$. The combinatorial factor $n(\mathfrak{T})$ is the number of diffeomorphisms of $\mathbf{S}_{\partial}$ stabilising $(\zeta, \partial\tau)$, modulo homotopy.

**12.5.2.** Rough upper bounds on the volume function associated to a type. Let us prove the following rough upper bound.

**Lemma 12.24.** Let $\mathfrak{T} = [\mathbf{S}, \zeta, \tau]_{\text{loc}}$ be a local topological type of lc-surface such that $\mathfrak{c}(\mathbf{S}) = 0$. For any large enough $g$, any $j, Q \ge 0$, any test function $F$, any $0 < \eta \le 1/9$,

$$ \left| \int_{\mathbb{R}_{>0}} F(\ell) V_{g,Q}^{\nu, \bar{\mathfrak{T}}, j}(\ell) \, d\ell \right| \le \text{fn}(\chi(\mathbf{S}), \eta) 2^{(n_{\mathbf{S}}+2j)(Q+n_{\mathbf{S}})} (e^{\kappa} - 1)^j f(j, \chi(\tau)) e^{13R\chi(\mathbf{S})} \frac{\|F(\ell) e^{(1+\eta)\ell}\|_{\infty}}{g^{\chi(\mathbf{S})}}. $$

*Proof.* We use the formula for the volume function and the upper bound from Lemma 12.21. In order to control the exponential terms, we note that, in each term appearing with associated lengths $(z_t)_{t \in \bar{\partial} \mathbf{S}}$, if $\mathbf{x} = (x_1, \dots, x_{n_{\mathbf{S}}})$ is the vector of boundary lengths of the surface $\mathbf{S}$ filled by $(\zeta, \tau)$ and $\vec{\alpha} = (\alpha_1, \dots, \alpha_j)$ the vector of the 1d-component, then

$$ (12.11) \qquad \frac{1}{2} \sum_{t \in \bar{\partial} \mathbf{S}} z_t = \sum_{j=1}^{j} \alpha_j + \frac{1}{2} \sum_{i=1}^{n_{\mathbf{S}}} x_i. $$

By a straightforward adaptation from Proposition 1.29, for every hyperbolic metric $Y$ on $\mathbf{S}$,

$$ (12.12) \qquad \frac{1}{2} \sum_{i=1}^{n_{\mathbf{S}}} x_i \le \ell_Y(\zeta) + \ell_Y(\partial\tau) \le \ell_Y(\zeta) + 12R\chi(\mathbf{S}) $$

on the support of $\nu$, by Remark 12.13. As a result, the integral $|\int_{\mathbb{R}_{>0}} F(\ell) V_{g,Q}^{\nu, \bar{\mathfrak{T}}, j}(\ell) \, d\ell|$ is bounded above by $2^{(n_{\mathbf{S}}+2j)(Q+n_{\mathbf{S}})} e^{12R\chi(\mathbf{S})} \|F(\ell) e^{(1+\eta)\ell}\|_{\infty} g^{-\chi(\mathbf{S})}$ times

$$ \int_{Y \in \mathcal{M}_{\mathbf{x}}(\mathbf{S}|\mathbf{S}_{\partial})} e^{-\eta \ell_Y(\zeta)} \prod_{i=1}^{j} e^{\alpha_i} \mathbb{1}_{[0,\kappa]}(\alpha_i) |\nu(\tau_{(\mathbf{x},Y)}, \vec{\alpha})| \, d\text{Vol}_{g_{\mathbf{S}}, n_{\mathbf{S}}}^{\text{WP}}(\mathbf{x}, Y) \, d\vec{\alpha}. $$

We now integrate $\vec{\alpha}$ on $[0, \kappa]^j$, using the upper bound (12.12), to bound the previous integral by

$$ (e^{\kappa} - 1)^j f(j, \chi(\tau)) \int_{\mathbf{x}, Y \in \mathcal{M}_{\mathbf{x}}(\mathbf{S}|\mathbf{S}_{\partial})} e^{-\eta \ell_Y(\zeta)} \, d\text{Vol}_{g_{\mathbf{S}}, n_{\mathbf{S}}}^{\text{WP}}(\mathbf{x}, Y) $$

where the integral runs over the set of $\mathbf{x} > 0$ and $Y \in \mathcal{M}_{\mathbf{x}}(\mathbf{S}|\mathbf{S}_{\partial})$. We conclude by noticing that the integral above is bounded by $\text{fn}(\chi(\mathbf{S}), \eta) e^{9\eta R\chi(\tau)}$. Indeed, we showed in [3] that, because $\tau$ is a derived tangle, we can construct a multi-loop $\gamma_{\tau}$, filling $\tau$, of total length $\le 9R\chi(\tau)$. We then conclude using an adaptation of Corollary 3.25, proving that the Weil-Petersson volume of the set

$$ \{(\mathbf{x}, Y), \mathbf{x} > 0, Y \in \mathcal{M}_{\mathbf{x}}(\mathbf{S}|\mathbf{S}_{\partial}), \ell_Y(\zeta) + \ell_Y(\gamma_{\tau}) \le \ell\} $$

grows at most polynomially in $\ell$, because $(\zeta, \gamma_{\tau})$ fills $\mathbf{S}$.

□
12.5.3. Preliminary step. We are now ready to detail the proof of Theorem 12.16, which we do in the next few paragraphs. First, [2, Lemma B.2] shows that we can restrict attention to realizations of rank $\le N$ in the sum (12.10), modulo an error

$$ \le \mathrm{fn}(Q, N, \chi(\mathbf{S})) \frac{2^{2j(Q+n_S)}}{g^{N+1}} \exp\left(\frac{1}{2} \sum_{t \in \partial \mathbf{S}} z_t\right). $$

Given $Q, N$ and $\mathbf{S}$, the number of such realizations is bounded: we are reduced to dealing with a finite number of terms in (12.10). We therefore pick a realisation $\mathfrak{R} \in R_{g,Q}(\rho_j \mathbf{S})$.

12.5.4. First step. We now use [2, Theorem B.1] to write an approximate expression (up to errors decaying like $1/g^{N+1}$) of each term $\frac{1}{V_g} \prod_{t \in \partial \mathbf{S}} z_t V_{\mathfrak{R}}(\mathbf{z})$ in the definition of $\Phi_{g,Q}^{\tilde{\tau},j}(\mathbf{z})$, see equation (12.10). More precisely, the quantity above is well-approximated by a function $F_{g,\mathfrak{R}}^{(N)}(\mathbf{z})$ of the form

$$ (12.13) \quad F_{g,\mathfrak{R}}^{(N)}(\mathbf{z}) := \sum_{V_+, V_-, V_0} P_{g,\mathfrak{R}}^{(N,V_\pm)}(\mathbf{z}) \prod_{t \in V_+} \cosh\left(\frac{z_t}{2}\right) \prod_{t \in V_-} \sinh\left(\frac{z_t}{2}\right) \prod_{\substack{\chi_i^t=0 \\ J_l^t=t, t'}} z_t \delta(z_t - z_{t'}) $$

where:

* $V_+, V_-, V_0$ runs over partitions of the set $V := \tilde{\partial} \mathbf{S} \setminus J^{\mathrm{cyl}}$;

* the function $P_{g,\mathfrak{R}}^{(N,V_\pm)}$ is a polynomial function in the variables $(z_t)_{t \in V}$, of total degree in $(z_t)_{t \in V_+ \cup V_-}$ is $\le 2N$ and partial degree with respect to each $(z_t)_{t \in V_0}$ less than a constant $a_{N+1}$ depending only on $N$.

The error made in this approximation can be bounded above by

$$ \mathrm{fn}(Q, N, \chi(\mathbf{S})) \tilde{a}_N^{2j} \frac{(\|\mathbf{z}\| + 1)^{3N+1}}{g^{N+1}} \exp\left(\frac{1}{2} \sum_{t \in \partial \mathbf{S}} z_t\right). $$

By [2, Theorem B.1] and [2, eq. (B.4)], the coefficients of the polynomials $P_{g,\mathfrak{R}}^{(N,V_\pm)}$ are bounded by

$$ \mathrm{fn}(Q, N, \chi(\mathbf{S})) \tilde{a}_N^{n_S+2j} \frac{V_{g_S, n_S}}{V_g} V_{\mathfrak{R}}(\mathbf{0}) \le \mathrm{fn}(Q, N, \chi(\mathbf{S})) \frac{\tilde{a}_N^{2j}}{g^{\chi(\mathbf{S})}}. $$

12.5.5. Second step: the Friedman-Ramanujan property. The leading term in Theorem 12.16 is obtained by replacing the function $\Phi_{g,Q}^{\tilde{\tau},j}(\mathbf{z})$ by its approximate expression up to errors $1/g^{N+1}$. It is thus of the form (12.9), with

$$ \phi(\mathbf{x}, \vec{\alpha}) = \frac{1}{\alpha_1 \dots \alpha_j} \tilde{\Phi}_{g,Q}^{\tilde{\tau},j}(\mathbf{x}, \alpha_1, \alpha_1, \dots, \alpha_j, \alpha_j), $$

where $\tilde{\Phi}_{g,Q}^{\tilde{\tau},j}(\mathbf{z})$ is a sum of finitely many terms of the form (12.13). We are left with the question of proving that each function of $\ell$ defined by

$$ (12.14) \quad \int_{\alpha, \mathbf{x}>0} \int_{Y \in \mathcal{M}_{\mathbf{x}}(\mathbf{S}|\mathbf{S}_{\partial})} \frac{1}{\alpha_1 \dots \alpha_j} F_{g,\mathfrak{R}}^{(N)}(\mathbf{x}, \alpha_1, \alpha_1, \dots, \alpha_j, \alpha_j) \nu(\tau_{(\mathbf{x},Y)}, \vec{\alpha}) d\vec{\alpha} \frac{\mathrm{dVol}_{g_S, n_S}^{\mathrm{WP}}(\mathbf{x}, Y)}{\mathrm{d}\ell} $$
is a Friedman-Ramanujan function, with estimates on its norm.

