where:

* $\pi = (\pi_j)_{0\le j\le k}$ is a family of disjoint subsets of $\{1, \dots, k\}$ such that $\pi_j$ contains only integers $\ge j+1$ for all $j$;

* if $B := \bigsqcup_{j=0}^k \pi_j$, then for any $1 \le j \le k$, $\Phi_j$ satisfies (6.13).

*Proof.* As announced, this is done by induction on $k$. For $k=1$, we see that the announced expression fits with Corollary 6.13. If $T_1$ is a multiplication operator by $\Phi_1 \in \{F_1^{K_1}, G_1, H_1^{t''}\}$, we take $\pi_0 = \pi_1 = \emptyset$. If $T_1 = I_1^{t''} \partial_1$, we take $\Phi_1 = I_1^{t''}, \pi_0 = \{1\}$ and $\pi_1 = \emptyset$.

Assume the result is known for $k$. We apply Corollary 6.13 with respect to the variable $x_{k+1}$, to express the image under $\mathcal{L}^{K_{k+1}}$ of the function (6.14), corresponding to a given data set $\pi = (\pi_0, \dots, \pi_k)$. Application of Corollary 6.13 gives rise to four different kind of terms, depending on the nature of an operator $T_{k+1}$ that modifies the function under the integral. For the various situations, we examine how the data evolves to a new set of data $\pi' = (\pi'_0, \pi'_1, \dots, \pi'_k, \pi'_{k+1})$, for which we check that the conditions of the theorem are still satisfied.

First, for the terms for which $T_{k+1}$ is a multiplication operator by a function $\Phi_{k+1}$ equal to $F_{k+1}^{K_{k+1}}$, $G_{k+1}$ or $H_{k+1}^{t''}$, we take $\pi'_0 = \pi_0$, $\pi'_j = \pi_j$ for $j \le k$ and $\pi'_{k+1} = \emptyset$.

The last terms are those for which $T_{k+1} = I_{k+1}^{t''} \partial_{k+1}$. We need to examine the effect of applying this operator to an expression of the form $\partial^{\pi_0} \varphi \prod_{j=1}^k \partial^{\pi_j} \Phi_j$. We use the Leibniz rule to write
(6.15)

$$I_{k+1}^{t''} \partial_{k+1} \left( \partial^{\pi_0} \varphi \prod_{j=1}^{k} \partial^{\pi_j} \Phi_j \right) = I_{k+1}^{t''} \partial_{k+1} \partial^{\pi_0} \varphi \prod_{j=1}^{k} \partial^{\pi_j} \Phi_j + I_{k+1}^{t''} \partial^{\pi_0} \varphi \partial_{x_{k+1}} \left( \prod_{j=1}^{k} \partial^{\pi_j} \Phi_j \right).$$

Then, we observe that those two terms can be expressed in the announced form, letting $\Phi_{k+1} = I_{k+1}^{t''}$, $\pi'_{k+1} = \emptyset$, and

* for the first term, $\pi'_0 = \pi_0 \cup \{k+1\}$ and $\pi'_j = \pi_j$ for $1 \le j \le k$;

* for the second term, we take $\pi'_0 = \pi_0$ and apply the Leibniz rule to the product $\prod_j \partial^{\pi_j} \Phi_j$, to write it as a sum of different contributions; then, for $1 \le j \le k$, we take one of the $\pi'_j$ to be equal to $\pi_j \cup \{k+1\}$ (corresponding to the index which is hit by the Leibniz rule) and leave all the others are untouched ($\pi'_j = \pi_j$).

□

**6.3. The classes of functions $E^{(a)}$ and $\mathcal{E}^{(a)}$.** Let us introduce classes of functions which will be helpful to quantify the distance between a $(h, \varphi)$ pseudo-convolution and the usual convolution.

**6.3.1. Definition.** For now, and the rest of the paper, we fix $a > 0$.

**Definition 6.20.** We let $E_n^{(a)}$ be the vector space of functions $h : [a, +\infty)^n \to \mathbb{R}$ such that:

* $h$ is of class $C^\infty$;
• for every multi-index $\alpha \in \{0, 1\}^n$ such that $\alpha \neq (0, \dots, 0)$,

$$ \sup_{x \in [a, +\infty)^n} \left\{ e^{\alpha \cdot x} |\partial^\alpha h(x)| \right\} < +\infty. $$

This space is endowed with the seminorm $\|h\|_{\alpha, a} := \sup\{e^{\alpha \cdot x} |\partial^\alpha h(x)|, x \in [a, +\infty)^n\}$.

Notice that, in this definition, we only control partial derivatives of $h$ that are of order at most one in each variable.

**Notation 6.21.** Define $L_n(x_1, \dots, x_n) = x_1 + \dots + x_n$ to be the linear functional associated to the usual convolution.

In what follows, we will prove results for pseudo-convolutions assuming that $h$ is close to $L_n$ in a certain sense; the definition below is used to quantify this.

**Definition 6.22.** We let $\mathcal{E}_n^{(a)}$ be the affine space of functions $h$ such that $h - L_n \in E_n^{(a)}$. Equivalently, $h$ belongs in $\mathcal{E}_n^{(a)}$ if and only if

• $h$ is of class $C^\infty$;

• for every multi-index $\alpha \in \{0, 1\}^n$ such that $|\alpha| \ge 2$,

$$ \sup_{x \in [a, +\infty)^n} \left\{ e^{\alpha \cdot x} |\partial^\alpha h(x)| \right\} < +\infty $$

• for every $j \in \{1, \dots, n\}$,

$$ \sup_{x \in [a, +\infty)^n} \left\{ e^{x_j} |\partial_j h(x) - 1| \right\} < +\infty. $$

### 6.3.2. Core examples.
Our main examples of such functions will be the following.

**Notation 6.23.** For $\beta = (\beta_\epsilon)_{\epsilon \in \{\pm\}^n}$ a family of real numbers, we define

$$ h_{\beta}(x) = 2 \log \left( \sum_{\epsilon \in \{\pm\}^n} \beta_{\epsilon} \operatorname{hyp}_{\epsilon} \left( \frac{x}{2} \right) \right). $$

**Proposition 6.24.** We assume that $\beta$ is not identically 0 and that, for all $\epsilon$, $\beta_\epsilon \ge 0$. Then, $h_\beta \in \mathcal{E}_n^{(a)}$ for any $a > 0$. Moreover, there exists $C = C(a, n)$, independent of $\beta$, such that

$$ (6.16) \qquad \sup_{\alpha \neq (0, \dots, 0)} \|h_{\beta} - L_n\|_{\alpha, a} \le C. $$

**Corollary 6.25.** The same result holds for the family

$$ (6.17) \qquad \tilde{h}_{\beta}(x_1, \dots, x_n) = 2 \operatorname{argcosh} \left( \sum_{\epsilon \in \{\pm\}^n} \beta_{\epsilon} \operatorname{hyp}_{\epsilon} \left( \frac{x}{2} \right) \right) $$

if we impose the additional condition that $\beta_{(1, \dots, 1)} \ge 1$.

**Proof of Proposition 6.24.** For $y \in \mathbb{R}$, introduce

$$ g_{\pm}(y) = e^{-y/2} \operatorname{hyp}_{\pm}(y) = \frac{1 \pm e^{-y}}{2}. $$
Let $g_{\epsilon}(x) = \prod_{j=1}^{n} g_{\epsilon_j}(x_j)$. We have

$$h_{\beta}(x) = L_{n}(x) + 2 \log \left( \sum_{\epsilon \in \{\pm\}^{n}} \beta_{\epsilon} g_{\epsilon}(x) \right).$$

Note that $|\partial g_{+}(y)| = \frac{e^{-y}}{2} \le e^{-y} g_{+}(y)$ and, if $y \ge a > 0$,

$$|\partial g_{-}(y)| = \frac{e^{-y}}{2} \le \frac{1}{1 - e^{-a}} e^{-y} g_{-}(y).$$

It follows that the constant $C = (1 - e^{-a})^{-n}$ satisfies, for all $\alpha \in \{0, 1\}^n$ and $x \in [a, +\infty)^n$,

$$(6.18) \qquad |\partial^{\alpha} g_{\epsilon}(x)| \le C e^{-\alpha \cdot x} g_{\epsilon}(x).$$

For $\alpha \in \{0, 1\}^n$ with $|\alpha| = k \ge 1$, the partial derivative $\partial^{\alpha} \log (\sum_{\epsilon \in \{\pm\}^n} \beta_{\epsilon} g_{\epsilon}(x))$ is a linear combination of terms of the form

$$(6.19) \qquad \frac{\prod_{j=1}^{k} \left( \sum_{\epsilon \in \{\pm\}^n} \beta_{\epsilon} \partial^{\pi_j} g_{\epsilon}(x) \right)}{\left( \sum_{\epsilon \in \{\pm\}^n} \beta_{\epsilon} g_{\epsilon}(x) \right)^k}$$

where $\pi_1, \dots, \pi_k$ form a partition of $\{i, \alpha_i = 1\}$. For each $j \in \{1, \dots, k\}$, (6.18) implies

$$\frac{\sum_{\epsilon \in \{\pm\}^n} \beta_{\epsilon} |\partial^{\pi_j} g_{\epsilon}(x)|}{\sum_{\epsilon \in \{\pm\}^n} \beta_{\epsilon} g_{\epsilon}(x)} \le C \exp \left( -\sum_{i \in \pi_j} \alpha_i x_i \right).$$

As a result, the term (6.19) is bounded by $C^k \exp(-\sum_{i=1}^n \alpha_i x_i)$, proving the proposition. $\square$

<figure><img src="image_25.png" /></figure>

Let us now prove the corollary.

*Proof of Corollary 6.25.* We write $\tilde{h}_{\beta} = V(h_{\beta})$ where

$$V(x) = 2 \operatorname{argcosh}(e^{\frac{x}{2}}) = x + 2 \log 2 + \omega(e^{-x})$$

is the reciprocal of the function $U$ that appeared in (4.22). By $\omega(e^{-x})$ we mean a function whose derivatives are all $O(e^{-x})$. It is straightforward to check that $\tilde{h}$ satisfies the same derivative estimates as $h$, provided we know beforehand that $\sum_{\epsilon \in \{\pm\}^n} \beta_{\epsilon} \operatorname{hyp}_{\epsilon}(\frac{x}{2})$ stays bounded away from 1 on $[a, +\infty)^n$ (to avoid the singularity of $\operatorname{argcosh}$ at 1). The condition $\beta_{(1,...,1)} \ge 1$ implies that

$$\sum_{\epsilon \in \{\pm\}^n} \beta_{\epsilon} \operatorname{hyp}_{\epsilon} \left(\frac{x}{2}\right) \ge \exp\left(\frac{1}{2} \sum_{i=1}^n x_i\right) \ge \exp\left(\frac{na}{2}\right) > 1$$

which is enough to conclude. $\square$

<figure><img src="image_26.png" /></figure>
**6.3.3. Useful properties.** First, we make the following immediate observation.

**Remark 6.26.** If $\pi = (\pi_1, \dots, \pi_N)$ is a partition of $\{1, \dots, n\}$, and if for all $j \in \{1, \dots, N\}$, the functions $(x_i)_{i \in \pi_j} \mapsto h_j((x_i)_{i \in \pi_j})$ are in $\mathcal{E}_{|\pi_j|}^{(a)}$, then following function is an element of $\mathcal{E}_n^{(a)}$:

$$ (x_1, \dots, x_n) \mapsto \sum_{j=1}^{N} h_j((x_i)_{i \in \pi_j}). $$

It may also be useful to note the following simple implication of the definition of $\mathcal{E}_n^{(a)}$:

**Proposition 6.27.** Assume that (6.16) holds for all $\alpha \in \{0, 1\}^n$ such that $|\alpha| = 1$. Then there exists $l_0 \in \mathbb{R}$ such that, for all $x = (x_1, \dots, x_n) \in [a, +\infty)^n$,

$$ |h(x) - L_n(x) - l_0| \le C \sum_{i=1}^{n} e^{-x_i}. $$

*Proof.* First, we observe that if a function $g: [a, +\infty) \to \mathbb{R}$ satisfies $|g(t) - g(s)| \le Ce^{-t}$ for all $t < s$, by Cauchy's rule, it admits a limit $g(+\infty)$ at infinity, which satisfies

$$ |g(t) - g(+\infty)| \le Ce^{-t}. $$

This property holds for instance if $|g'(t)| \le Ce^{-t}$, and is stable under taking limits of sequences of functions. Applied to $g = h - L_n$ successively in each variable, denoting $l_0 = g(+\infty, \dots, +\infty)$, this yields

$$ |g(x) - l_0| \le \sum_{i=1}^{n} |g(x_1, \dots, x_{i-1}, x_i, +\infty, \dots, +\infty) - g(x_1, \dots, x_{i-1}, +\infty, +\infty, \dots, +\infty)| $$

$$ \text{which implies that } |g(x) - l_0| \le C \sum_{i=1}^{n} e^{-x_i}. \quad \square $$

<figure><img src="image_27.png" /></figure>

**6.4. Stability of the Friedman-Ramanujan hypothesis by pseudo-convolution.** We now have the ingredients to state a general result, according to which $f_1 \star \dots \star f_n|_{\varphi}^h$ is a Friedman-Ramanujan function if $f_1, \dots, f_n$ are so, under certain assumptions on $h$ on $\varphi$. In particular, we will prove Theorem 6.28 – which is not yet the result used to prove Theorem 1.22, but may be considered as a preliminary version, and provides a nice abstract statement:

**Theorem 6.28.** Let $a > 0$. We assume that $h \in \mathcal{E}_n^{(a)}$ and that $\varphi \in E_n^{(a)}$ with $\text{supp} \varphi \subset (a, +\infty)^n$. If $(f_j)_{1 \le j \le n}$ are continuous functions in $\mathcal{F}$, then $f_1 \star \dots \star f_n|_{\varphi}^h \in \mathcal{F}$.

Recall that we always assume that the conditions of Notation 6.2 hold.

**6.4.1. Quantitative hypotheses on the functions $h, \varphi$ and $f_j$.** As we prove Theorem 6.28, we will be able to give estimates on the Friedman-Ramanujan norms of $f_1 \star \dots \star f_n|_{\varphi}^h$ as a function of those of $f_1, \dots, f_n$ (the precise bound is given in (6.27)). To this end, we introduce more specific notation which will be useful to track everything in the proof.

**Notation 6.29.** Let $n \ge 1$ and $a > 0$. We assume the following.
(h) The function $h$ lies in $\mathcal{E}_n^{(a)}$. We denote $C_1, C_2 \ge 1$ two constants such that

$$ \forall \alpha \in \{0, 1\}^n \setminus \{(0, \dots, 0)\}, \quad \|h - L_n\|_{\alpha, a} \le C_1 $$

$$ \forall j \in \{1, \dots, n\}, \quad \left| \frac{\partial h}{\partial x_j} \right| \le C_2. $$

By the hypothesis from Notation 6.2 there exists a constant $C'_2$ such that

$$ \forall j \in \{1, \dots, n\}, \quad 0 < \left| \frac{\partial x_j}{\partial h} \right| \le C'_2. $$

We let $C_0 = \max(C_1, C_2, C'_2)$. By Proposition 6.27, there exists a $l_0 \ge 1$ such that

$$ (6.20) \qquad \forall x = (x_1, \dots, x_n) \in [a, +\infty)^n, \quad \sum_{i=1}^n x_i \le h(x) + l_0. $$

(φ) The function $\varphi$ is supported in $(a, +\infty)^n$ and $\varphi \in E_n^{(a)}$. Let $C_3 \ge 0$ be a constant such that, for every multi-index $\alpha \in \{0, 1\}^n$ with $\alpha \ne (0, \dots, 0)$, $\|\varphi\|_{\alpha, a} \le C_3$.

(f) For $j \in \{1, \dots, n\}$, $f_j \in \mathcal{F}^{K_j, N_j}$ is a continuous function. We assume w.l.o.g. that $K_j \le N_j$ and $\|f_j\|_{\mathcal{F}^{K_j, N_j}} = 1$. By Remark 1.28, for $x_j \ge 0$ and $0 \le m < K_j$,

$$ |\mathcal{L}^m f_j(x_j)| \le M_{K_j, N_j} (1 + x_j)^{K_j - 1} e^{x_j} $$

$$ |\mathcal{L}^{K_j} f_j(x_j)| \le M_{K_j, N_j} (1 + x_j)^{N_j - 1} e^{\frac{x_j}{2}}. $$

We let $K = \sum_{j=1}^n K_j$, $N = \sum_{j=1}^n N_j$ and $M_{K,N} := \prod_{j=1}^n M_{K_j,N_j}$.

**Remark 6.30.** Comparison estimates such as (6.20), comparing $\sum_{i=1}^n x_i$ with $h(x_1, \dots, x_n)$, are crucial throughout the paper.

6.4.2. *Estimates on the functions* $F_j^{K_j}$, $G_j$, $H_j^t$ and $I_j^t$. We can deduce from our hypotheses some bounds on the functions introduced in Notation 6.12, which appear when computing $\mathcal{L}_\varphi^K(f_1 \star \dots f_n|_\varphi^h)$. This will allow us to foresee some key elements of the proof of Theorem 6.28.

**Lemma 6.31.** Under the assumptions (h) and (f) from Notation 6.29, for $j \in \{1, \dots, n\}$, for any set $\pi_j \subset \{1, \dots, n\}$ such that $j \notin \pi_j$ and any $x \in [a, +\infty)^n$,

* if $\Phi_j = F_j^{K_j}, G_j$ or $H_j^t$ with $0 \le t < K_j$,

$$ (6.21) \qquad |\partial^{\pi_j} \Phi_j(x)| \le C_0^2 M_{K_j, N_j} (1 + x_j)^{N_j} e^{\frac{x_j}{2}} \prod_{i \in \pi_j} e^{-x_i} $$

* if $\Phi_j = I_j^t$ with $0 \le t < K_j$,

$$ (6.22) \qquad |\partial^{\pi_j} \Phi_j(x)| \le C_0^2 M_{K_j, N_j} (1 + x_j)^{N_j} e^{x_j} \prod_{i \in \pi_j} e^{-x_i}. $$

*Proof.* If $\pi_j = \emptyset$, it is a straightforward computation that

$$ \left\{ \begin{array}{l} |F_j^{K_j}(x)| \le C_2 M_{K_j, N_j} (1 + x_j)^{N_j - 1} e^{\frac{x_j}{2}}, \\ |G_j(x)| \le C_1 M_{K_j, N_j} (1 + x_j)^{K_j - 1}, \\ |H_j^t(x)| \le C_1 C_2 M_{K_j, N_j} (1 + x_j)^{K_j} \end{array} \right. $$
and

$$|I_j^t(x)| \le C_2 M_{K_j, N_j} (1+x_j)^{K_j-1} e^{x_j}.$$

We then use the fact that $K_j \le N_j$ and $C_1, C_2 \le C_0$. When $\pi_j$ is non-empty, the additional exponential factors come from the hypothesis on the derivatives of $h$. $\square$

We here see the crucial phenomenon at play for the proof of Theorem 6.28: compared to the original upper bound $|f_j(x_j)| \le M_{K_j, N_j} (1+x_j)^{K_j-1} e^{x_j}$, we have gained in (6.21) a factor $e^{-x_j/2}$ when replacing the function $f_j$ by the function $F_j^{K_j}$, $G_j$ or $H_j^t$. Unfortunately, the estimate we obtain on the function $I_j^t$ does not constitute an improvement compared to the initial growth-rate of $f_j$.

In this case however, an exponential factor $e^{-x_j/2}$ will be gained elsewhere, from the derivatives of the other functions. Remember that we have noticed in Remark 6.17 that the set of “bad” indices $B$ for which the function $I_j^t$ appears corresponds to the set of derivatives that appear elsewhere in the integral. We have seen in the lemma above that such derivatives will create a decay reducing the growth of $I_j^t$.

We obtain the following bound on the quantity $\mathcal{L}^K(f_1 \star \dots \star f_n \|_{\varphi}^h)$.

**Proposition 6.32.** Under the assumptions (h) and (f) from Notation 6.29,

$$(6.23) \qquad |\mathcal{L}^K(f_1 \star \dots \star f_n \|_{\varphi}^h)(\ell)| \le \sum_{\pi} \sum_{0 \le t+t' \le K} |\mathcal{L}_{\ell}^t \mathcal{P}_{\ell}^{t'} \mathrm{Int}_{\pi}(\ell)|$$

and $|\mathrm{Int}_{\pi}(\ell)|$ is bounded above by

$$(6.24) \quad C_0^{2n} M_{K,N}^{\times} \int_{h(x)=\ell} |\partial^{\pi_0} \varphi(x)| \prod_{j \in B \setminus \pi_0} e^{-x_j} \prod_{j \in B} (1+x_j)^{N_j} e^{x_j} \prod_{j \notin B} (1+x_j)^{N_j} e^{\frac{x_j}{2}} \frac{dx}{d\ell}.$$

*Proof.* The bound (6.23) is a direct consequence of the expression (6.12) together with the triangle inequality. Recall that

$$\mathrm{Int}_{\pi}(\ell) = \int_{h(x)=\ell} \partial^{\pi_0} \varphi(x) \prod_{j=1}^{n} \partial^{\pi_j} \Phi_j(x) \frac{dx}{d\ell}$$

where $\pi_0, \pi_1, \dots, \pi_n$ are disjoint subsets of $\{1, \dots, n\}$ such that $j \notin \pi_j$, and $B = \bigsqcup_{j=0}^n \pi_j$. We then use Lemma 6.31 on each term, remembering that $\Phi_j$ is in the first situation if $j \notin B$ and the second if $j \in B$. We conclude by noticing that $\prod_{j=1}^n \prod_{i \in \pi_j} e^{-x_i} = \prod_{j \in B \setminus \pi_0} e^{-x_j}$. $\square$

**Remark 6.33.** The previous statement is not completely optimal, but we have decided to line up all the estimates on the worst possible case. For later use, it will also be useful to lose again in optimality in (6.24), and write that:

$$(6.25) \qquad |\mathrm{Int}_{\pi}(\ell)| \le C_0^{2n} M_{K,N}^{\times} \int_{h(x)=\ell} |\partial^{\pi_0} \varphi(x)| \prod_{j \notin \pi_0} e^{-\frac{x_j}{2}} \prod_{j=1}^{n} (1+x_j)^{N_j} e^{x_j} \frac{dx}{d\ell}.$$

If we compare with (6.24), we note that we have accepted to waste a factor $e^{-x_j/2}$ for $j \in B \setminus \pi_0$. The resulting upper bound only depends on $\pi_0$ and not on $B$ (both coincide if $\pi_0 = B$).
6.4.3. *End of the proof of Theorem 6.28.* We are now ready to finish the proof of Theorem 6.28 by using Assumption $(\varphi)$ to bound (6.24).

*Proof.* Given that we now assume that $|\partial^{\pi_0} \varphi(x)| \le C_3 e^{-\sum_{j \in \pi_0} x_j}$, we obtain that for any term $\pi$ in (6.23),

$$ (6.26) \qquad |\mathrm{Int}_{\pi}(\ell)| \le C_3 C_0^{2n} M_{K,N}^{\times} \int_{h(x)=\ell} \prod_{j=1}^{n} (1+x_j)^{N_j} e^{\frac{x_j}{2}} \frac{dx}{d\ell}. $$

We now explicit the level-set integration and use the comparison estimate (6.20). We obtain that

$$ |\mathrm{Int}_{\pi}(\ell)| \le C_3 C_0^{2n} M_{K,N}^{\times} \int_{\sum_{j=2}^n x_j \le \ell+l_0} \prod_{j=1}^n (1+x_j)^{N_j} e^{\frac{x_j}{2}} \left|\frac{\partial x_1}{\partial h}\right| \prod_{j=2}^n dx_j $$

which, by assumption on the inverse derivatives of $h$, implies

$$ |\mathrm{Int}_{\pi}(\ell)| \le C_3 C_0^{2n} C_2' M_{K,N}^{\times} (1+\ell+l_0)^N e^{\frac{\ell+l_0}{2}} \int_{\sum_{j=2}^n x_j \le \ell+l_0} \prod_{j=2}^n dx_j. $$

As a consequence,

$$ |\mathrm{Int}_{\pi}(\ell)| \le C_3 C_0^{2n+1} M_{K,N}^{\times} (1+\ell+l_0)^{N+n-1} e^{\frac{\ell+l_0}{2}}. $$

We have thus proven that the functions $\mathrm{Int}_{\pi}$ are in the class $\mathcal{R}^{N+n}$. From Proposition 6.32, $\mathcal{L}_{\ell}^{\mathcal{K}}(f_1 \star \dots \star f_n|_{\varphi}^{h})$ is a sum of such functions (or images thereof under powers of $\mathcal{P}_{\ell}$ or $\mathcal{L}_{\ell}$), hence $\mathcal{L}_{\ell}^{\mathcal{K}}(f_1 \star \dots \star f_n|_{\varphi}^{h}) \in \mathcal{R}^{N+n}$. Proposition 1.26 then implies that $f_1 \star \dots \star f_n|_{\varphi}^{h}$ belongs in $\mathcal{F}^{\mathcal{K},N+n}$. The proof furthermore gives

$$ (6.27) \qquad \|f_1 \star \dots \star f_n|_{\varphi}^h\|_{\mathcal{F}^{\mathcal{K},N+n}} \le C_3 C_0^{2n+1} \mathrm{fn}(n, \mathcal{K}) (1+l_0)^{N+n-1} e^{\frac{l_0}{2}} \prod_{j=1}^n \|f_j\|_{\mathcal{F}^{\mathcal{K}_j, N_j}}. $$

□

6.5. **Problems in the geometric application of Assumption (h) and (φ).** Having noticed at the end of §5 that the integral (5.13) is the $(h, \varphi_0)$-convolution of Friedman-Ramanujan functions (with $\varphi_0$ and $h$ specified in (5.15) and (5.16) respectively), we now need to examine the possibility to apply Theorem 6.28. Unfortunately, the functions $h$ and $\varphi_0$ do not satisfy the required assumptions, which explains why we will need two additional sections §7 and §8 to solve these issues.