The proof of this fact is the same as the proof of Theorem 9.4, applied to the multi-loop $(\zeta, \partial\tau, c_1, \dots, c_j)$, with the set $W$ corresponding to loops in $\partial\tau \cup (c_1, \dots, c_j)$, $\kappa_i := \kappa$ for loops in $(c_1, \dots, c_j)$, and $\sum_{i \in \partial\tau} \kappa_i \le 12R\chi(\mathbf{S})$.

The set of a priori neutral variables is enlarged to include the parameters $q \in \Theta$ such that the simple portion $\mathcal{I}_q$ is contained in the interior of the 2d-part of $\tau$ (indeed, those simple portions cannot be included in a boundary component of $\mathbf{S}$).

Theorem 9.4 then shows that (12.14) is a Friedman-Ramanujan function, with the same parameters as in Remarks 1.17. Hence, the parameters $K(N, \mathbf{S}) = 2N+2\chi(\mathbf{S})$ and $N(N, \mathbf{S}) = n_{\mathbf{S}}(a_{N+1} + 1) + 3\chi(\mathbf{S}) + 1$ satisfy the claim of Theorem 12.16. We obtain an estimate of the Friedman-Ramanujan norm of the function (12.14) by:

$$ \mathrm{fn}(Q, N, \chi(\mathbf{S})) \tilde{a}_N^{2j} f(j, \chi(\tau)) e^{6R\chi(\mathbf{S})}, $$

where the term $f(j, \chi(\tau))$ comes from the requirements on the function $\nu$ given in Notation 12.12.

### 12.5.6. Third step: estimate of the remainder term.
As in the proof of Lemma 12.24, using (12.11), (12.12) and the requirements on the function $\nu$ coming from Notation 12.12, we see that the remainder term in Theorem 12.16 is bounded by

$$ \begin{aligned} & \mathrm{fn}(Q, N, \chi(\mathbf{S})) \frac{\|F(\ell) e^{(1+\eta)\ell}\|_{\infty}}{g^{N+1}} (2^{2j}(Q+n_{\mathbf{S}}) + \tilde{a}_N^{2j}) e^{12R\chi(\mathbf{S})} \\ & \int_{\alpha, \mathbf{x}>0} \int_{Y \in \mathcal{M}_{\mathbf{x}}(\mathbf{S}|\mathbf{S}_{\partial})} e^{-\eta\ell Y(\zeta)} \prod_{i=1}^{j} e^{\alpha_i} \mathbb{1}_{[0,\kappa]}(\alpha_i) |\nu(\tau_Y, \vec{\alpha})| d\vec{\alpha} \, d\mathrm{Vol}_{g_{\mathbf{S},n_{\mathbf{S}}}}^{\mathrm{WP}}(\mathbf{x}, Y). \end{aligned} $$

As in the proof of Lemma 12.24, we conclude by integrating $\vec{\alpha}$ on $[0, \kappa]^j$ and bounding the final integral, introducing a multi-loop filling $\tau$.

## 13. PROOF OF THE SPECTRAL GAP RESULT

Theorem 12.16 was the last brick in the proof of Theorem 1.6, namely that, given $\alpha < 1/2$ and arbitrary small $\epsilon > 0$,

$$ (13.1) \qquad \lim_{g \to \infty} \mathbb{P}_g^{\mathrm{WP}} \left( \lambda_1(X) \le \frac{1}{4} - \alpha^2 - \epsilon \right) = 0. $$

The full argument leading to Theorem 1.6 from Theorem 12.16 is given in our exposition paper [4]. Here, we outline the main steps.

### 13.1. Trace method.
The trace method consists in controlling the expectation of a sum over the set $\mathcal{G}(X)$ of all primitive closed geodesics of $X$:

$$ (13.2) \qquad \langle F \rangle_{g, \kappa, R} = \mathbb{E}_g^{\mathrm{WP}} \left[ \sum_{\gamma \in \mathcal{G}(X)} F(\ell(\gamma)) \mathbb{1}_{\mathrm{TF}_g^{\kappa, R}}(X) \right] $$

where we restricted the Weil-Petersson measure to the set $\mathrm{TF}_g^{\kappa, R}$ of all $(\kappa, R)$-tangle-free surfaces, following the definition of [11, 3]. The test function $F = F_{m,L}$ will be of the
form

$$ (13.3) \qquad F_{m,L}(\ell) = \ell e^{-\ell/2} \mathcal{D}^m H_L(\ell) $$

where $H_L$ is itself a smooth function whose support is $[-L, L]$, constructed by dilation of the function for $L = 1$, i.e. $H_L(\ell) := H_1(\ell/L)$ and $\mathcal{D}$ is the differential operator $\frac{1}{4} - \partial^2$.

The introduction of the operator $\mathcal{D}$ is one of the keys of our approach, as it will give rise to cancellations in (13.2), leading to subexponential growth. The parameters $\kappa$, $R$ and $L$ will be determined later; $\kappa$ is a fixed, arbitrary small number, and $R = \kappa \log g$; similarly, $L$ will grow logarithmically with the genus $g$. The important point is that the tangle-free set $\text{TF}_{g}^{\kappa,R}$ is a set of large probability (see [4, Lemma 4.4]), namely that, for $\kappa < \kappa_0 = \min(2/3, 2 \operatorname{argsinh} 1)$,

$$ (13.4) \qquad 1 - \mathbb{P}_{g}^{\text{WP}}(\text{TF}_{g}^{\kappa,R}) \leq C_{\text{tf}}(\kappa^2 + g^{\frac{3}{2}\kappa-1}). $$

From [4], it suffices to show that the expectation (13.2) is $o(e^{(\alpha+\epsilon)L})$ as $g \to +\infty$. More precisely, the Selberg trace formula together with the Markov inequality allows to write

$$ (13.5) \qquad \mathbb{P}_g^{\text{WP}} \left( \delta \le \lambda_1 \le \frac{1}{4} - \alpha^2 - \epsilon \right) \le \frac{C}{e^{(\alpha+\epsilon)L}} \left( \langle F_{m,L} \rangle_{g,\kappa,R} + g \log(g)^2 \right) + C_{\text{tf}}(\kappa^2 + g^{\frac{3}{2}\kappa-1}) $$

for all $\delta > 0$ (see [4, Lemma 2.7]). We therefore need to have $e^{(\alpha+\epsilon)L} \ge g^{1+\eta}$ for a $\eta > 0$, in order to balance the terms growing linearly with $g$ (including, in particular, the topological term from the Selberg trace formula). From now on, we take $L = A \log g$ for an integer $A \ge \alpha + \epsilon$.

**13.2. Restriction to a finite number of types.** We now restrict the number of topological configurations in the average $\langle F_{m,L} \rangle_{g,\kappa,R}$ by using asymptotic expansions in powers of $1/g$ and the tangle-free hypothesis. First, letting $\chi_+ = 2A + 1$, we observe that we can replace the average $\langle F_{m,L} \rangle_{g,\kappa,R}$ by a truncated version $\langle F_{m,L} \rangle_{g,\kappa,R,\chi_+}$, where for a general test function $F$,

$$ \langle F \rangle_{g,\kappa,R,\chi_+} := \mathbb{E}_g^{\text{WP}} \left[ \sum_{\mathbf{T}:\chi(\mathbf{T}) \le \chi_+} \sum_{\substack{\gamma \in \mathcal{G}(X) \\ \gamma \sim \mathbf{T}}} F(\ell(\gamma)) \mathbb{1}_{\text{TF}_g^{\kappa,R}}(X) \right]. $$

Indeed, the possibility to restrict the sum to local topological types $\mathbf{T}$ of absolute Euler characteristic $\chi(\mathbf{T}) \le \chi_+$ comes from standard estimates presented in [2, Lemma 2.2 and Proposition A.1], which allow to write

$$ (13.6) \qquad \langle F \rangle_g^{\text{all}} = \sum_{\mathbf{T}:\chi(\mathbf{T}) \le \chi_+} \langle F \rangle_g^{\mathbf{T}} + \mathcal{O}_{\chi_+} \left( \frac{L^{c(\chi_+)+1} e^{2L}}{g^{\chi_+}} \|F\|_{\infty} \right) $$

if $F$ is supported on $[-L, L]$, the remainder being $\mathcal{O}_A (\|F\|_{\infty})$ with $L = A \log g$ and $\chi_+ = 2A + 1$.