6.5.1. *Problem with Assumption (h).* The explicit expression of $\ell_Y(\mathbf{c})$ was obtained in (4.12). By virtue of Corollary 6.25 and Remark 6.26, for each parameter $\vec{L}$, we have that the function

$$ h: \vec{\theta} \mapsto \ell_Y(\mathbf{c}) $$

lies in the set $\mathcal{E}_{2r}^{(a)}$ provided we restrict all the $\theta_q$ to be $\ge a$. If we now fix the neutral parameters $\vec{\theta}_{nt} \in \mathbb{R}_{\ge 0}^{\Theta_{nt}}$ and consider

$$ (6.28) \qquad h: \vec{\theta}_{ac} \mapsto \ell_Y(\mathbf{c}), $$
then it is easy to check that the restriction of this function to $\mathbb{R}_{\ge a}^{\Theta_{ac}}$ belongs to $\mathcal{E}_{\#\Theta_{ac}}^{(a)}$.

We have already remarked that the non-crossing variables $\theta_q$ stay positive, as consequence of the Useful Remark 2.11. As a consequence, in the purely non-crossing case (when all variables are non-crossing), Assumption (h) is satisfied.

If there exists some crossing variables, then they may take positive or negative values, in which case we are out of the domain of application of Corollary 6.25.

~~Section 7 will be devoted to solving this issue by choosing a different representative of the homotopy class of c. This will provide an expression of h which falls in a class $\mathcal{E}$ with respect to newly defined variables. The reader might wish to focus upon first read on the purely non-crossing case, and hence skip §7.2 and §7.3.~~

### 6.5.2. Problem with Assumption $(\varphi)$.
The coefficients arising in the expression of $Q_{m(\lambda)}$, obtained in (4.19) and (4.21), are equal to $\pm 1$, thus not of constant sign. As a consequence, we cannot use Proposition 6.24 nor Corollary 6.25 to deduce that $Q_{m(\lambda)}$ belongs to the class $\mathcal{E}_{m(\lambda)}$ with respect to the variables $(\tau_1^\lambda, \dots, \tau_{m(\lambda)}^\lambda)$. We cannot either deduce that the functions $\log F_{m(\lambda)}$ are in the class $E_{m(\lambda)}$.

~~Section 8 will be devoted to estimating the derivatives of $F_{m(\lambda)}$. Again this will necessitate changing the representative of the homotopy class of $\Gamma_\lambda$.~~

### 6.6. Variants of Theorem 6.28.
In the geometric applications of these techniques, we shall not use Theorem 6.28 directly, but several variants. Because these variants become quite technical, we present them by gradually modifying the assumptions and conclusions. The reader is invited to skip this section at first read, and come back to it when the necessity for these variants becomes clear.

#### 6.6.1. First variant: comparison estimates.
First, we loosen the decay hypothesis on $\varphi$, and replace it by a new assumption.

**Notation 6.34 (Assumption $(\varphi$-CE)).** Assume that the smooth function $\varphi$ is supported on $(a, +\infty)^n$ and there exists positive functions $Z_j : \mathbb{R}_{>0} \to \mathbb{R}_{>0}$, $1 \le j \le n$, such that:

* for all $\pi_0 \subseteq \{1, \dots, n\}$, all $x = (x_1, \dots, x_n) \in (a, +\infty)^n$,

$$|\partial^{\pi_0} \varphi(x)| \le C_3 \prod_{j \in \pi_0} \frac{1}{Z_j^{1/2}(x_j)};$$

* there exists $l_0 > 0$ satisfying (6.20) and such that, for any $\pi_0 \subseteq \{1, \dots, n\}$,

$$ (6.29) \qquad 2 \sum_{j=1}^{n} x_j \le h(x) + \sum_{j \notin \pi_0} x_j + \sum_{j \in \pi_0} \log Z_j + l_0. $$

The important point, here, is that the functions $(Z_j)_{1 \le j \le n}$ controlling the decay of the derivatives of $\varphi$ can be incorporated in a comparison estimate for the function $h$.

**Remark 6.35.** The initial result, Theorem 6.28, may be recovered by taking $Z_j = e^{x_j}$, in which case the comparison estimate (6.29) coincides with (6.20).

**Theorem 6.36.** Let $a > 0$. We assume that the assumptions (h), (f) and $(\varphi$-CE) hold. Then, $f_1 \star \dots \star f_n|_{\varphi}^h \in \mathcal{F}^{K,N+n}$, and the estimate (6.27) holds.
**Proof.** We do not reproduce the proof, completely identical to the proof of Theorem 6.28. A crucial point is again the comparison estimate (6.29), which may be rewritten on the level-set {$h(x) = \ell$} in exponential form

$$ \prod_{j=1}^{n} e^{x_j} \prod_{j \in \pi_0} \frac{1}{Z_j^{1/2}} \prod_{j \notin \pi_0} e^{-\frac{x_j}{2}} \le e^{\frac{l_0}{2}} e^{\frac{\ell}{2}}. $$

This means that the gain $\prod_{j \in \pi_0} Z_j^{-1/2} \prod_{j \notin \pi_0} e^{-x_j/2}$ on the left-hand side results in a growth of size $e^{\ell/2}$ instead of the expected $e^{\ell}$ on the right-hand side, leading to the fact that the resulting function is a Friedman-Ramanujan remainder. □

**6.6.2. Second variant: Linear forms.** In our geometric applications, $\varphi(x_1, \dots, x_n)$ will actually be split into a product involving linear forms, as follows:

**Notation 6.37.** Let $\varphi$ be of the form:

$$ (6.30) \qquad \varphi(x_1, \dots, x_n) = \phi(\tau_1, \dots, \tau_m) \psi(x_1, \dots, x_n) $$

where, for all $1 \le i \le m$, $\tau_i$ is a linear form, and more precisely $\tau_i(x) = \sum_{j \in \Theta(i)} x_j$ for a subset $\Theta(i) \subseteq \{1, \dots, n\}$. For $\tilde{\pi} \subseteq \{1, \dots, m\}$, we denote $\Theta(\tilde{\pi}) := \bigcup_{i \in \tilde{\pi}} \Theta(i)$.

**Remark 6.38.** The original result corresponds to the case where $m = n$, $\Theta(i) = \{i\}$ so that $\tau_i(x) = x_i$, and $\psi = 1$.

Let us adapt Theorem 6.16 to this new form of the density $\varphi$. The motivation for this upcoming variant may be roughly explained as follows: suppose that the derivatives $\partial^{\tilde{\pi}} \phi(\tau_1, \dots, \tau_m)$ are bounded by

$$ (6.31) \qquad |\partial^{\tilde{\pi}} \phi(\tau_1, \dots, \tau_m)| \le C_3 e^{-\sum_{i \in \tilde{\pi}} \tau_i} = C_3 e^{-\sum_{i \in \tilde{\pi}} \sum_{j \in \Theta(i)} x_j} $$

for any $\tilde{\pi} \subseteq \{1, \dots, m\}$. Then, once the term $\partial^{\tilde{\pi}} \phi$ appears in the application of Corollary 6.13, there is no further need to gain exponential decay with respect to the variables $x_j$ with $j \in \Theta(\tilde{\pi})$. We wish to avoid this extra work, not only to optimize the proof, but also because in the geometric application the higher order derivatives of $\phi$ will be difficult to calculate. Thus, we copy the proof of Theorem 6.16, but once $\phi$ has been differentiated with respect to a variable $x_j$ (say, for $j \in \Theta(i)$), we stop applying our operators to the other variables $x_j'$ with $j' \in \Theta(i)$. This leads to the following statement.

**Theorem 6.39.** 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 $\mathcal{L}_{\ell}^K(f_1 \star \dots \star f_n)^{\frac{h}{\varphi}}$ can be expressed as the sum of the following functions, or their images under $\mathcal{L}_{\ell}^{t_j} \mathcal{P}_{\ell}^{t_j'}$ for $0 \le t \le K-n$ and $0 \le t' \le n$,

$$ (6.32) \qquad \mathrm{Int}_{\tilde{\pi}_0, \tilde{\pi}, \mathcal{V}, \pi}(\ell) := \int_{h(x)=\ell} \partial^{\tilde{\pi}} \phi(\tau_1, \dots, \tau_m) \partial^{\pi_0} \psi(x) \prod_{j \in \mathcal{V}} \partial^{\pi_j} \Phi_j(x) \prod_{j \notin \mathcal{V}} f_j(x_j) \frac{dx}{d\ell} $$

where:

* $\tilde{\pi}_0 \subseteq \{1, \dots, n\}$ and $\tilde{\pi} \subseteq \{1, \dots, m\}$ are two sets of same cardinal, with a bijection $\tilde{\pi}_0 \ni j \mapsto i_j \in \tilde{\pi}$ such that $j \in \Theta(i_j)$ for all $j \in \tilde{\pi}_0$;

* $\mathcal{V}$ is a subset of $\{1, \dots, n\}$ containing $\tilde{\pi}_0$, and so that $\mathcal{V} \cup \Theta(\tilde{\pi}) = \{1, \dots, n\}$;

* $(\pi_j)_{j \in \mathcal{V} \cup \{0\}}$ is a family of disjoint subsets of $\mathcal{V} \setminus \tilde{\pi}_0$ such that $j \notin \pi_j$ for all $j$;
• if $B := \tilde{\pi}_0 \sqcup \pi_0 \sqcup \bigsqcup_{j \in \mathcal{V}} \pi_j$, then for any $j \in \mathcal{V}$, $\Phi_j$ satisfies (6.13).

**Remark 6.40.** To go from Theorem 6.39 to Theorem 6.16, let $\tilde{\pi}_0 = \tilde{\pi} = \emptyset$, $\mathcal{V} = \{1, \dots, n\}$.

We prove, once again, this result by proving the following (stronger) lemma for by induction on the integer $0 \le k \le n$. This leads to the claim taking $k = n$.

**Lemma 6.41.** For any $0 \le k \le n$, the function $\mathcal{L}_{\ell}^{\mathcal{K}}(f_1 \star \dots \star f_n|_{\varphi}^{\mathcal{h}})$ can be expressed as the sum of the following functions, or their images under $\mathcal{L}_{\ell}^{t} \mathcal{P}_{\ell}^{t'}$ for $0 \le t \le K-k$ and $0 \le t' \le k$,

$$ (6.33) \quad \mathrm{Int}_{\tilde{\pi}_0, \tilde{\pi}, \mathcal{V}, \pi}(\ell) := \int_{h(x)=\ell} \partial^{\tilde{\pi}} \phi(\tau_1, \dots, \tau_m) \partial^{\pi_0} \psi(x) \prod_{j \in \mathcal{V}} \partial^{\pi_j} \Phi_j(x) \prod_{j \notin \mathcal{V}} f_j(x_j) \frac{dx}{d\ell} $$

where:

• $\tilde{\pi}_0 \subseteq \{1, \dots, k\}$ and $\tilde{\pi} \subseteq \{1, \dots, m\}$ are two sets of same cardinal, with a bijection $\tilde{\pi}_0 \ni j \mapsto i_j \in \tilde{\pi}$ such that $j \in \Theta(i_j)$ for all $j \in \tilde{\pi}_0$;

• $\mathcal{V}$ is a subset of $\{1, \dots, k\}$ containing $\tilde{\pi}_0$, and so that $\{1, \dots, k\} \subseteq \mathcal{V} \cup \Theta(\tilde{\pi})$;

• $(\pi_j)_{j \in \mathcal{V} \cup \{0\}}$ is a family of disjoint subsets of $\mathcal{V} \setminus \tilde{\pi}_0$ such that $\pi_j$ contains only integers $\ge j+1$ for all $j$;

• if $B := \tilde{\pi}_0 \sqcup \pi_0 \sqcup \bigsqcup_{j \in \mathcal{V}} \pi_j$, then for any $j \in \mathcal{V}$, $\Phi_j$ satisfies (6.13).

*Proof.* For $k=0$, we take $\tilde{\pi} = \tilde{\pi}_0 = \pi_0 = \mathcal{V} = \emptyset$. Now assume the formula known at rank $k$ and let us prove it for $k+1$, by applying $\mathcal{L}^{K_{k+1}}$ to an expression of the form (6.33).

First, let us observe that, if $k+1 \in \Theta(\tilde{\pi})$, then we can simply apply $\mathcal{L}^{K_{k+1}}$ to the integral (6.32) and leave it unchanged; the result will satisfy the requirements of the theorem at the rank $k+1$, with the same $\tilde{\pi}_0$, $\tilde{\pi}$, $\mathcal{V}$, $(\pi_j)_{j \in \mathcal{V}}$.

Let us now assume that $k+1 \notin \Theta(\tilde{\pi})$. As before, we apply Corollary 6.13 to (6.33) with respect to the variable $x_{k+1}$. For each term, we explicit how the data $(\tilde{\pi}_0, \tilde{\pi}, \mathcal{V}, (\pi_j)_{j \in \mathcal{V}})$ evolves to a new set of data $(\tilde{\pi}'_0, \tilde{\pi}', \mathcal{V}', (\pi'_j)_{j \in \mathcal{V}'})$ which satisfies the conditions at the next rank. Note that, here, we will always take $\mathcal{V}' = \mathcal{V} \cup \{k+1\}$ and $\pi'_{k+1} = \emptyset$.

In the three first cases, $T_{k+1}$ is a multiplication operator by some function $\Phi_{k+1}$ equal to $F_{k+1}^{K_{k+1}}$, $G_{k+1}$ or $H_{k+1}^t$. We take $\tilde{\pi}'_0 = \tilde{\pi}_0$, $\tilde{\pi}' = \tilde{\pi}$, and $\pi'_j = \pi_j$ for $j \in \mathcal{V}$.

Otherwise, $T_{k+1} = I_{k+1}^t \partial_{k+1}$. In this case, we let $\Phi_{k+1} = I_{k+1}^t$. When we apply $\partial_{k+1}$ to an expression of the form $\partial^{\tilde{\pi}} \phi(\tau_1, \dots, \tau_m) \partial^{\pi_0} \psi(x) \prod_{j \in \mathcal{V}} \partial^{\pi_j} \Phi_j(x)$ and then multiply it by $I_{k+1}^t = \partial^{\pi'_{k+1}} \Phi_{k+1}$, by the Leibniz rule we obtain three types of terms.

• Terms of the form

$$ (6.34) \quad \partial^{\tilde{\pi}} \partial_i \phi(\tau_1, \dots, \tau_m) \partial^{\pi_0} \psi(x) \prod_{j \in \mathcal{V}'} \partial^{\pi_j} \Phi_j(x) $$

for $i \in \{1, \dots, m\}$ with $k+1 \in \Theta(i)$. We note that by hypothesis $i \notin \tilde{\pi}$. We take $\tilde{\pi}'_0 = \tilde{\pi}_0 \cup \{k+1\}$, $\tilde{\pi}' = \tilde{\pi} \cup \{i\}$, and leave the $(\pi_j)_{j \in \mathcal{V}}$ unchanged.

• A term of the form

$$ (6.35) \quad \partial^{\tilde{\pi}} \phi(\tau_1, \dots, \tau_m) \partial^{\pi_0} \partial_{k+1} \psi(x) \prod_{j \in \mathcal{V}'} \partial^{\pi_j} \Phi_j(x) $$
for which we set $\tilde{\pi}'_0 = \tilde{\pi}_0$, $\tilde{\pi}' = \tilde{\pi}$, $\pi'_0 = \pi_0 \cup \{k+1\}$, and leave the $(\pi_j)_{j \in \mathcal{V}}$ unchanged.

* Finally, a term of the form

$$ (6.36) \qquad \partial^{\tilde{\pi}} \phi(\tau_1, \dots, \tau_m) \partial^{\pi_0} \psi(x) \partial^{k+1} \left( \prod_{j \in \mathcal{V}} \partial^{\pi_j} \Phi_j \right) (x) \Phi_{k+1}(x) $$

which is itself a sum of terms coming from re-applying the Leibniz rule. For each of these terms, we leave $\tilde{\pi}_0$, $\tilde{\pi}$ and $\pi_0$ unchanged, add $k+1$ to exactly one of the $\pi_j$ with $j \in \mathcal{V}$ (corresponding to the index which is hit by the Leibniz rule), leaving the others unchanged. $\square$

We deduce from Theorem 6.39 the following proposition replacing Proposition 6.32:

**Lemma 6.42.** Under the assumptions (h) and (f),

$$ |\mathcal{L}^K(f_1 \star \dots \star f_n|_{\varphi}^h)(\ell)| \le \sum_{\tilde{\pi}_0, \tilde{\pi}, \mathcal{V}, \pi} \sum_{0 \le t+t' \le K+n} |\mathcal{L}^t \mathcal{P}^{t'} \mathrm{Int}_{\tilde{\pi}_0, \tilde{\pi}, \mathcal{V}, \pi}(\ell)| $$

and $|\mathrm{Int}_{\tilde{\pi}_0, \tilde{\pi}, \mathcal{V}, \pi}(\ell)|$ can be bounded above by

$$ C_0^{2n} M_{K,N}^{\times} \int_{h(x)=\ell} |\partial^{\tilde{\pi}} \phi(\tau_1, \dots, \tau_m)| |\partial^{\pi_0} \psi(x)| \prod_{j \in \mathcal{V} \setminus (\tilde{\pi}_0 \cup \pi_0)} e^{-\frac{x_j}{2}} \prod_{j=1}^{n} (1+x_j)^{N_j} e^{x_j} \frac{dx}{d\ell}. $$

The following proposition inputs an assumption about $\varphi$ to find upper bounds on the terms appearing in Lemma 6.42.

**Notation 6.43 (Assumption (φ-CE-form)).** Assume that $\varphi(x) = \phi(\tau_1, \dots, \tau_m)\psi(x)$ where:

* $\psi$ satisfies Assumption (φ), namely it is supported in $(a, +\infty)^n$ and there exists a constant $C_3$ such that, for any $\pi \subseteq \{1, \dots, n\}$,

$$ \sup_{x \in (a, +\infty)^n} e^{\sum_{j \in \pi} x_j} |\partial^{\pi} \psi| \le C_3. $$

* $\phi$ satisfies Assumption (φ-CE) w.r.t. its variables $(\tau_i)_{1 \le i \le m}$, i.e. there exists positive functions $Z_i: \mathbb{R}_{>0} \to \mathbb{R}_{>0}$, $1 \le i \le m$, such that:

- for all $\tilde{\pi} \subseteq \{1, \dots, m\}$,

$$ (6.37) \qquad |\partial^{\tilde{\pi}} \phi(\tau_1, \dots, \tau_m)| \le C_4 \prod_{i \in \tilde{\pi}} \frac{1}{Z_i^{1/2}(\tau_i)}; $$

- there exists a function $\mathcal{H}(x)$ such that for all $\tilde{\pi} \subseteq \{1, \dots, m\}$, we have the comparison estimate

$$ (6.38) \qquad 2\mathcal{H}(x) \le h(x) + \sum_{j \notin \Theta(\tilde{\pi})} x_j + \sum_{i \in \tilde{\pi}} \log Z_i(\tau_i). $$

**Remark 6.44.** On the level-set $\{h(x) = \ell\}$, the comparison estimate can be rewritten under the multiplicative form

$$ (6.39) \qquad \prod_{j \notin \Theta(\tilde{\pi})} e^{-\frac{x_j}{2}} \prod_{i \in \tilde{\pi}} \frac{1}{Z_i^{1/2}(\tau_i)} \le e^{\ell/2} e^{-\mathcal{H}(x)}. $$
**Proposition 6.45.** Assume that $(f), (h)$ from Notation 6.29 and Assumption (φ-CE-form) are satisfied. Then a term of the form (6.32) may be bounded by

$$C_0^{2n} C_3 C_4 M_{K,N}^{\times} \prod_{j=1}^{n} \|f_j\|_{\mathcal{H}^{K_j,N_j}} (1+\ell+l_0)^N e^{\frac{\ell}{2}} \int_{h(x)=\ell} e^{-\mathcal{H}(x)} e^{\sum_{j=1}^n x_j} \frac{dx}{d\ell}$$

As a consequence, in order to prove the Friedman-Ramanujan property, it will be enough to bound the terms appearing in Proposition 6.45.

## 7. ERASING OF CROSSING PARAMETERS IN THE LENGTH OF c

In this section, a hyperbolic metric $Y$ on $S$ is fixed. We consider our multi-loop $c = (c_1, \dots, c_n)$. The section is devoted to solving the first issue raised in §6.5.

We recall that, for $q \in \Theta$, if $\text{Sign}(q)\text{Sign}(\sigma q) = -1$, we call $q$ a U-turn index, and otherwise a crossing index. We denote as $\mathcal{C} \subseteq \Theta$ the set of crossing indices. We shall solve issues related to situations when some crossing parameters $\theta_q$, with $q \in \mathcal{C}$, assume negative values.

**7.1. Idea and organization of this section.** An example of a crossing variable is represented in Figure 21. In Figure 21a, the crossing parameter $\theta_q$ is positive, and the length-function $\ell_Y(c)$ belongs to an appropriate class $\mathcal{E}$, which means that the argument we have set up in §6 can be used to establish Theorem 1.15. This is not the case in Figure 21b unfortunately, when $\theta_q \le 0$.

<figure><img src="image_28.png" /><figcaption>FIGURE 21. A simple example with a crossing parameter $\theta_1^+$. The dotted curve is the element of the homotopy class of $c_i$ constructed in §3 to express $\ell_Y(c_i)$.</figcaption></figure>

In order to solve this issue, depending on the metric $Y$, we shall find a new representative of the homotopy class of a component $c_i$ of $c$, and introduce new positive variables $\tilde{\theta}_q$, such that the expression of $\ell_Y(c_i)$ in those new variables now belongs to the desired class of functions $\mathcal{E}$. The idea is that, when a crossing variable is negative, we can rectify the problem by a basic move: if $\text{Sign}(q)\text{Sign}(\sigma q) = +1$ and $\theta_q < 0$, the basic move consists in replacing $\bar{B}_{\sigma^{-1}q,q} \cdot \bar{I}_q \cdot \bar{B}_{q,\sigma q}$ by the homotopic orthogeodesic $\bar{B}_{\sigma^{-1}q,\sigma q}$ from
$\beta_{\sigma^{-1}q}$ to $\beta_{\sigma q}$. The segment $\bar{\mathcal{I}}_q$ is then said to be erased, and the variable $\theta_q$ is an erased variable. On Figure 22 we show a basic move in the case $\text{Sign}(q) = \text{Sign}(\sigma q) = +1$. We then perform these basic moves iteratively to get rid of all negative crossing variables.

<figure><img src="image_29.png" /><figcaption>FIGURE 22. The basic move consists in replacing the dotted path with the dashed line in the homotopy class of $c_i$.</figcaption></figure>

We start this section by studying families of aligned geodesics in $\mathbb{H}^2$ and their orthogeodesics. We prove a key lemma called the *staircase inequality* in §7.2.2. We then construct the erasing procedure in §7.2.3; see Proposition 7.4. Once we lift our closed geodesic $c_i$, loops $\beta$ and bars $B$ to the hyperbolic plane $\mathbb{H}^2$ as done in §4.2.2, we can apply Proposition 7.4 to them. This yields a partition of the Teichmüller space, as well as a new set of parameters $\tilde{\theta}_q$ on each element of this partition, as detailed in §7.3. The new length $\ell_Y(c_i)$ is expressed in these new parameters in §7.4. We then introduce a partition of unity adapted to this construction in §7.6.

For readers who decide to focus on the purely non-crossing case, it suffices to read §7.5-7.6.

**7.2. Families of aligned geodesics and their orthogeodesics.** Let us group together a few key observations about families of aligned geodesics and their orthogeodesics, which will be key to this section. In this whole subsection, we shall consider the following set-up. For an integer $M \ge 2$, $\gamma_0, \dots, \gamma_M$ are $M+1$ oriented geodesics in the hyperbolic plane. We assume that they are pairwise aligned, and that $\gamma_{i+1}$ is on the left of $\gamma_i$. We define $\bar{B}_{ij}$ to be the oriented orthogeodesics from $\gamma_i$ to $\gamma_j$, of length denoted by $L_{ij} > 0$, and $z_{ij} \in \gamma_j$ (resp. $z_{ji} \in \gamma_i$) the endpoint (resp. origin) of $\bar{B}_{ij}$.