We now decompose this average according to the local types it contains, i.e. we write

$$ (13.7) \qquad \langle F \rangle_{g,\kappa,R,\chi_+} = \sum_{\mathbf{T} \in \text{Loc}_{\chi_+}^{\kappa,R,L}} \langle F \rangle_{g,\kappa,R}^{\mathbf{T}} $$
where $\mathrm{Loc}_{\chi_+}^{\kappa,R,L}$ is the set of local topological types of loops, filling a surface of Euler characteristic bounded by $\chi_+$, that can be realized by a closed geodesic of length $\le L$ in a $(\kappa, R)$-tangle-free surface, and for $\mathbf{T} \in \mathrm{Loc}_{\chi_+}^{\kappa,R,L}$

$$ (13.8) \qquad \langle F \rangle_{g,\kappa,R}^{\mathbf{T}} := \mathbb{E}_g^{\mathrm{WP}} \left[ \sum_{\substack{\gamma \in \mathcal{G}(X) \\ \gamma \sim \mathbf{T}}} F(\ell(\gamma)) \mathbb{1}_{\mathrm{TF}_g^{\kappa,R}}(X) \right]. $$

We know from [3] that for $R = \kappa \log g$ and $L = A \log g$,

$$ (13.9) \qquad \#\mathrm{Loc}_{\chi_+}^{\kappa,R,L} \le \mathrm{fn}(\kappa, A, \chi_+)(\log g)^{n_{\mathrm{typ}}} $$

for a constant $n_{\mathrm{typ}} = \mathrm{fn}(\kappa, A, \chi_+) > 0$ depending only on $\kappa, A$ and $\chi_+$.

**13.3. Expression of the indicator function using the Moebius function.** Thanks to the result of [3], we can express the indicator function $\mathbb{1}_{\mathrm{TF}_g^{\kappa,R}}$ in terms of the Moebius function $\mu_{\kappa,R}$: for every hyperbolic surface $X$,

$$ (13.10) \qquad 1 - \mathbb{1}_{\mathrm{TF}_g^{\kappa,R}}(X) = \sum_{\tau \in \mathcal{S}(X)} \mu_{\kappa,R}(\tau) $$

where the sum runs over all the geodesic sub-c-surfaces of $X$.

This formula will allow to rewrite (13.7) as

$$ (13.11) \qquad \sum_{\mathbf{T} \in \mathrm{Loc}_{\chi_+}^{\kappa,R,L}} \langle F \rangle_g^{\mathbf{T}} - \sum_{\chi(\mathbf{S}) < \chi_+''} \sum_{\substack{\mathfrak{T} \in \mathrm{Loc}_{\mathbf{S},\chi_+,\chi_+}^{\kappa,R,L} \\ \chi_+(\mathfrak{T}) = \chi_+''}} \langle \mu_{\kappa,R} \otimes F \rangle_{g,Q}^{\mathfrak{T}} $$

up to a small error term. The set $\mathrm{Loc}_{\mathbf{S},\chi_+,\chi_+}^{\kappa,R,L}$ now is the set of local topological types $\mathfrak{T}$ of lc-surfaces $(\mathbf{S}, \zeta, \tau)$ such that:

* the loop $\zeta$ fills a surface of absolute Euler characteristic $\le \chi_+$, and has a topology that can be realised in a tangle-free surface with a geodesic having length less than $L$;

* the absolute Euler characteristic of $\tau$ is less than $\chi'_+$;

* the topology of the pair $(\zeta, \tau)$ is constrained by the fact that it can be realised in a hyperbolic surface $Z$, in which $\ell_Z(\zeta) \le L$ and $\tau$ is a $(\kappa, R)$-derived tangle.

We recall from (12.2) that

$$ (13.12) \qquad \langle \mu_{\kappa,R} \otimes F \rangle_{g,Q}^{\mathfrak{T}} := \mathbb{E}_g^{\mathrm{WP}} \left[ \sum_{\substack{\gamma \in \mathcal{G}(X) \\ \tau \in \mathcal{S}_Q(X) \\ (\gamma, \tau) \sim \mathfrak{T}}} \mu_{\kappa,R}(\tau) F(\ell(\gamma)) \right]. $$

Let us bound the error made going from (13.7) to (13.11). We prove the following, a generalization of [2, Lemma 10.11] to the general inclusion-exclusion using the Moebius function.

**Proposition 13.1.** Let $\mathbf{T} \in \mathrm{Loc}_{\chi_+}^{\kappa,R,L}$, for $L = A \log g$, $R = \kappa \log g$, $\chi'_+ = 2A + 4$, $Q = 2A + 5 + 6\chi'_+$. Provided that $\kappa(U_2(\chi'_+) + 37\chi'_+) < 1$, with $U_2$ the sequence in (12.5),
we have:

$$ (13.13) \quad \langle F \rangle_{g, \kappa, R}^{\mathbf{T}} = \langle F \rangle_{g}^{\mathbf{T}} - \mathbb{E}_{g}^{\text{WP}} \left[ \sum_{\substack{(c, \sigma) \in S(X) \\ \chi(\sigma) < \chi'_{+} \\ c \in \text{MC}_X(Q)}} \sum_{\substack{\gamma \in \mathcal{G}(X) \\ \gamma \sim \mathbf{T}}} \mu_{\kappa, R}(c, \sigma) F(\ell(\gamma)) \right] + \mathcal{O}_A (\|F\|_{\infty}). $$

*Proof.* By definition, we have that $TF_g^{\kappa,R} \subset \mathcal{B}_g^{\kappa,Q}$ for any $Q$. Furthermore, if $\tau_{\kappa,R}(X)$ denotes the maximal $(\kappa,R)$-derived tangle of $X$, defined in [3], we have $TF_g^{\kappa,R} \subset \{\chi(\tau_{\kappa,R}(X)) < \chi'_+\}$ because, on $TF_g^{\kappa,R}$, the property “all derived tangles have absolute Euler characteristic less than $\chi'_+$” is trivially true. We can rewrite these inclusions in the form

$$ \mathbb{1}_{TF_g^{\kappa,R}} = \mathbb{1}_{TF_g^{\kappa,R}} \mathbb{1}_{\mathcal{B}_g^{\kappa,Q}} \mathbb{1}_{[0, \chi'_+)}(\chi(\tau_{\kappa,R}(X)). $$

The Moebius inversion formula from [3] allows to rewrite the error term in (13.13) as $E_1 + E_2$, where

$$ (13.14) \quad E_1 = \mathbb{E}_g^{\text{WP}} \left[ \mathbb{1}_{[\chi'_+, +\infty)}(\chi(\tau_{\kappa,R}(X))) \left( 1 - \sum_{\substack{(c,\sigma) \in S(X) \\ \chi(\sigma) < \chi'_+ \\ c \in \text{MC}_X(Q)}} \mu_{\kappa,R}(c, \sigma) \right) \sum_{\substack{\gamma \in \mathcal{G}(X) \\ \gamma \sim \mathbf{T}}} F(\ell(\gamma)) \right] $$

and
(13.15)

$$ E_2 = \mathbb{E}_g^{\text{WP}} \left[ \left( \mathbb{1}_{\mathcal{B}_g^{\kappa,Q}}(X) - 1 \right) \mathbb{1}_{[0, \chi'_+)}(\chi(\tau_{\kappa,R}(X))) \left( 1 - \sum_{\substack{(c,\sigma) \in S(X) \\ \chi(\sigma) < \chi'_+ \\ c \in \text{MC}_X(Q)}} \mu_{\kappa,R}(c, \sigma) \right) \sum_{\substack{\gamma \in \mathcal{G}(X) \\ \gamma \sim \mathbf{T}}} F(\ell(\gamma)) \right]. $$

We shall bound these two terms separately, using the bound (12.5) on the Moebius function, [2, Lemma 2.2], as well as our probabilistic bounds on the events $\{\chi(\tau_{\kappa,R}(X)) \ge \chi'_+\}$ and $\mathcal{B}_g^{\kappa,Q}$ proven in [2, Appendix A]. This is done in the two following lemmas.

**Lemma 13.2.** We have

$$ |E_1| \le \mathrm{fn}(Q, \chi'_+, \kappa) \|F\|_{\infty} \frac{\mathrm{Le}^{L+R(36\chi'_+ + U_2(\chi'_+))}}{g^{\chi'_+/2-1}} $$

and, in particular, if we specify $L = A \log g$, $R = \kappa \log g$, $\chi'_+ = 2A+4$ and fix a value of $\kappa$ such that $\kappa < (U_2(\chi'_+) + 36\chi'_+)^{-1}$, then $|E_1| \le \mathrm{fn}(Q, A) \|F\|_{\infty}$.