**7.2.1. The 3-geodesic lemma.** First, we prove the following elementary observation.

**Lemma 7.1.** Let $j \in \{1, \dots, M-1\}$. Then,

$$ (7.1) \quad \begin{array}{l} \bar{d}(z_{j,j+1}, z_{j-1,j+1}) \ge 0 \\ \Leftrightarrow \bar{d}(z_{j+1,j}, z_{j-1,j}) \ge 0 \\ \Leftrightarrow \bar{d}(z_{j+1,j-1}, z_{j,j-1}) \ge 0 \end{array} \quad \text{and, in terms of absolute distances,} $$

$$ (7.2) \quad |\bar{d}(z_{j+1,j}, z_{j-1,j})| > |\bar{d}(z_{j+1,j-1}, z_{j,j-1})| + |\bar{d}(z_{j,j+1}, z_{j-1,j+1})|. $$

*Proof.* These results stem from the elementary study of the polygon of consecutive vertices $z_{j-1,j+1}$, $z_{j,j+1}$, $z_{j+1,j}$, $z_{j-1,j}$, $z_{j,j-1}$, $z_{j+1,j-1}$ represented in Figure 23. Equa-
<figure><img src="image_30.png" /><figcaption>FIGURE 23. Illustration of the proof of Lemma 7.1.</figcaption></figure>

tion (7.1) is obtained by observing that there exists no hyperbolic hexagon with 5 inner angles equal to $\pi/2$ and one equal to $3\pi/2$, and hence the polygon is a self-intersecting right-angled hexagon. Then, (7.2) is a direct consequence of the classic trigonometric formula for self-intersecting right-angled hexagons of consecutive side-lengths ($\ell_i$)$_{1\le i\le 6}$ (see e.g. [5, Theorem 2.4.4]):

$$ \cosh(\ell_3) = \cosh(\ell_1) \cosh(\ell_5) + \cosh(\ell_6) \sinh(\ell_1) \sinh(\ell_5) > \cosh(\ell_1 + \ell_5). $$

□

7.2.2. *The staircase inequality.* Let us now prove an inequality which we will refer to as the staircase inequality, illustrated in Figure 24.

<figure><img src="image_31.png" /><figcaption>FIGURE 24. The geometric situation for the staircase inequality.</figcaption></figure>

**Lemma 7.2.** Let $M \ge 2$. We introduce the distances

$$ \theta_j = \vec{d}(z_{j-1,j}, z_{j+1,j}), \quad \mu_j = \vec{d}(z_{M,j}, z_{j+1,j}), \quad \nu_M = \vec{d}(z_{M-1,M}, z_{0,M}). $$
Assume that $\theta_j \ge 0$ for $1 \le j \le M-2$, that $\theta_{M-1} < 0$, and that $\mu_j > 0$ for $0 \le j \le M-2$.
Then,

$$ (7.3) \qquad \left|\theta_{M-1}\right| > \sum_{j=1}^{M-2} \theta_j + \mu_0 + \nu_M. $$

**Remark 7.3.** The inequality (7.3) can also be written in the form

$$ (7.4) \qquad -\sum_{j=1}^{M-1} \theta_j > \mu_0 + \nu_M. $$

*Proof.* We apply Lemma 7.1 to the geodesics $\gamma_{j-1}$, $\gamma_j$ and $\gamma_M$. We obtain that

$$ (7.5) \qquad u_j := \vec{d}(z_{M,j}, z_{j-1,j}) > \vec{d}(z_{M,j-1}, z_{j,j-1}) + \vec{d}(z_{j,M}, z_{j-1,M}), $$

all of these quantities being positive thanks to the hypothesis $\mu_{j-1} > 0$ and (7.1).
For $2 \le j \le M$, going along $\gamma_{j-1}$, we have the identity

$$ \vec{d}(z_{M,j-1}, z_{j,j-1}) = \vec{d}(z_{M,j-1}, z_{j-2,j-1}) + \vec{d}(z_{j-2,j-1}, z_{j,j-1}) = u_{j-1} + \theta_{j-1}, $$

and for $j = 1$, $\vec{d}(z_{M,0}, z_{1,0}) = \mu_0$. As a consequence, (7.5) can be rewritten in terms of
the sequence $(u_j)_j$ as:

$$ \left\{ \begin{array}{ll} u_j > u_{j-1} + \theta_{j-1} + \vec{d}(z_{j,M}, z_{j-1,M}) & (j \ge 2) \\ u_1 > \mu_0 + \vec{d}(z_{1,M}, z_{0,M}) & (j = 1). \end{array} \right. $$

The result follows by induction, using that $u_{M-1} = |\theta_{M-1}|$ and

$$ \nu_M = \sum_{j=1}^{M-1} \vec{d}(z_{j,M}, z_{j-1,M}). $$

□

**7.2.3. Erasing the negative crossings.** We are now ready to prove the main result of this section, the construction of the erasing procedure. We denote as $\vec{\theta} \in \mathbb{R}^{M-1}$ the vector $\theta_j = \vec{d}(z_{j-1,j}, z_{j+1,j})$ for all $1 \le j \le M-1$, and $\vec{L} \in \mathbb{R}_{>0}^M$ the vector of components $L_{j,j+1}$, $0 \le j < M$. We note that the geometry of our family of aligned geodesics is entirely determined by the values of the parameters $\vec{L}, \vec{\theta}$ (which belong in a subset of $\mathbb{R}_{>0}^M \times \mathbb{R}^{M-1}$).

**Proposition 7.4.** Let $M \ge 2$ and $\vec{L} \in \mathbb{R}_{>0}^M$ be a fixed length-vector. There exists a partition $(P_\xi)_{\xi \in \Xi_M}$ of $\mathbb{R}^{M-1}$ and a family $(I(\xi))_{\xi \in \Xi_M}$ of subsets

$$ I(\xi) = \{0 = \varphi_{\xi}(0) < \varphi_{\xi}(1) < \dots < \varphi_{\xi}(K_{\xi} + 1) = M\} $$

such that the following holds. Let $\xi \in \Xi_M$. For any $\vec{\theta} \in P_\xi$,

$$ (a) \text{ for any } 1 \le j \le K_{\xi}, \vec{d}(z_{\varphi_{\xi}(j-1), \varphi_{\xi}(j)}, z_{\varphi_{\xi}(j+1), \varphi_{\xi}(j)}) \ge 0; $$
(b) for $0 \le j \le K_\xi$, if we set

$$ \mu_{\varphi_{\xi}(j)} := \vec{d}(z_{\varphi_{\xi}(j+1),\varphi_{\xi}(j)}, z_{\varphi_{\xi}(j)+1,\varphi_{\xi}(j)}) $$

$$ \nu_{\varphi_{\xi}(j+1)} := \vec{d}(z_{\varphi_{\xi}(j+1)-1,\varphi_{\xi}(j+1)}, z_{\varphi_{\xi}(j),\varphi_{\xi}(j+1)}) $$

then both of these distances are positive;

(c) for $0 \le j \le K_\xi$,

$$ (7.6) \qquad - \sum_{\varphi_{\xi}(j) < k < \varphi_{\xi}(j+1)} \theta_k \ge \mu_{\varphi_{\xi}(j)} + \nu_{\varphi_{\xi}(j+1)}. $$

Intuitively, this statement uses a combination of basic moves, together with the stair-case inequality, to extract a path homotopic to the initial path, but with only positive turns. This is illustrated in Figure 25. Moving forward, we will use Proposition 7.4 to erase the variables $\theta_k$ with $k \notin I(\xi)$. This includes all of the negative $\theta_k$, but also the $\theta_k$ that are “not positive enough” to compensate for the existence of negative ones. Inequality 7.6 says that the sum of erased $\theta_k$ is negative, and gives a lower bound on its absolute value.

<figure><img src="image_32.png" /><figcaption>FIGURE 25. Illustration of the statement of Proposition 7.4.</figcaption></figure>

**Remark 7.5.** The partition is indexed by some set $\Xi_M$, the definition of which will be determined in the proof, but is of no particular interest. It is worth noting from the proof that the elements $P_\xi$ of the partition are products of intervals.

**Remark 7.6.** We have fixed the lengths $\vec{L}$ because the whole discussion depends implicitly on the value of $\vec{L}$, although not apparent in the notation. The dependency on $\vec{L}$ shall not be relevant to our purposes.
*Proof.* We shall prove the proposition by induction on the integer $M$, using a succession of basic moves. If $M=2$, the partition of $\mathbb{R}$ is $\mathbb{R}_{<0} \cup \mathbb{R}_{\ge 0}$. More precisely, the sets $I$ are as follows.

* If $\theta_1 \ge 0$, there is nothing to be done, and we take $I = \{0, 1, 2\}$ (i.e. $K = M - 1 = 1$ and $\varphi$ is the identity).

* If $\theta_1 < 0$, then we perform a basic move, that is, we take $I = \{0, 2\}$. Inequality (7.2) then implies the claim that $|\theta_1| > \vec{d}(z_{2,0}, z_{1,0}) + \vec{d}(z_{1,2}, z_{0,2}) = \mu_0 + \nu_2$.

Assume now that the construction has been achieved at the rank $M$, and extend it to the next rank $M + 1$, i.e. to a family of aligned geodesics $\gamma_0, \dots, \gamma_{M+1}$. We apply the construction to the geodesics $\gamma_0, \dots, \gamma_M$ and obtain a partition $(P_\xi)_{\xi \in \Xi_M}$ of $\mathbb{R}^{M-1}$ and a family of sets

$$I(\xi) = \{0 = \varphi_{\xi}(0) < \varphi_{\xi}(1) < \dots < \varphi_{\xi}(K_{\xi} + 1) = M\}$$

satisfying the properties (a), (b), (c). Let us now construct a partition $(P'_{\xi'})_{\xi' \in \Xi_{M+1}}$ of $\mathbb{R}^M$ and a family of sets $I'(\xi')$ (associated to a function $\varphi'_{\xi'}$ and an integer $K'_{\xi'}$) satisfying (a), (b), (c) at the rank $M + 1$.

We shall partition

$$\mathbb{R}^M = \mathbb{R}^{M-1} \times \mathbb{R} = \bigcup_{\xi \in \Xi_M} (P_\xi \times \mathbb{R})$$

by partitionning each individual $P_\xi \times \mathbb{R}$. Let $\xi \in \Xi_M$ be fixed – to simplify notations, we drop the mention of $\xi$ in $\varphi_\xi$ and $K_\xi$. Let us cut the $\mathbb{R}$-component of $P_\xi \times \mathbb{R}$ (related to the value of $\theta_M$) depending on the relative positions of the points $z_{\varphi(K),M}$ and $z_{M+1,M}$ along $\gamma_M$.

* If the segment $[z_{\varphi(K),M}, z_{M+1,M}]$ is positively oriented, there is nothing to be done, and we simply let $I' = I(\xi) \cup \{M+1\}$ (in which case $K' = K+1$).

* Otherwise, using (7.1), we observe that for any $k \in \{1, \dots, K\}$,

$$\vec{d}(z_{M+1,\varphi(k)}, z_{\varphi(k+1),\varphi(k)}) \ge 0 \Rightarrow \forall j \ge k, \quad \vec{d}(z_{M+1,\varphi(j)}, z_{\varphi(j-1),\varphi(j)}) > 0$$

$$\vec{d}(z_{\varphi(k-1),\varphi(k)}, z_{M+1,\varphi(k)}) \ge 0 \Rightarrow \forall j \le k, \quad \vec{d}(z_{\varphi(j+1),\varphi(j)}, z_{M+1,\varphi(j)}) > 0.$$

We use this to establish the existence of a uniquely defined $K' \in \{0, \dots, K\}$, represented in Figure 26, such that the following two conditions hold:

$$\begin{array}{l} (7.7) \qquad \forall 0 \le j \le K, \quad (\vec{d}(z_{M+1,\varphi(j)}, z_{\varphi(j+1),\varphi(j)}) \ge 0 \Leftrightarrow j \ge K') \\ \qquad \forall 1 \le j \le K, \quad (\vec{d}(z_{\varphi(j-1),\varphi(j)}, z_{M+1,\varphi(j)}) > 0 \Leftrightarrow j \le K'). \end{array}$$

We then let $I' = \{\varphi(0), \dots, \varphi(K'), M+1\}$, a set associated to the integers $K'$ and the function $\varphi'$ which coincides with $\varphi$ for $j \le K'$ and is equal to $M+1$ at $K'+1$.