*Proof.* We use the triangle inequality and the deterministic bound from [2, Lemma 2.2] to obtain

$$ (13.16) \quad |E_1| \le 10^4 L g e^L \|F\|_{\infty} \mathbb{E}_g^{\text{WP}} \left[ \mathbb{1}_{[\chi'_+, +\infty)}(\chi(\tau_{\kappa,R}(X))) \left( 1 + \sum_{\substack{(c,\sigma) \in S(X) \\ \chi(\sigma) < \chi'_+ \\ c \in \text{MC}_X(Q)}} |\mu_{\kappa,R}(c, \sigma)| \right) \right]. $$
Let us now bound the contribution of the Moebius function. We first use the fact that, if $(c, \sigma)$ with $c = (c_1, \dots, c_j)$, by [3],

$$|\mu_{\kappa,R}(c, \sigma)| = |\mu_{\kappa,R}(c)||\mu_{\kappa,R}(\sigma)| = \frac{1}{2j!} \mathbb{1}_{[0,\kappa]}(\ell_X^{\max}(c))|\mu_{\kappa,R}(\sigma)|.$$

Introducing the function

$$\mathcal{Y}_{\kappa,Q}(X) = \sum_{j=1}^{\infty} \frac{1}{j!} \sum_{\substack{c \in \text{MC}_X(Q) \\ c(c)=j}} \mathbb{1}_{[0,\kappa]}(\ell_X^{\max}(c))$$

as in [2, §A.2], we obtain

$$1 + \sum_{\substack{(c,\sigma) \in \mathcal{S}(X) \\ \chi(\sigma) < \chi'_+ \\ c \in \text{MC}_X(Q)}} |\mu(c, \sigma)| \le (1 + \mathcal{Y}_{\kappa,Q}) \left(1 + \sum_{\substack{\sigma \in \mathcal{S}(X) \\ \chi(\sigma) < \chi'_+ \\ c(\sigma) = 0}} |\mu(\sigma)|\right).$$

This allows us to reduce ourselves to bounding the sum of the Moebius function for purely 2d c-surfaces. This is done in [3], and we obtain a bound of the form

$$(13.17) \qquad \sum_{\substack{\sigma \in \mathcal{S}(X) \\ \chi(\sigma) < \chi'_+ \\ c(\sigma) = 0}} |\mu(\sigma)| \le \text{fn}(\chi'_+) g^{3\chi'_+} e^{RU_2(\chi'_+)}.$$

The factor $g^{3\chi'_+}$ in (13.17) is inconvenient. In order to compensate it, we split the event $\{\chi(\tau_{\kappa,R}(X)) \ge \chi'_+\}$ depending on the relative position of $\chi(\tau_{\kappa,R})$ and $8\chi'_+$:

$$\mathbb{1}_{[\chi'_+, +\infty)}(\chi(\tau_{\kappa,R}(X))) = \mathbb{1}_{[\chi'_+, 8\chi'_+)}(\chi(\tau_{\kappa,R}(X))) + \mathbb{1}_{[8\chi'_+, +\infty)}(\chi(\tau_{\kappa,R}(X))).$$

This is helpful because, now, we can improve (13.17) on the set where $\chi(\tau_{\kappa,R}(X)) < 8\chi'_+$:

$$\mathbb{1}_{[\chi'_+, 8\chi'_+)}(\chi(\tau_{\kappa,R}(X))) \sum_{\substack{\sigma \in \mathcal{S}(X) \\ \chi(\sigma) < \chi'_+ \\ c(\sigma) = 0}} |\mu(\sigma)| \le \text{fn}(\chi'_+) e^{RU_2(\chi'_+)}.$$

Altogether, we obtain

$$|E_1| \le \text{fn}(\chi'_+) \|F\|_{\infty} L g e^{L+RU_2(\chi'_+)}$$

$$\mathbb{E}_g^{\text{WP}} \left[ \left( 1 + \mathcal{Y}_{\kappa,Q} \left( \mathbb{1}_{[\chi'_+, +\infty)}(\chi(\tau_{\kappa,R}(X))) + g^{3\chi'_+} \mathbb{1}_{[8\chi'_+, +\infty)}(\chi(\tau_{\kappa,R}(X))) \right) \right) \right].$$

We then use the Cauchy–Schwarz inequality to bound the expectation above by

$$(1 + \mathbb{E}_g^{\text{WP}}[\mathcal{Y}_{\kappa,Q}^2]^{1/2}) \left( \mathbb{P}_g^{\text{WP}} \left( \chi(\tau_{\kappa,R}(X)) \ge \chi'_+ \right)^{1/2} + g^{3\chi'_+} \mathbb{P}_g^{\text{WP}} \left( \chi(\tau_{\kappa,R}(X)) \ge 8\chi'_+ \right)^{1/2} \right)$$

which, by [2, Propositions A.1 and A.5], is bounded by

$$\text{fn}(Q, \chi'_+, \kappa) R^{c(\chi'_+)} \left( \frac{e^{9R\chi'_+/2}}{g^{\chi'_+/2}} + g^{3\chi'_+} \frac{e^{36R\chi'_+}}{g^{4\chi'_+}} \right) \le \text{fn}(Q, \chi'_+, \kappa) \frac{e^{36R\chi'_+}}{g^{\chi'_+/2}}$$

which is our claim. □
Now, with exactly the same proof, now using [2, Proposition A.4] to bound the probability of the complement of $B_g^{\kappa,Q}$, we obtain:

**Lemma 13.3.** We have:

$$|E_2| \le \mathrm{fn}(Q, \chi'_+) \|F\|_{\infty} \frac{Le^{L+RU_2(\chi'_+)}}{g^{(Q-1)/2-1-3\chi'_+}}.$$

In particular, if we specify $L = A \log g$, $R = \kappa \log g$, $Q = 2A + 5 + 6\chi'_+$ and fix $\kappa$ such that $\kappa < U_2(\chi'_+)^{-1}$, then $|E_2| \le \mathrm{fn}(A, \chi'_+) \|F\|_{\infty}$.

Combining the two lemmas allows us to conclude. □

The following proposition shows that we can further restrict the Euler characteristic of $S$, the surface filled by $(\zeta, \tau)$:

**Proposition 13.4.** If $L = A \log g$ and $R = \kappa \log g$, we may find $\chi''_+ = \mathrm{fn}(A, \chi'_+)$ such that, for all $\kappa < 1/10$,

$$\sum_{S} \sum_{\mathcal{T} \in \mathrm{Loc}_{S, \chi_+, \chi'_+}^{\kappa, R, L}} \langle \mu_{\kappa, R} \otimes F \rangle_{g, Q}^{\mathcal{T}} = \sum_{S} \sum_{\chi(S) < \chi''_+} \langle \mu_{\kappa, R} \otimes F \rangle_{g, Q}^{\mathcal{T}} + \mathcal{O}_{A, Q}(1).$$

We omit the proof, a medley of all the previous arguments.

Note that the sum $\sum_S$ in equation (13.11) now runs over all c-surfaces of absolute Euler characteristic less than some $\chi''_+$ to be chosen later (we assume without loss of generality that $\chi_+, \chi'_+ \le \chi''_+$). The set of such c-surfaces is infinite, because $S$ could contain an arbitrary number of 1d-components (which do not contribute to the Euler characteristic). Thus we prefer to rewrite the second sum in (13.11) as

$$(13.18) \qquad \sum_{S} \sum_{\substack{\chi(S) < \chi''_+ \\ c(S) = 0}} \sum_{\mathcal{T} \in \mathrm{Loc}_{S, \chi_+, \chi'_+}^{\kappa, R, L}} \sum_{j=0}^{+\infty} \langle \mu_{\kappa, R} \otimes F \rangle_{g, Q}^{\rho_j \mathcal{T}},$$

where the first sum, running over purely 2d c-surfaces of bounded Euler characteristic, is now finite. Recall that the operation $\rho_j$ consists in adding $j$ 1d components to a c-surface.

Now, we showed in [3] that, if $L = A \log g$, $R = \kappa \log g$, $c(S) = 0$ and $\chi(S) < \chi''_+$, then

$$(13.19) \qquad \#\mathrm{Loc}_{S, \chi_+, \chi'_+}^{\kappa, R, L} \le \mathrm{fn}(\kappa, A, \chi''_+) g^{10\chi'_+\chi''_+\kappa}.$$

Therefore, we shall take $\kappa$ very small so that this polynomial contribution in $g$ does not interfere with our method.

The important thing here is that we have removed the indicator function $1_{\mathrm{TF}_g^{\kappa,R}}$ from the average we consider. This allows to use Theorems 1.15 and 12.16 that describe the special structure of the volume functions associated to $\langle F \rangle_g^T$ and $\langle \mu_{\kappa,R} \otimes F \rangle_{g,Q}^\mathcal{T}$.
**13.4. Cancellation and the Friedman-Ramanujan hypothesis.** Theorems 1.15 and 12.16 can be combined with the integration by parts phenomenon presented in [2, Proposition 3.19] to obtain the following propositions. The underlying principle is that $\mathcal{D}^m$ vanishes on $\mathbb{C}_{m-1}[\ell]e^{\ell/2}$, hence the choice of our test function (13.3).

First, Theorem 1.15 implies the following, related to the average $\langle F_{m,L} \rangle_g^\mathbf{T}$.

**Proposition 13.5 ([2]).** Let $\mathbf{T} = [\mathbf{S}, \mathbf{c}]_{\text{loc}}$ be a local topological type with $\chi(\mathbf{S}) \le \chi_+$. For any integer $N \ge 0$, there exists constants $c, K = \text{fn}(N, \chi_+)$ such that, for any large enough $g$, any $m \ge K$, any $\eta > 0$ and any $L \ge 1$,

$$ \langle F_{m,L} \rangle_g^\mathbf{T} \le \text{fn}(N, \chi_+, \eta, m) \left( L^c + \frac{e^{L(\frac{1}{2}+\eta)}}{g^{N+1}} \right). $$

Summing over $\mathbf{T} \in \text{Loc}_{\chi_+}^{\kappa,R,L}$, and using (13.9) to bound the number of terms in the sum, we obtain that for $R = \kappa \log g$ and $L = A \log g$,

$$ (13.20) \qquad \sum_{\mathbf{T} \in \text{Loc}_{\chi_+}^{\kappa,R,L}} \langle F_{m,L} \rangle_g^\mathbf{T} \le \text{fn}(N, \kappa, A, \chi_+, m, \eta) L^{n_{\text{typ}}} \left( L^c + \frac{e^{L(\frac{1}{2}+\eta)}}{g^{N+1}} \right) $$

where we recall that $c = \text{fn}(N, \chi_+)$ and $n_{\text{typ}} = \text{fn}(\kappa, A, \chi_+)$.

We now state a similar property for the average (13.18), now using Theorem 12.16.

**Proposition 13.6.** Let $\mathcal{T} \in \text{Loc}_{\mathbf{S},\chi_+,\chi'_+}^{\kappa,R,L}$ for a filling type $\mathbf{S}$ such that $c(\mathbf{S}) = 0$ and $\chi(\mathbf{S}) \le \chi''_+$. For any integer $N \ge 0$, there exists constants $c, K = \text{fn}(N, \chi''_+)$ such that, for any large enough $g$, any $m \ge K$, any $\eta > 0$ and any $L \ge 1$, $R = \kappa \log(g)$, $U_3(\chi) := U_2(\chi) + 13\chi$,

$$ \sum_{j=0}^{+\infty} \langle \mu_{\kappa,R} \otimes F_{m,L} \rangle_{g,Q}^{\rho_j \mathcal{T}} \le \text{fn}(Q, N, \chi''_+, \kappa, m, \eta) g^{\kappa U_3(\chi'_+)} \left( L^c + \frac{e^{L(\frac{1}{2}+\eta)}}{g^{N+1}} \right). $$

*Proof.* First, we observe that we can truncate the sum to values $j \le \log(g)$ due to the decay in $j$, as done in [2, Lemma 10.12]. We then use our cancellation argument with the Friedman-Ramanujan function $h_{g,Q,N}^{\mu_{\kappa,R},\mathcal{T},j}$, and sum the contribution associated to each $j$, using the fact that, for the Moebius function $\mu_{\kappa,R}$, $f(j, \chi) = U_1(\chi)e^{RU_2(\chi)/j!}$ and hence the constant appearing is smaller than

$$ (13.21) \qquad \text{fn}(Q, N, \chi''_+) e^{R(U_2(\chi'_+) + 13\chi''_+)} \sum_{j=0}^{+\infty} \frac{M^j}{j!} = \text{fn}(Q, N, \chi''_+) g^{\kappa U_3(\chi'_+)} e^M $$

$$ \text{where } M = \text{fn}(Q, N, \chi''_+, \kappa). \quad \square $$

Summing over all filling types $\mathbf{S}$ such that $c(\mathbf{S}) = 0$ and $\chi(\mathbf{S}) \le \chi''_+$ and all local topological types $\mathcal{T} \in \text{Loc}_{\mathbf{S},\chi_+,\chi'_+}^{\kappa,R,L}$ (the cardinality of which is bounded by (13.19)), we
obtain an integer K such that, for any $m \ge K$,
(13.22)

$$ \sum_{\substack{\mathbf{S} \\ \chi(\mathbf{S})<\chi_+'' \\ \mathfrak{c}(\mathbf{S})=0}} \sum_{\mathfrak{T} \in \text{Loc}_{\kappa,R,L}} \sum_{\mathbf{S},\chi_+,\chi_+'}^{+\infty} \langle \mu_{\kappa,R} \otimes F_{m,L} \rangle_{g,Q}^{\rho_j \mathfrak{T}} \le \text{fn}(Q,N,\chi_+'',\kappa,m,\eta) g^{\kappa U_4(\chi_+'')} \left( L^c + \frac{e^{L(\frac{1}{2}+\eta)}}{g^{N+1}} \right) $$

where $U_4(\chi) := U_2(\chi) + 13\chi + 10\chi^2$. The important thing here is that the bound is subexponential in $L$: there is a polynomial term $L^c$, and an exponential term $e^{L(\frac{1}{2}+\eta)}$ which is compensated by the factor $g^{-(N+1)}$. Balancing these terms, we can conclude the proof of Theorem 1.6.

13.5. **Final choice of parameters.** Let $\alpha \in (0, 1/2)$ and $0 < \epsilon < \frac{1}{4} - \alpha^2$. We pick our parameters the following way.

* The order $N$ of the expansion is chosen to be an integer so that $2\alpha(N+2) > 1$.

* We take $L = A \log g$ with $A = 2(N+2)$; this ensures that $e^{L/2}/g^{N+1} = g$, to match the topological term of the trace formula.

* $\chi_+ = 2A+1$ to be able to limit the local types of loops filling a surface of Euler characteristic $< \chi_+$ (as prescribed after (13.6)).

* $\chi'_+ = 2A+4$ and $Q = 2A+5+6\chi'_+$, as recommended in Proposition 13.1, to reduce the possibilities to derived tangles $\tau$ with $\chi(\tau) < \chi'_+$ with a 1d-component separating at most $Q$ connected components.

* $\chi''_+ = \text{fn}(A, \chi_+, \chi'_+)$ is prescribed in Proposition 13.4, implying that we can restrict attention to c-filling types $\mathbf{S}$ with $\chi(\mathbf{S}) < \chi''_+$.

* We can now apply Propositions 13.5 and 13.6 with a parameter $\eta < \epsilon$.

* This allows to fix the integer $m$ in the definition of $F_{m,L}$: it is chosen as the maximum of the integers given by Propositions 13.5 and 13.6, for the finite number of c-filling types $\mathbf{S}$ with $\chi(\mathbf{S}) < \chi''_+$ and $\mathfrak{c}(\mathbf{S}) = 0$.

* As announced, we take $R = \kappa \log g$, where we now pick the parameter $\kappa$ so that

$$ \begin{cases} 0 < \kappa < \min(1/10, 2 \operatorname{argsinh} 1) \\ (U_2(\chi'_+) + 36\chi'_+)\kappa < 1 \\ U_4(\chi''_+)\kappa < \alpha A - 1 \end{cases} $$

as recommended in Proposition 13.1 and to ensure that the upper bound obtained in equation (13.21) is $O(e^{(\alpha+\eta)L})$.

Then, for $F_{m,L}(\ell) = \ell D^m H_L(\ell) e^{-\ell/2}$, the expected trace (13.2) is $o(e^{(\alpha+\epsilon)L})$ as $g \to +\infty$. By the inequality (13.5), this allows to conclude that

$$ \limsup_{g \to +\infty} \mathbb{P}_g^{\text{WP}} \left( \delta \le \lambda_1 \le \frac{1}{4} - \alpha^2 - \epsilon \right) \le C_{\text{tf}} \kappa^2 $$

for the universal constant $C_{\text{tf}}$ from (13.4). Since $\kappa$ can be taken to be arbitrarily small, this concludes the proof of Theorem 1.6.