Let us now construct the partition in more detail. We temporarily revert to notations depending on $\xi$ for clarity. For any $K' \in \{0, \dots, K_\xi + 1\}$, we denote as $E_{K'}$ the set of values of $\theta_M \in \mathbb{R}$ such that:

* $\vec{d}(z_{\varphi_\xi(K_\xi),M}, z_{M+1,M}) \ge 0$ if $K' = K_\xi + 1$;

* the set of conditions (7.7) holds if $K'$ is an integer in $\{0, \dots, K_\xi\}$.
<figure><img src="image_33.png" /><figcaption>FIGURE 26. In order to highlight the construction of the integer $K'$, we have only represented the geodesics $\gamma_{\varphi(j)}$ for $0 \le j \le K+1$ as well as $\gamma_{M+1}$, and the relevant orthogeodesics.</figcaption></figure>

We can then write $\mathbb{R} = \bigsqcup_{K'=0}^{K_\xi+1} E_{K'}$ for each individual $\xi \in \Xi_M$. This yields the partition

$$ \mathbb{R}^M = \bigsqcup_{\xi' \in \Xi_{M+1}} P_{\xi'}', \quad \text{where} \quad \Xi_{M+1} = \{(\xi, K') : \xi \in \Xi_M, 0 \le K' \le K_\xi + 1\} $$

and $P_{\xi'}' := P_{\xi} \times E_{K'}$ for $\xi' = (\xi, K') \in \Xi_{M+1}$.

Our choice of $K'$ straightforwardly implies that (a) and (b) are satisfied. The condition (c) is automatic in the case $K' = K + 1$, and is the last thing that we need to check when $K' \le K$.

By the induction hypothesis and the construction of $I'$, we already know that (7.6) holds for $j = 1, \dots, K' - 1$, and we just need to check it for $j = K'$. That is, we need to prove that

$$ - \sum_{\varphi(K')<k<M+1} \theta_k \ge \vec{d}(z_{M+1,\varphi(K')}, z_{\varphi(K')+1,\varphi(K')}) + \vec{d}(z_{M,M+1}, z_{\varphi(K'),M+1}). $$

In order to do so, we write

$$ (7.8) \qquad \sum_{\varphi(K')<k<M+1} \theta_k = \sum_{j=K'+1}^{K+1} \theta_{\varphi(j)} + \sum_{j=K'}^{K} \sum_{\varphi(j)<k<\varphi(j+1)} \theta_k. $$

In order to bound the first term, we apply the staircase inequality, Lemma 7.2 (under the form (7.4)), to the geodesics $\gamma_{\varphi(j)}$ for $K' \le j \le K$ and $\gamma_{M+1}$. The conditions (a) and (b) allow to verify the hypotheses. We obtain

$$ (7.9) \qquad \begin{aligned} & - \sum_{j=K'+1}^{K} \vec{d}(z_{\varphi(j-1),\varphi(j)}, z_{\varphi(j+1),\varphi(j)}) - \vec{d}(z_{\varphi(K),M}, z_{M+1,M}) \\ & > \vec{d}(z_{M+1,\varphi(K')}, z_{\varphi(K'+1),\varphi(K')}) + \vec{d}(z_{M,M+1}, z_{\varphi(K'),M+1}). \end{aligned} $$
We then relate each of the terms on the left-hand-side to a $\theta_{\varphi(j)}$ by writing for $j \le K$

$$ \begin{aligned} \theta_{\varphi(j)} &= \vec{d}(z_{\varphi(j)-1,\varphi(j)}, z_{\varphi(j)+1,\varphi(j)}) \\ &= \vec{d}(z_{\varphi(j)-1,\varphi(j)}, z_{\varphi(j-1),\varphi(j)}) + \vec{d}(z_{\varphi(j-1),\varphi(j)}, z_{\varphi(j+1),\varphi(j)}) + \vec{d}(z_{\varphi(j+1),\varphi(j)}, z_{\varphi(j)+1,\varphi(j)}) \\ &= \nu_{\varphi(j)} + \vec{d}(z_{\varphi(j-1),\varphi(j)}, z_{\varphi(j+1),\varphi(j)}) + \mu_{\varphi(j)} \end{aligned} $$

and similarly

$$ \theta_{\varphi(K+1)} = \theta_M = \vec{d}(z_{M-1,M}, z_{M+1,M}) = \nu_{\varphi(K+1)} + \vec{d}(z_{\varphi(K),M}, z_{M+1,M}). $$

Equation (7.9) then becomes

$$ (7.10) \qquad - \sum_{j=K'+1}^{K+1} \theta_{\varphi(j)} + \sum_{j=K'+1}^{K} \mu_{\varphi(j)} + \sum_{j=K'}^{K} \nu_{\varphi(j+1)} \\ > \vec{d}(z_{M+1,\varphi(K')}, z_{\varphi(K'+1),\varphi(K')}) + \vec{d}(z_{M,M+1}, z_{\varphi(K'),M+1}). $$

Now, to bound the second sum in (7.8), we apply the induction hypothesis to $K' \le j \le K$, which yields:

$$ (7.11) \qquad - \sum_{\varphi(j) < k < \varphi(j+1)} \theta_k > \mu_{\varphi(j)} + \nu_{\varphi(j+1)}. $$

Summing (7.10) with (7.11) for all $K' \le j \le K$ allows to obtain, due to (7.8), that

$$ - \sum_{\varphi(K') < k < M+1} \theta_k > \vec{d}(z_{M+1,\varphi(K')}, z_{\varphi(K'+1),\varphi(K')}) + \vec{d}(z_{M,M+1}, z_{\varphi(K'),M+1}) + \mu_{\varphi(K')}. $$

(we notice that all but one of the terms $\mu_{\varphi(j)}$ and $\nu_{\varphi(j+1)}$ vanish). The conclusion then follows, observing that

$$ \vec{d}(z_{M+1,\varphi(K')}, z_{\varphi(K'+1),\varphi(K')}) + \mu_{\varphi(K')} = \vec{d}(z_{M+1,\varphi(K')}, z_{\varphi(K')+1,\varphi(K')}), $$

□

**Remark 7.7.** We can apply a similar construction to families of aligned geodesics ordered in the opposite order, i.e. such that $\gamma_{i+1}$ is now on the right of $\gamma_i$. Importantly, in this case, we do not reverse the direction of the hypotheses (b) (otherwise, the construction would be exactly the same and of no additional interest). See Figure 27.

<figure><img src="image_34.png" /><figcaption>FIGURE 27. Two examples with $M = 3$ and positive or negative crossings.</figcaption></figure>
**7.3. Crossing intervals and erasing of negative crossing parameters.** In this section, we lift the geodesics $\beta$ to the hyperbolic plane, and apply Proposition 7.4 to erase all negative crossing variables. This yields newly defined positive variables $(\tilde{\theta}_q)_{q \in I(\xi)}$. We will rely on the periodic re-indexation of $\Theta$ introduced in §2.5 as well as the lifting from §4.2.2.

The forthcoming discussion, in the case of purely non-crossing eights, is essentially empty.

**7.3.1. Decomposition of $\Theta^i$ in crossing intervals.** Let $i \in \{1, \dots, n\}$. We decompose the set $\Theta^i \sim \mathbb{Z}_{n_i}$ according to the signs of its elements.

**Notation 7.8.** We let $I_1^+ , \dots, I_p^+ \subseteq \Theta^i$ denote the connected components (maximal subintervals) of $\{q \in \Theta^i, \text{Sign}(q) = +\},$ and $I_1^- , \dots, I_p^- \subseteq \Theta^i$ the connected components of the complement. We call the intervals $I_j^\delta$ crossing intervals ($j = 1, \dots, p, \delta = \pm$). Actual crossings correspond to intervals $I_j^\delta$ of length $\ge 2$.

**Remark 7.9.** If the function Sign is constant on $\Theta^i$, then $p=1$ (there is only one crossing interval), and we either have $I_1^+ = \emptyset$ or $I_1^- = \emptyset$. In all other cases, due to the cyclicity of $\Theta^i \sim \mathbb{Z}_{n_i}$, the crossing intervals alternate in signs, and hence the number of positive and negative crossing intervals are equal.

This decomposition of $\Theta^i$ in crossing intervals yields the partition

$$ \Theta^i = \bigsqcup_{j=1}^{p} \bigsqcup_{\delta=\pm} I_j^\delta, $$

which allows us to identify

$$ \mathbb{R}^{\Theta^i} \approx \prod_{j=1}^{p} \prod_{\delta=\pm} \mathbb{R}^{I_j^\delta}. $$

Note that crossing intervals $I_j^\delta$ ($j \in \mathbb{Z}$) can also be defined as subsets of $\Theta^i$, after pre-composing with the projection map $\mathbb{Z} \to \mathbb{Z}_{n_i}$. In that case we have $I_{j+p}^\delta = I_j^\delta + n_i$, where $p$ is the number of positive crossing intervals (equivalently, of negative crossing intervals).

**7.3.2. Application of the erasing procedure to one crossing interval.** Let $j \in \{1, \dots, p\}$ be an actual crossing. We write $I_j^\delta = \{q_0 + 1, \dots, q_0 + M\}$, where $M = \#I_j^\delta \ge 2$. By Lemma 4.1, we can apply Proposition 7.4 (and Remark 7.7 if $\delta = -$) to the family of $M+1$ aligned geodesics $\tilde{\beta}(q)$ with $q_0 \le q \le q_0 + M$, and obtain:

* a partition $(P_\xi^o)_{\xi \in \Xi_M}$ of $\mathbb{R}^{\{q_0+1, \dots, q_0+M-1\}}$;

* subsets $\{q_0 = \varphi_\xi(0) < \varphi_\xi(1) < \dots < \varphi_\xi(K_\xi + 1) = q_0 + M\}$ of $I_j^\delta \cup \{q_0\}$

satisfying the properties (a), (b), (c). We then define:

* $\Xi(I_j^\delta)$ as a copy of $\Xi_M$;

* for $\xi = \xi_j^\delta \in \Xi(I_j^\delta)$, $P_{\xi_j^\delta} = P_\xi^o \times \mathbb{R}$, so that $(P_{\xi_j^\delta})_{\xi_j^\delta \in \Xi(I_j^\delta)}$ is a partition of $\mathbb{R}^{I_j^\delta}$;

* for $\xi \in \Xi(I_j^\delta)$, $I_j^\delta(\xi) := \{\varphi_\xi(1) < \dots < \varphi_\xi(K_\xi + 1)\}$, which is a subinterval of $I_j^\delta$.
We extend this definition to the case where the crossing is not an actual crossing, by setting $(P_{\xi_j^\delta})_{\xi_j^\delta \in \Xi(I_j^\delta)}$ to be the trivial partition $\mathbb{R}^{I_j^\delta} = \mathbb{R}$ indexed by a singleton set $\Xi(I_j^\delta)$, and $I_j^\delta(\xi) = I_j^\delta$ in that case.

**Remark 7.10.** Keep in mind that by construction, for any $\xi \in \Xi(I_j^\delta)$, the interval $I_j^\delta(\xi)$ always contains the right endpoint $q_0+M$ of $I_j^\delta$, which is by definition a U-turn. The sets $P_{\xi_j^\delta} \subseteq \mathbb{R}^{I_j^\delta}$ all contain a full copy of $\mathbb{R}$ for the rightmost point of $I_j^\delta$, because the U-turn variables do not need to be controlled, by virtue of the Useful Remark, Remark 2.11. Indeed, it is automatic that the segments $[z_{q_0-1,q_0}, z_{\varphi(1),q_0}]$ and $[z_{\varphi(K),q_0+M}, z_{q_0+M+1,q_0+M}]$ are positively oriented.

**Remark 7.11.** In the case described in Remark 7.9, the construction is slightly different at the last step of the induction. If $[z_{\varphi(K),n_i}, z_{n_i+1,n_i}]$ is positively oriented, we proceed as normal, but if $[z_{\varphi(K),n_i}, z_{n_i+1,n_i}]$ is negatively oriented, then we simply take $I(\xi) = \emptyset$.

**Remark 7.12.** The construction is done in the universal cover. Since the covering group acts by isometries, if we replace $I$ by $I+n_i$, then the picture is translated by a hyperbolic isometry, namely the translation of axis $\tilde{c}_i$ and of translation length $\ell(c_i)$. This remark implies that the construction done in the universal cover projects down to $S$, and that the subsets $I_j^\delta(\xi_j^\delta)$ satisfy the invariance property $I_{j+p}^\delta(\xi_{j+p}^\delta) = I_j^\delta(\xi_j^\delta) + n_i$.