APPENDIX A. LIST OF NOTATIONS
<table>
  <thead>
    <tr>
      <th>fn((u<sub>i</sub>)<sub>i∈V</sub>)</th>
      <th>Function of the parameters (u<sub>i</sub>)<sub>i∈V</sub>.</th>
      <th>Nota 1.13</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>S<sub>g</sub>, g ≥ 2</td>
      <td>Fixed compact surface of genus g.</td>
      <td></td>
    </tr>
    <tr>
      <td>X ∈ M<sub>g</sub></td>
      <td>A metric X on S<sub>g</sub>, up to isometry.</td>
      <td></td>
    </tr>
    <tr>
      <td>G(X)</td>
      <td>Set of primitive closed geodesics of X.</td>
      <td>Def 1.1</td>
    </tr>
    <tr>
      <td>ℓ<sub>X</sub>(γ)</td>
      <td>Length of the geodesic representative of γ in X.</td>
      <td>Def 1.1</td>
    </tr>
    <tr>
      <td>P<sup>WP</sup><sub>g</sub>, E<sup>WP</sup><sub>g</sub></td>
      <td>Weil-Petersson probability measure and expectation.</td>
      <td></td>
    </tr>
    <tr>
      <td>λ<sub>1</sub>(X)</td>
      <td>First non-trivial eigenvalue of Laplacian.</td>
      <td>Thm 1.6</td>
    </tr>
    <tr>
      <td>T = [S, c]<sub>loc</sub></td>
      <td>Local topological type T, i.e. a surface S filled by a multi-loop c.</td>
      <td>Def 1.1</td>
    </tr>
    <tr>
      <td>⟨F⟩<sup>T</sup><sub>g</sub></td>
      <td>Average E<sup>WP</sup><sub>g</sub>(∑<sub>γ of type T</sub> F(ℓ<sub>X</sub>(γ))).</td>
      <td>Def 1.1</td>
    </tr>
    <tr>
      <td>V<sup>T</sup><sub>g</sub>(ℓ)</td>
      <td>Volume function corresponding to the average ⟨·⟩<sup>T</sup><sub>g</sub>.</td>
      <td>Prop 1.2</td>
    </tr>
    <tr>
      <td>f<sup>T</sup><sub>k</sub>(ℓ)</td>
      <td>k-th coefficient of the asymptotic expansion of V<sup>T</sup><sub>g</sub>(ℓ).</td>
      <td>Thm 1.3</td>
    </tr>
    <tr>
      <td>F and R</td>
      <td>Friedman-Ramanujan functions and remainders.</td>
      <td>Def 1.7</td>
    </tr>
    <tr>
      <td>F<sup>K</sup>, F<sup>K,N</sup>, R<sup>N</sup></td>
      <td>FR functions and remainders of bounded growth.</td>
      <td>Def 1.7</td>
    </tr>
    <tr>
      <td>F<sub>w</sub> and R<sub>w</sub></td>
      <td>Weak forms of F and R.</td>
      <td>Def 1.10</td>
    </tr>
    <tr>
      <td>‖·‖<sup>R<sup>N</sup></sup>, ‖·‖<sup>F<sup>K,N</sup></sup></td>
      <td>Norms on those spaces (see also weak forms version).</td>
      <td>Def 1.12</td>
    </tr>
    <tr>
      <td>(g<sub>S</sub>, n<sub>S</sub>), χ(S), ∂S</td>
      <td>Signature of S, absolute value of Euler characteristic and boundary of S.</td>
      <td>Nota 1.16</td>
    </tr>
    <tr>
      <td>T<sup>*</sup><sub>g<sub>S</sub>,n<sub>S</sub></sub></td>
      <td>Teichmüller space of S (with free > 0 boundary length).</td>
      <td>§1.3.1</td>
    </tr>
    <tr>
      <td>T<sub>g<sub>S</sub>,n<sub>S</sub></sub>(x)</td>
      <td>Teichmüller space of S (with fixed boundary lengths x).</td>
      <td>§1.3.1</td>
    </tr>
    <tr>
      <td>dVol<sup>WP</sup><sub>g<sub>S</sub>,n<sub>S</sub></sub></td>
      <td>Weil-Petersson measure on T<sup>*</sup><sub>g<sub>S</sub>,n<sub>S</sub></sub>.</td>
      <td>§1.3.1</td>
    </tr>
    <tr>
      <td>ρ[T]</td>
      <td>Number of unshielded simple portions of T.</td>
      <td>Def 1.20</td>
    </tr>
    <tr>
      <td>V</td>
      <td>A subset of ∂S.</td>
      <td>Nota 1.21</td>
    </tr>
  </tbody>
</table>
<table>
  <tr>
    <td>m</td>
    <td>A matching of the complement of V as ⋃<sub>j</sub>{i<sub>j</sub>, i'<sub>j</sub>}. Nota 1.21</td>
  </tr>
  <tr>
    <td>χ<sub>z</sub></td>
    <td>A smooth function vanishing at 0. Nota 1.21</td>
  </tr>
  <tr>
    <td>J = J<sub>T,V,m,f</sub></td>
    <td>The main integral to understand. Eq (1.12)</td>
  </tr>
  <tr>
    <td>f<sub>j</sub>, ˜g<sub>j</sub></td>
    <td>Functions appearing in J. Nota 1.21</td>
  </tr>
  <tr>
    <td>K<sub>j</sub>, N<sub>j</sub>, K, N</td>
    <td>Friedman-Ramanujan exponents. Nota 1.21</td>
  </tr>
  <tr>
    <td>P = P<sub>x</sub>, L = L<sub>x</sub></td>
    <td>Operators ∫<sub>0</sub><sup>x</sup> and id - ∫<sub>0</sub><sup>x</sup>. §1.3.7</td>
  </tr>
  <tr>
    <td>c = (c<sub>1</sub>, ..., c<sub>n</sub>)</td>
    <td>A multi-loop filling S. Def 2.2.</td>
  </tr>
  <tr>
    <td>J, o(J), t(J)</td>
    <td>A segment J, its origin and terminus. Nota 2.5</td>
  </tr>
  <tr>
    <td>p·q</td>
    <td>Concatenation of the paths p and q. Nota 2.6</td>
  </tr>
  <tr>
    <td>J&#773;</td>
    <td>Orthogeodesic homotopic to J with gliding endpoints. Rem 2.8</td>
  </tr>
  <tr>
    <td>H<sup>2</sup></td>
    <td>The hyperbolic plane.</td>
  </tr>
  <tr>
    <td>d&#773;</td>
    <td>The algebraic distance along a geodesic. Def 2.10</td>
  </tr>
  <tr>
    <td>l(γ), l<sub>Y</sub>(γ)</td>
    <td>Length of a path v.s. the geodesic in a free homotopy class. §2.1.5</td>
  </tr>
  <tr>
    <td>r</td>
    <td>Number of intersections of c. Nota 2.12</td>
  </tr>
  <tr>
    <td>β, (B<sub>1</sub>, ..., B<sub>r</sub>)</td>
    <td>Simple multi-loop and bars obtained when opening the intersections of c. §2.2.1</td>
  </tr>
  <tr>
    <td>c<sup>OP</sup></td>
    <td>The representative of c obtained after opening. §2.2.1</td>
  </tr>
  <tr>
    <td>(β<sub>λ</sub>)<sub>λ∈Λ<sup>β</sup></sub></td>
    <td>The components of the multi-loop β. Rem 2.13</td>
  </tr>
  <tr>
    <td>B<sub>j</sub><sup>±</sup></td>
    <td>Choice of orientation of the bars. §2.2.2</td>
  </tr>
  <tr>
    <td>Θ</td>
    <td>The set of q = (j, ε) ∈ {1, ..., r} × {+, -}. §2.2.3</td>
  </tr>
  <tr>
    <td>I<sub>q</sub>, q ∈ Θ</td>
    <td>Labelling of the simple portions. §2.2.3</td>
  </tr>
  <tr>
    <td>Sign(q), ι(q)</td>
    <td>Sign function and sign switching involution. Def 2.15</td>
  </tr>
  <tr>
    <td>λ(q)</td>
    <td>The component of β in which B<sub>q</sub> terminates. Nota 2.16</td>
  </tr>
</table>
<table>
  <tr>
    <td>Θ<sub>t</sub>(λ), Θ<sub>t</sub>(W)</td>
    <td>The set of q ∈ Θ s.t. λ(q) = λ or λ(q) ∈ W.</td>
    <td>Nota 2.16</td>
  </tr>
  <tr>
    <td>β<sub>λ,init</sub>, I<sub>q,init</sub></td>
    <td>Paths on c corresponding to β<sub>λ</sub> and I<sub>q</sub>.</td>
    <td>Rem 2.18</td>
  </tr>
  <tr>
    <td>D</td>
    <td>A diagram (β, B).</td>
    <td>Def 2.19</td>
  </tr>
  <tr>
    <td>D<sup>(1)</sup></td>
    <td>Diagram obtained by opening all intersections the first way.</td>
    <td>§2.2.4</td>
  </tr>
  <tr>
    <td>p(q)</td>
    <td>The concatenation of B<sub>q</sub> and I<sub>q</sub>.</td>
    <td>Eq (2.5)</td>
  </tr>
  <tr>
    <td>σ</td>
    <td>The permutation of Θ allowing to go along c.</td>
    <td>Eq (2.6)</td>
  </tr>
  <tr>
    <td>n<sub>i</sub></td>
    <td>Length of the cycle of σ describing c<sub>i</sub>.</td>
    <td>Eq (2.6)</td>
  </tr>
  <tr>
    <td>Z<sub>n</sub></td>
    <td>Cyclic set Z/nZ.</td>
    <td>§2.5.1</td>
  </tr>
  <tr>
    <td>Comp(q)</td>
    <td>Label of the component c<sub>i</sub> of c in which q appears.</td>
    <td>§2.5.2</td>
  </tr>
  <tr>
    <td>Θ<sup>i</sup> = (q<sup>i</sup><sub>k</sub>)<sub>k∈Z<sub>n<sub>i</sub></sub></sub></td>
    <td>Set of q ∈ Θ of component Comp(q) = i.</td>
    <td>§2.5.2</td>
  </tr>
  <tr>
    <td>Θ, Θ<sup>i</sup></td>
    <td>Periodic extensions of Θ and Θ<sup>i</sup>.</td>
    <td>§2.5.3</td>
  </tr>
  <tr>
    <td>C, U</td>
    <td>Set of crossing v.s. non-crossing indices.</td>
    <td>Def 2.26</td>
  </tr>
  <tr>
    <td>L&#773;, θ&#773;</td>
    <td>Coordinate system on T<sup>*</sup><sub>g<sub>S</sub>,n<sub>S</sub></sub>.</td>
    <td>§3.1</td>
  </tr>
  <tr>
    <td>L(q), εL(q)</td>
    <td>Length of B<sub>q</sub> and its signed version.</td>
    <td>Nota 3.3</td>
  </tr>
  <tr>
    <td>P<sub>1</sub>,...,P<sub>r</sub></td>
    <td>Decomposition of S in pairs of pants.</td>
    <td>§3.2</td>
  </tr>
  <tr>
    <td>Σ<sub>0</sub>,...,Σ<sub>r</sub></td>
    <td>Auxilliary families of multi-curves and surfaces.</td>
    <td>§3.2</td>
  </tr>
  <tr>
    <td>S<sub>0</sub>,...,S<sub>r</sub></td>
    <td></td>
    <td></td>
  </tr>
  <tr>
    <td>(Γ<sub>λ</sub>)<sub>λ∈Λ</sub></td>
    <td>Set of boundary components of (P<sub>1</sub>,...,P<sub>r</sub>).</td>
    <td>§3.3.1</td>
  </tr>
  <tr>
    <td>Λ<sub>BC</sub>, Λ<sub>in</sub></td>
    <td>Indices λ such that Γ<sub>λ</sub> is a boundary / inner curve.</td>
    <td>§3.3.2</td>
  </tr>
  <tr>
    <td>Λ<sup>β</sup>, Λ<sup>Γ</sup></td>
    <td>Indices λ such that Γ<sub>λ</sub> is / isn't a component of β.</td>
    <td>§3.3.2</td>
  </tr>
  <tr>
    <td>Λ<sup>β</sup><sub>BC</sub>, Λ<sup>β</sup><sub>in</sub>, Λ<sup>Γ</sup><sub>BC</sub>, Λ<sup>Γ</sup><sub>in</sub></td>
    <td>Intersection of the above sets.</td>
    <td>§3.3.2</td>
  </tr>
  <tr>
    <td>step(λ), λ &lt; λ'</td>
    <td>Step when Γ<sub>λ</sub> is constructed, and induced order on Λ.</td>
    <td>Def 3.8</td>
  </tr>
</table>
<table>
  <tbody>
    <tr>
      <td>y<sub>λ</sub>, α<sub>λ</sub></td>
      <td>Fenchel-Nielsen parameters associated to P<sub>1</sub>, ..., P<sub>r</sub>.</td>
      <td>§3.4</td>
    </tr>
    <tr>
      <td>Q</td>
      <td>Image of the map Y ↦→ (L̃, θ̃).</td>
      <td>Thm 3.11</td>
    </tr>
    <tr>
      <td>V<sup>β</sup>, V<sup>Γ</sup></td>
      <td>Intersection of V with Λ<sup>β</sup> and Λ<sup>Γ</sup>.</td>
      <td>Cor 3.13</td>
    </tr>
    <tr>
      <td>T<sub>V,m</sub></td>
      <td>Expression of a Dirac-distribution in (L̃, θ̃).</td>
      <td>Cor 3.13</td>
    </tr>
    <tr>
      <td>a<sup>t</sup>, w<sup>t</sup>, k<sup>θ</sup></td>
      <td>Matrices in PSL<sub>2</sub>(R) representing different moves in T<sup>1</sup>H<sup>2</sup>.</td>
      <td>§4.1</td>
    </tr>
    <tr>
      <td>p̃(q) = B̃<sub>q</sub> • Ĩ<sub>q</sub></td>
      <td>Path associated to q ∈ Θ.</td>
      <td>§4.2.2</td>
    </tr>
    <tr>
      <td>β̃<sub>q</sub>, B̃<sub>q</sub>, Ĩ<sub>q</sub>, p̃(q)</td>
      <td>Lifts of β<sub>λ(q)</sub>, B̃<sub>q</sub>, Ĩ<sub>q</sub> and p̃(q) to H<sup>2</sup>, for q ∈ Θ<sup>i</sup>.</td>
      <td>§4.2.2</td>
    </tr>
    <tr>
      <td>B̃<sub>mn</sub>, L<sub>mn</sub>, z<sub>mn</sub></td>
      <td>Orthogeodesic from β̃<sub>m</sub> to β̃<sub>n</sub>, length and terminus.</td>
      <td>§4.2.2</td>
    </tr>
    <tr>
      <td>hyp<sub>+</sub> and hyp<sub>-</sub></td>
      <td>The hyperbolic functions cosh and sinh.</td>
      <td>Nota 4.5</td>
    </tr>
    <tr>
      <td>hyp<sub>δ</sub>(t)</td>
      <td>The product of the hyperbolic functions hyp<sub>δ<sub>i</sub></sub>(t<sub>i</sub>).</td>
      <td>Nota 4.5</td>
    </tr>
    <tr>
      <td>E<sub>±</sub>(s)</td>
      <td>Useful matrices.</td>
      <td>Eq (4.8)</td>
    </tr>
    <tr>
      <td>M<sub>n</sub>(x, t)</td>
      <td>Function used to compute ℓ<sub>Y</sub>(c<sub>i</sub>) as a function of L̃, θ̃.</td>
      <td>Nota 4.7</td>
    </tr>
    <tr>
      <td>K<sub>qq'</sub>, K̄<sub>qq'</sub>, τ<sub>qq'</sub></td>
      <td>Segment between two bars of the same sign, and its length.</td>
      <td>§4.3</td>
    </tr>
    <tr>
      <td>K<sub>q</sub>, τ<sub>q</sub></td>
      <td>Special case where q' is the next bar of the same side.</td>
      <td>§4.3</td>
    </tr>
    <tr>
      <td>Θ<sub>τ</sub>(qq')</td>
      <td>Set of indices needed to write τ<sub>qq'</sub> in terms of θ̃.</td>
      <td>§4.3</td>
    </tr>
    <tr>
      <td>Θ<sub>τ</sub>(W)</td>
      <td>Set of indices needed to write (τ<sub>q</sub>)<sub>q∈W</sub> in terms of θ̃.</td>
      <td>§4.3</td>
    </tr>
    <tr>
      <td>p̃<sup>λ</sup><sub>k</sub>, q<sup>λ</sup><sub>k</sub>, q'<sup>λ</sup><sub>k</sub></td>
      <td>Segments appearing when we describe Γ<sub>λ</sub> and their indices.</td>
      <td>§4.4.2</td>
    </tr>
    <tr>
      <td>m(λ)</td>
      <td>Number of segments in the writing of Γ<sub>λ</sub>.</td>
      <td>§4.4.2</td>
    </tr>
    <tr>
      <td>L̃<sup>λ</sup>, τ̃<sup>λ</sup></td>
      <td>Length parameters appearing in Γ<sub>λ</sub>.</td>
      <td>§4.4.2</td>
    </tr>
    <tr>
      <td>∇α, δα</td>
      <td>Sequences of indices associated to α.</td>
      <td>Thm 4.11<br>Nota (4.13)</td>
    </tr>
    <tr>
      <td>Q<sub>m</sub>(x, t), F<sub>m</sub>(x, t)</td>
      <td>Functions used to compute ℓ<sub>Y</sub>(Γ<sub>λ</sub>) as a function of L̃, θ̃.</td>
      <td>Nota 4.13</td>
    </tr>
  </tbody>
</table>
<table>
  <tbody>
    <tr>
      <td>Θ<sub>ℓ</sub>(λ), Θ<sub>ℓ</sub>(W)</td>
      <td>Sets of indices needed to write ℓ<sub>Y</sub>(Γ<sub>λ</sub>), or (ℓ<sub>Y</sub>(Γ<sub>λ</sub>))<sub>λ∈W</sub>.</td>
      <td>Nota 4.15</td>
    </tr>