### 7.3.3. New representative of the homotopy class of $c_i$. 
Let us show how our construction will be used to resolve the issue caused by the crossing interval $I_j^\delta$, raised in §6.5.

**Notation 7.13.** Let $\xi \in \Xi(I_j^\delta)$. For $q = \varphi_\xi(k) \in I_j^\delta(\xi)$, we consider the points

$$G_q := z_{\varphi_\xi(k-1), \varphi_\xi(k)} \quad \text{and} \quad D_q := z_{\varphi_\xi(k+1), \varphi_\xi(k)}.$$

We denote as $\tilde{I}_q'$ the segment $[G_q, D_q]$ on $\tilde{\beta}(q)$.

By property (a), the segment $\tilde{I}_q'$ is positively oriented for every $q \in I_j^\delta(\xi)$. We observe that the portion $\tilde{p}_{q_0+1} \cdot \dots \cdot \tilde{p}_{q_0+M}$ of the infinite path $\cdot_{q \in \Theta^i} \tilde{p}(q)$ is homotopic, with endpoints gliding along $\tilde{\beta}(q_0)$ and $\tilde{\beta}(q_0+M)$, to the path

$$(7.12) \qquad \tilde{B}_{q_0, \varphi_\xi(1)} \cdot \mathcal{I}_{\varphi_\xi(1)}' \cdot \dots \cdot \tilde{B}_{\varphi_\xi(k-1), \varphi_\xi(k)} \cdot \mathcal{I}_{\varphi_\xi(k)}' \cdot \tilde{B}_{\varphi_\xi(j), \varphi_\xi(j+1)} \cdot \dots \cdot \mathcal{I}_{\varphi_\xi(K)}' \cdot \tilde{B}_{\varphi_\xi(K), q_0+M}.$$

This is the path represented in Figure 26. We therefore see that Proposition 7.4 allows to replace our original representation of $\tilde{c}_i$, where the segments $\tilde{I}(q)$ could be oriented in any direction, to a new one where the segments $\mathcal{I}_{\varphi_\xi(k)}'$ are positively oriented.

### 7.3.4. Erasing of all crossing intervals in $\Theta^i$. 
Let us now apply the erasing procedure simultaneously to all crossing intervals in $\Theta^i$.

**Notation 7.14.** We define for the component $c_i$ the set

$$(7.13) \qquad \Xi^i := \prod_{1 \le j \le p} \prod_{\delta=\pm} \Xi(I_j^\delta).$$
We denote as $\xi^i = (\xi_j^\delta)_{\substack{1\le j\le p \\ \delta=\pm}}$ the elements of $\Xi^i$, and for such a $\xi^i$, we let

$$ (7.14) \qquad P_{\xi^i} := \prod_{j=1}^{p} \prod_{\delta=\pm} P_{\xi_j^\delta} \subseteq \prod_{j=1}^{p} \prod_{\delta=\pm} \mathbb{R}^{I_j^\delta} = \mathbb{R}^{\Theta^i} $$

and

$$ (7.15) \qquad I(\xi^i) = \bigsqcup_{j=1}^{p} \bigsqcup_{\delta=\pm} I_j^\delta(\xi_j^\delta), \qquad \tilde{I}(\xi^i) = \bigsqcup_{j \in \mathbb{Z}} \bigsqcup_{\delta=\pm} I_j^\delta(\xi_j^\delta). $$

We denote $n_{\xi^i} = \#I(\xi^i)$.

The collection $(P_{\xi^i})_{\xi^i \in \Xi^i}$ then forms a partition of $\mathbb{R}^{\Theta^i}$. The set $I(\xi^i)$ is a subset of $\Theta^i$ while $\tilde{I}(\xi^i)$ is the corresponding $n_i\mathbb{Z}$-periodic subset of $\Theta^i$. Because $I_j^\delta(\xi_j^\delta)$ always has the same right endpoint as $I_j^\delta$, $I(\xi^i)$ contains all the U-turn variables in $\Theta^i$.

**7.3.5. New variables $\tilde{\theta}_q$ and crossing inequality.** For $\xi^i \in \Xi^i$, the sets $I(\xi^i)$ and $\tilde{I}(\xi^i)$ can be described by an increasing function $(\varphi_k)_{k \in \mathbb{Z}}$ with values in $\Theta^i$ such that $\varphi_{k+\#I(\xi^i)} = \varphi_k + n_i$.

**Notation 7.15.** Let $\xi^i \in \Xi^i$. For $q = \varphi_k \in I(\xi^i)$, we denote as:

* $G_q := z_{\varphi_{k-1}, \varphi_k}$ and $D_q := z_{\varphi_{k+1}, \varphi_k}$;

* $\mathcal{I}'_q$ the segment $[G_q, D_q]$ on $\tilde{\beta}(q)$, and $\tilde{\theta}_q$ its length;

* $\tilde{L}_q = L_{\varphi_{k-1}, \varphi_k}$ the length of the orthogeodesic $\tilde{B}_{\varphi_{k-1}, \varphi_k}$.

By property (a), the segment $\mathcal{I}'_q$ is always positively oriented and hence $\tilde{\theta}_q \ge 0$.

We can now adapt (7.12) and now simultaneous erase all crossing intervals. Indeed, the infinite concatenation $\cdot_{k \in \mathbb{Z}}(\tilde{B}_{\varphi_{k-1}, \varphi_k} \cdot \mathcal{I}'_{\varphi_k})$ gives an infinite curve in the hyperbolic plane, which projects down to a curve homotopic to the lift $\tilde{c}_i$ of $c_i$ to $\mathbb{H}$. The fact that the new path does no longer go along $\tilde{I}(q)$ for $q \in \tilde{I}(\xi^i)$ motivates the following terminology.

**Definition 7.16.** The variables $\theta_q$ for $q \in \Theta^i \setminus \tilde{I}(\xi^i)$ are said to be erased.

We can then restate the property (c) from Proposition 7.4 in the following equivalent way.

**Lemma 7.17.** For any component $i$,

$$ (7.16) \qquad \sum_{q \in \Theta^i} \theta_q < \sum_{q \in I(\xi^i)} \tilde{\theta}_q. $$

*Proof.* We simply notice that property (c) states that

$$ (7.17) \qquad \sum_{q \in \Theta^i \setminus I(\xi^i)} \theta_q < - \sum_{k=1}^{\#I(\xi^i)} (\nu_{\varphi_{k+1}} + \mu_{\varphi_k}) $$

with the notation $\mu, \nu$ from Proposition 7.4, and observe that by definition,

$$ (7.18) \qquad \nu_{\varphi_k} + \tilde{\theta}_{\varphi_k} + \mu_{\varphi_k} = \theta_{\varphi_k}. $$

□
**Remark 7.18.** In the case described in Remarks 7.9 and 7.11, when the sign function is constant on $\Theta^i$, if $\xi$ is such that $I(\xi) = \emptyset$, then (7.16) means that

$$ \sum_{q \in \Theta^i} \theta_q < 0. $$

In this case it is sufficient to consider the geodesic representative of $c_i$, and we won't need to express the length $\ell_Y(c_i)$ in coordinates.

**7.3.6. Product over all components $c_i$.** Let us now group our constructions for the different components $c_1, \dots, c_n$ of the multiloop $c$. In order to do so, we let $\Xi := \prod_{i=1}^n \Xi^i$. We shall consider the cartesian product of the partitions $P_{\xi^i}$, that is, for $\xi = (\xi^1, \dots, \xi^n) \in \Xi$, we let

$$ (7.19) \qquad P_{\xi} = \prod_{i=1}^{n} P_{\xi^i} \quad \text{and} \quad I(\xi) = \bigsqcup_{i=1}^{n} I(\xi^i) \subseteq \Theta. $$

The sets $(P_{\xi})_{\xi \in \Xi}$ then form a partition of $\prod_{i=1}^n \mathbb{R}^{\Theta^i} = \mathbb{R}^{\Theta}$. Each $P_{\xi}$ is a product of intervals.

We shall say that the variables $\theta_q$ for $q \notin I(\xi)$ are erased. The set $I(\xi)$ contains all U-turn variables. As a consequence, in the purely non-crossing case, we have $I(\xi) = \Theta$ for all $\xi$, and all the previous discussion is empty.

**7.4. New expression for the length.** The erasing procedure immediately yields a new expression for $\ell_Y(c)$, true for $\vec{\theta} \in P_{\xi}$, which replaces (4.12).

**Proposition 7.19.** Let $\xi \in \Xi$. As soon as $\vec{\theta} \in P_{\xi}$, we have

$$ (7.20) \qquad \ell_Y(\mathbf{c}) = \sum_{i=1}^{n} M_{n_{\xi^i}}((\epsilon \tilde{L}_q)_{q \in I(\xi^i)}, (\tilde{\theta}_q)_{q \in I(\xi^i)}) $$

where $n_{\xi^i} = \#I(\xi^i)$, $\epsilon \tilde{L}_q := \text{Sign}(q) \tilde{L}_q$ and $M_n$ is defined in (4.10).

*Proof.* For each $1 \le i \le n$, on the subset $P_{\xi^i}$ of $\mathbb{R}^{\Theta^i}$, we apply Proposition 4.6 to the representative $\cdot_{k \in \mathbb{Z}}(\tilde{B}_{\varphi_{k-1}, \varphi_k} \cdot \mathcal{I}'_{\varphi_k})$ of the homotopy class of $\tilde{c}_i$. $\square$

Note that, as long as $\vec{\theta} \in P_{\xi}$, each variable $\tilde{\theta}_q$ with $q \in I(\xi)$ is positive. As a consequence, this new formula solves the first issue raised in §6.5.

When using formula (7.20) later, we shall consider all the variables $\theta_q$ with $q \notin I(\xi)$ as additional neutral parameters. The following observation will then be useful.

**Lemma 7.20.** Let $\xi \in \Xi$. On the set $P_{\xi}$, for any $q \in I(\xi)$,

$$ (7.21) \qquad \tilde{L}_q = \text{fn}((\theta_{q'})_{q' \notin I(\xi)}, \vec{L}) \quad \text{and} \quad \tilde{\theta}_q = \theta_q + \text{fn}((\theta_{q'})_{q' \notin I(\xi)}, \vec{L}). $$

*Proof.* Let $\xi \in \Xi$. For $1 \le i \le n$, let us represent $\xi^i \in \Xi^i$ by a function $(\varphi_k)_{k \in \mathbb{Z}}$. For $q \in I(\xi^i)$, the length $\tilde{L}_q$ is the length of the orthogeodesic from $\tilde{\beta}(\varphi_{k-1})$ to $\tilde{\beta}(\varphi_k)$, and therefore can be expressed as a function of $\vec{L}$ and the $\theta_{q'}$ with $q' \in \Theta^i$ such that $\varphi_{k-1} < q' < \varphi_k$. This implies that $q' \notin I(\xi)$, which was our claim.
Similarly, the quantities $\nu_{\varphi_k}, \mu_{\varphi_k}$ can be expressed solely in terms of the $\theta_{q'}$ with $q' \notin I(\xi)$. However, we have seen in (7.18) that $\theta_{\varphi_k} = \nu_{\varphi_k} + \tilde{\theta}_{\varphi_k} + \mu_{\varphi_k}$, which allows us to conclude. $\square$

As a consequence, if we fix $(\theta_q)_{q \notin I(\xi^i)}$ as well as $\vec{L}$, and view the expression (7.20) as a function of $(\tilde{\theta}_q)_{q \in I(\xi^i)}$, then Proposition 6.24 holds for this new length function.

**Remark 7.21.** An intuition as to why the variables $\theta_q$ taking negative values yield additional neutral parameters can be found by examining Figure 21. Recall that we have motivated in §5.3 the fact that the lengths $\vec{L}$ are neutral parameters by observing that, when we describe the path $c_i$ as the concatenation $c_i^{\text{op}} = \cdot_{q \in \Theta^i} p(q)$, each bar is visited twice, which means that there is a factor of 2 in front of each $L_j$ in our comparison estimates.

We see in the example of Figure 21 that, when $\theta_1^+ < 0$, the portion $I_1^+$ is visited three times by the path $c_i^{\text{op}}$, instead of once when $\theta_1^+ \ge 0$. This means that we do not need to apply the operator $\mathcal{L}$ to cancel exponential growth related to the parameter $\theta_1^+$, or, in other words, that it is a neutral parameter.

**7.5. The purely non-crossing case.** The reader only interested in the purely non-crossing case can read the rest of the paper, considering that $\Xi$ is a set with one element denoted $\xi = (\xi^1, \dots, \xi^n)$, and that in what follows we let $I(\xi) = \Theta$, $I(\xi^i) = \Theta^i$ for $i \in \{1, \dots, n\}$. In this case we have $\tilde{\theta}_q = \theta_q$ for all $q \in \Theta$, as well as $I'_q = \bar{I}_q = [G_q, D_q]$.