    <tr>
      <td>V<sup>β</sup>, V<sup>Γ</sup>, V<sub>+</sub><sup>β</sup>, V<sub>+</sub></td>
      <td>Sets of variables of interest in Λ.</td>
      <td>§5.1</td>
    </tr>
    <tr>
      <td>W, W<sup>β</sup>, W<sup>Γ</sup>, r<sub>λ</sub></td>
      <td>Set of variables for which we took a remainder term r<sub>λ</sub>.</td>
      <td>§5.2.3</td>
    </tr>
    <tr>
      <td>d<sub>λ</sub>, K<sub>q</sub></td>
      <td>Exponents appearing in the integral J.</td>
      <td>§5.2.3</td>
    </tr>
    <tr>
      <td>Θ<sub>ac</sub>, Θ<sub>nt</sub></td>
      <td>Set of active and neutral parameters.</td>
      <td>Nota 5.1</td>
    </tr>
    <tr>
      <td>θ&#773;<sub>ac</sub>, θ&#773;<sub>nt</sub></td>
      <td>Corresponding subvectors of θ&#773;.</td>
      <td>Nota 5.1</td>
    </tr>
    <tr>
      <td>f<sub>1</sub> ★ ★ ★ f<sub>n</sub>|<sub>ϕ</sub><sup>h</sup></td>
      <td>Pseudo-convolution of (f<sub>i</sub>)<sub>1≤i≤n</sub> with height h and weight ϕ.</td>
      <td>Nota 6.4</td>
    </tr>
    <tr>
      <td>dx/dℓ = dx<sub>1</sub>...dx<sub>n</sub>/dℓ</td>
      <td>Desintegration of Lebesgue measure on the level-sets of h.</td>
      <td>Eq (6.2)</td>
    </tr>
    <tr>
      <td>F<sub>j</sub><sup>K</sup>, G<sub>j</sub>, H<sub>j</sub><sup>t</sup>, I<sub>j</sub><sup>t</sup></td>
      <td>Auxilliary functions associated to f<sub>j</sub>.</td>
      <td>Nota 6.12</td>
    </tr>
    <tr>
      <td>Int<sub>π</sub>, B, Φ<sub>j</sub></td>
      <td>Integrals we obtain when we apply our operator L, and associated quantities.</td>
      <td>Thm 6.16</td>
    </tr>
    <tr>
      <td>E<sub>n</sub><sup>(a)</sup>, E<sub>n</sub><sup>(a)</sup></td>
      <td>Functional spaces in which we hope to find ϕ and h.</td>
      <td>Def 6.20, 6.22</td>
    </tr>
    <tr>
      <td>||·||<sub>α,a</sub></td>
      <td>Semi-norms on E<sub>n</sub><sup>(a)</sup>.</td>
      <td>Def 6.20</td>
    </tr>
    <tr>
      <td>(h), (ϕ), (f)</td>
      <td>Assumptions on the functions h, ϕ, (f<sub>j</sub>)<sub>1≤j≤n</sub>.</td>
      <td>Nota 6.29</td>
    </tr>
    <tr>
      <td>(ϕ-CE), Z<sub>j</sub></td>
      <td>Assumption as a comparison estimate and height function.</td>
      <td>Nota 6.34</td>
    </tr>
    <tr>
      <td>B&#773;<sub>ij</sub>, L<sub>ij</sub>, z<sub>ij</sub></td>
      <td>Orthogeodesics, their lengths and endpoints.</td>
      <td>§7.2</td>
    </tr>
    <tr>
      <td>I<sub>j</sub><sup>δ</sup></td>
      <td>The crossing intervals.</td>
      <td>Nota 7.8</td>
    </tr>
    <tr>
      <td>Ξ(I<sub>j</sub><sup>δ</sup>), P<sub>ξ<sub>j</sub><sup>δ</sup></sub>, I<sub>j</sub><sup>δ</sup>(ξ)</td>
      <td>The partitioning associated to the crossing interval I<sub>j</sub><sup>δ</sup>.</td>
      <td>§7.3.2</td>
    </tr>
    <tr>
      <td>Ξ<sup>i</sup>, P<sub>ξ<sub>i</sub></sub>, I(ξ<sup>i</sup>), n<sub>ξ<sub>i</sub></sub></td>
      <td>The partitioning associated to c<sub>i</sub>.</td>
      <td>Nota 7.14</td>
    </tr>
    <tr>
      <td>I'<sub>q</sub> = [G<sub>q</sub>, D<sub>q</sub>], θ&#773;<sub>q</sub>, L&#773;<sub>q</sub></td>
      <td>The new segments and lengths after erasing.</td>
      <td>Nota 7.15</td>
    </tr>
    <tr>
      <td>Ξ, P<sub>ξ</sub>, I(ξ)</td>
      <td>Product over all partitions for 1 ≤ i ≤ n.</td>
      <td>Nota 7.3.6</td>
    </tr>
    <tr>
      <td>χ<sub>C,0</sub>, χ<sub>C,∞</sub></td>
      <td>Functions used to construct partition of unity.</td>
      <td>Nota 7.22</td>
    </tr>
    <tr>
      <td>(Ψ<sub>ξ,Q</sub>)<sub>ξ∈Ξ, Q⊆I(ξ)</sub></td>
      <td>Partition of unity to deal with crossing parameters.</td>
      <td>Def 7.23</td>
    </tr>
  </tbody>
</table>
<table>
  <thead>
    <tr>
      <th>Θ<sub>nt</sub>(ξ, Q)</th>
      <th>Neutral parameters associated to (ξ, Q).</th>
      <th>Nota 7.24</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>Γ = (J<sub>j</sub>, K<sub>j</sub>)<sub>j∈Z<sub>m</sub></sub></td>
      <td>Polygonal curve, its bridges and cells.</td>
      <td>Def 8.1</td>
    </tr>
    <tr>
      <td>Cell(Γ), Br(Γ)</td>
      <td>Cells and bridges of Γ.</td>
      <td>Def 8.2</td>
    </tr>
    <tr>
      <td>|Γ|</td>
      <td>Combinatorial length of Γ.</td>
      <td>Nota 8.3</td>
    </tr>
    <tr>
      <td>L<sub>j</sub>, τ<sub>j</sub></td>
      <td>Length of geodesic representatives of the bridges and cells.</td>
      <td>§8.3.1</td>
    </tr>
    <tr>
      <td>Z<sup>Γ</sup><sub>K</sub></td>
      <td>Height of a cell.</td>
      <td>Def 8.8</td>
    </tr>
    <tr>
      <td>χ<sub>Γ,0</sub>, χ<sub>Γ,∞</sub>, a</td>
      <td>Functions used to construct partition of unity, cutoff value.</td>
      <td>Nota 8.14</td>
    </tr>
    <tr>
      <td>(Γ, d)</td>
      <td>Decorated polygonal curve.</td>
      <td>Def 8.18</td>
    </tr>
    <tr>
      <td>(Γ, d)</td>
      <td>Exhaustion.</td>
      <td>Def 8.19</td>
    </tr>
    <tr>
      <td>(Γ<sup>∞</sup>, d<sup>∞</sup>)</td>
      <td>Terminal step of the exhaustion.</td>
      <td>Nota 8.21</td>
    </tr>
    <tr>
      <td>Term(Γ, d)</td>
      <td>Set of terminal cells of the exhaustion.</td>
      <td>Nota 8.35</td>
    </tr>
    <tr>
      <td>(Ψ<sub>Γ,d</sub>)<sub>(Γ,d)∈C</sub></td>
      <td>Partition of unity associated to (Γ<sub>λ</sub>)<sub>λ∈Λ</sub>.</td>
      <td>§8.6</td>
    </tr>
    <tr>
      <td>Term<sub>BC</sub>(Γ, d)</td>
      <td>Set of boundary terminal cells of an exhaustion in C.</td>
      <td>Nota 8.38</td>
    </tr>
    <tr>
      <td>Θ<sub>surv</sub>, τ&#773;<sub>surv</sub></td>
      <td>Set of surviving boundary segments and their lengths.</td>
      <td>Def 8.39</td>
    </tr>
    <tr>
      <td>Z<sub>Γ,d</sub>(K<sub>q</sub>)</td>
      <td>Height of a surviving boundary segments.</td>
      <td>Def 8.39</td>
    </tr>
    <tr>
      <td>Θ<sub>nt</sub>(Γ, d), θ&#773;<sub>nt</sub>(Γ, d)</td>
      <td>Neutral variables associated to (Γ, d) and their vector.</td>
      <td>Nota 8.40</td>
    </tr>
  </tbody>
</table>
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[12] Xin Nie, Yunhui Wu, and Yuhao Xue. Large genus asymptotics for lengths of separating closed geodesics on random surfaces. *Journal of Topology*, 16(1):106–175, 2023.

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