**7.6. A partition of unity on Teichmüller space.** We recall that the space of interest in this article is the Teichmüller space $\mathcal{T}_{g_S, n_S}^*$, which we have equipped with a new set of coordinates $(\vec{L}, \vec{\theta}) \in \mathbb{R}_{>0}^r \times \mathbb{R}^{\Theta}$. We have introduced a partition $\mathbb{R}^{\Theta} = \bigsqcup_{\xi \in \Xi} P_{\xi}$ that we wish to use on the parameter $\vec{\theta}$. Because our methods require some smoothness, it will be more convenient to partition the values of $\vec{\theta}$ using a partition of unity, which we shall now construct. Note that the entire discussion above is dependent on the length vector $\vec{L}$, although this has not been made explicit in the notation.

**Notation 7.22.** Let $\chi_{C,0}$ and $\chi_{C,\infty}$ denote two smooth functions on $\mathbb{R}_{\ge 0}$ such that:

* $\chi_{C,0} + \chi_{C,\infty} \equiv 1$;

* $\chi_{C,0}$ is supported in $[0, 2 \log 2]$ and identically equal to 1 on $[0, \log 2]$.

**Definition 7.23.** For $\xi \in \Xi$ and $Q \subseteq I(\xi)$, we define a cut-off function on Teichmüller space,

$$ (7.22) \qquad \Psi_{\xi, Q}(\vec{L}, \vec{\theta}) = \mathbb{1}_{P_{\xi}}(\vec{L}, \vec{\theta}) \prod_{q \in Q} \chi_{C, \infty}(\tilde{\theta}_q) \prod_{q' \notin Q} \chi_{C, 0}(\tilde{\theta}_{q'}) $$

By construction,

$$ \sum_{\substack{\xi \in \Xi \\ Q \subseteq I(\xi)}} \Psi_{\xi, Q} \equiv 1. $$

As announced in §7.4, we associate to each partition element some additional neutral variables.
**Notation 7.24** (Neutral variables associated with $(\xi, \mathcal{Q})$). For $\xi \in \Xi$ and $\mathcal{Q} \subseteq I(\xi)$, we let

$$ (7.23) \qquad \Theta_{\mathrm{nt}}(\xi, \mathcal{Q}) = \Theta \setminus \mathcal{Q}. $$

Note that, on the support of the cut-off function $\Psi_{\xi, \mathcal{Q}}$, the variables $(\tilde{\theta}_q)_{q \in \Theta \setminus \mathcal{Q}}$ stay in $[0, 2 \log 2]$, so that we do not need to gain exponential decay in those variables. They can therefore be considered to be neutral variables.

## 8. THE LENGTHS OF BOUNDARY CURVES

8.1. **Objective and plan of the section.** Recall that, in Proposition 4.11, we have obtained expressions for the lengths of the pair of pants decomposition $(\Gamma_\lambda)_{\lambda \in \Lambda}$ in our new coordinates $(\vec{L}, \vec{\theta})$ on the Teichmüller space $\mathcal{T}_{g_S, n_S}^*$. This new expression reads $\ell_Y(\Gamma_\lambda) = Q_m(\vec{L}^\lambda, \vec{\tau}^\lambda)$, where $\vec{L}^\lambda = (L_1^\lambda, \dots, L_m^\lambda)$ and $\vec{\tau}^\lambda = (\tau_1^\lambda, \dots, \tau_m^\lambda)$ are the parameters described in §4.4.2, $m = m(\lambda)$, and the functions $F_m, Q_m$ are defined in (4.20) by

$$ (8.1) \qquad F_m(\vec{L}, \vec{\tau}) = \frac{\cosh(\frac{1}{2} Q_m(\vec{L}, \vec{\tau}))}{\exp(\frac{1}{2} \sum_{j=1}^m \tau_j)} = \frac{1}{2} \sum_{\alpha \in \{0,1\}^m} (-1)^{|\alpha|} \operatorname{hyp}_{-\partial \alpha} \left(\frac{\vec{L}}{2}\right) e^{-\alpha \cdot \vec{\tau}} $$

with $(\partial \alpha)_j = 1$ if $\alpha_j = \alpha_{j-1}$ and $-1$ if $\alpha_j \neq \alpha_{j-1}$.

The aim of this section is to study the function $F_m$, and in particular its logarithmic derivatives, which will appear when we apply our key argument to the integral (5.13). The formula (8.1) is problematic because of its alternating signs: it is not clear which terms dominate the sum, nor is it that the formula gives a positive function. In fact, the domain $\mathcal{D}$ of $(\vec{L}, \vec{\theta})$ is precisely defined by the fact that the formula giving $\cosh(\ell_Y(\Gamma_\lambda)/2)$ is greater than 1 for all $\lambda \in \Lambda$, as seen in Proposition 3.21.

We work around this problem introducing a *favorable region*, i.e. a region of $\mathcal{D}$ on which $F_m$ is well-approximated by its term with $\alpha = (0, \dots, 0)$, equal to $\prod_{j=1}^m \sinh(L_j/2)$. This domain is defined in terms of the *heights of cells* introduced in §8.4.1. We provide estimates on the derivatives of $F_m$ true inside the favorable region in Proposition 8.17. Outside the favorable region, the height of one (or several) cells must be bounded – we *neutralize* such cells. The neutralization procedure is to low-height cells what the erasing procedure was to negative $\theta$ parameters in §7. The description of the neutralization procedure while dealing simultaneously with all of the boundary components $(\Gamma_\lambda)_{\lambda \in \Lambda}$ requires some technical considerations, addressed in §8.5. In §8.6, we finally obtain a partition of unity of the Teichmüller space so that we understand the length of $\Gamma_\lambda$ and its derivatives on every piece of the partition.

8.2. **Polygonal curves, neutralization, derivation.** In §4.4, we have used a key observation on the structure of the curves $\Gamma_\lambda$ to compute their lengths: the fact that they can be viewed as a succession of perpendicular geodesic segments of alternating lengths $L_j^\lambda, \tau_j^\lambda$, turning the same direction each time (see Figure 19). We shall introduce a notion of polygonal curve, which encapsulates this key property of $\Gamma_\lambda$. Let us first provide this definition without reference to a hyperbolic metric. Subsequently in §8.3, a hyperbolic metric will be given, and we will work with preferred “geodesic representatives”.
**8.2.1. Definition of polygonal curves.** In the following, we fix a base surface **S** and an oriented multi-curve **β** on **S**.

**Definition 8.1.** A polygonal curve $\Gamma$ (on the surface $S$ and based on $\beta$) is an oriented closed path without self-intersections, formed by a cyclic sequence $(J_j, K_j)_{j \in \mathbb{Z}_m}$ of oriented segments where, for all $j \in \mathbb{Z}_m$,

• $J_j$ meets the multi-curve $\beta$ at its endpoints and not in its interior, and is homotopically non-trivial with gliding endpoints along $\beta$;

• $K_j$ is a subsegment of $\beta$, possibly with reversed orientation;

• $J_j$ arrives at $K_j$ on the right at the point $t(J_j) = o(K_j)$, and $K_j$ arrives at $J_{j+1}$ on the right at the point $t(K_j) = o(J_{j+1})$.

Note that the multi-curve $\beta$ is essential to this definition. The segments $J_j$ and $K_j$ play different roles and will therefore be named differently.

**Definition 8.2.** The segments $K_j$ are called the *cells* of the polygonal curve $\Gamma$, and the $J_j$ its *bridges*. We denote $\text{Cell}(\Gamma) = \{K_j, j \in \mathbb{Z}_m\}$ and $\text{Br}(\Gamma) = \{J_j, j \in \mathbb{Z}_m\}$.

For a cell $K = K_j \in \text{Cell}(\Gamma)$, the bridges *adjacent to* $K$ refer to the bridges $J_j$ and $J_{j+1}$. We respectively call these bridges the *bridge arriving on* $K$ and the *bridge coming out of* $K$.

**Notation 8.3.** We call $m$ the combinatorial length of $\Gamma$, denoted $|\Gamma|$.

By convention, for $m=0$, a polygonal curve of combinatorial length 0 is a simple curve that does not intersect the family $\beta$ (but may be homotopic to a component of $\beta$). In particular, polygonal curves of combinatorial length 0 do not have any cells.

**Example 8.4.** We have seen in §4.4.2 that each of the curves $(\Gamma_\lambda)_{\lambda \in \Lambda}$ in the pair of pants decomposition defined in §3.2 is a polygonal curve based on $\beta$, where:

• the bridges $J_j$ are among the bars $B_q$;

• the cells $K_j$ are among the segments $K_{q'}$ defined in §4.3.

The family $(\Gamma_\lambda)_{\lambda \in \Lambda}$ will be the main example to which the whole discussion applies. For $\lambda \in \Lambda_{BC}$, we can even say that the corresponding cells are among the segments $(K_q)_{q \in \Theta}$.
Note that the components $(\beta_\lambda)_{\lambda \in \Lambda^\beta}$ are polygonal curves of combinatorial length 0.

**8.2.2. Neutralizing a cell.** Let $\Gamma = (J_j, K_j)_{j \in \mathbb{Z}_m}$ be a polygonal curve on $S$ of combinatorial length $m \ge 2$. Let $j \in \mathbb{Z}_m$. We may construct a new polygonal curve $\Gamma'$ by replacing the sequence $(J_j, K_j, J_{j+1})$ in $\Gamma$ by one single bridge $\tilde{J}_j := J_j \cdot K_j \cdot J_{j+1}$ (or a smoothened version thereof). This operation is represented in Figure 28a. We observe that the derived polygonal curve $\Gamma'$ we obtain is homotopic to $\Gamma$ and that its combinatorial length is $m-1$.

**Definition 8.5.** The polygonal curve $\Gamma'$ is said to be *derived from* $\Gamma$ by neutralizing the cell $K_j$. We call $\tilde{J}_j$ the *roof over* $K_j$. We denote $\tilde{K}_{j-1} := K_{j-1}$, $\tilde{K}_{j+1} := K_{j+1}$ the *new cells of* $\Gamma'$.

The motivation for the name “new cells” will become apparent when we take geodesic representatives. The new polygonal curve $\Gamma'$ satisfies

• $\text{Br}(\Gamma') = \{\tilde{J}_j\} \cup (\text{Br}(\Gamma) \setminus \{J_j, J_{j+1}\})$;
<figure><img src="image_35.png" /><figcaption>FIGURE 28. Neutralization of one or several cells.</figcaption></figure>

$$
\bullet \ Cell(\Gamma') = \{\tilde{K}_{j-1}, \tilde{K}_{j+1}\} \cup (Cell(\Gamma) \setminus \{K_{j-1}, K_j, K_{j+1}\}).
$$

If now $\Gamma$ is a polygonal curve of combinatorial length 1, we define $\tilde{J}_1$ as the simple closed curve $J_1 \cdot K_1$ (or a smoothened version thereof). Then $\Gamma' = \tilde{J}_1$ is a polygonal curve of combinatorial length 0 homotopic to $\Gamma$.

8.2.3. *Simultaneous neutralization of several cells.* We now consider an interval $I = \{a, \dots, b\}$ of $\mathbb{Z}_m$ of length $1 \le \#I < m$. Let us define $\tilde{J}_I := J_a \cdot K_a \cdot \dots \cdot K_b \cdot J_{b+1}$ (or a smoothened version thereof) and call $\tilde{J}_I$ the *roof over* $(K_j)_{j \in I}$. A new polygonal curve $\Gamma'$ may be obtained by replacing the sequence $(J_a, K_a, \dots, K_b, J_{b+1})$ by $\tilde{J}_I$, as represented in Figure 28b. We then say that $\Gamma'$ is *derived from* $\Gamma$ *by neutralizing simultaneously all the* $(K_j)_{j \in I}$.

We denote $\tilde{K}_{a-1} := K_{a-1}$, $\tilde{K}_{b+1} := K_{b+1}$ and call them *new cells* of $\Gamma'$. The new polygonal curve $\Gamma'$ then satisfies:

$$
\bullet \ Br(\Gamma') = \{\tilde{J}_I\} \cup (Br(\Gamma) \setminus \{J_a, J_{a+1}, \dots, J_{b+1}\});
$$

$$
\bullet \ Cell(\Gamma') = \{\tilde{K}_{a-1}, \tilde{K}_{b+1}\} \cup (Cell(\Gamma) \setminus \{K_{a-1}, K_a, \dots, K_{b+1}\}).
$$

We extend these definitions to the case $I = \mathbb{Z}_m$, now taking $\Gamma'$ to be a simple curve freely homotopic to the concatenation $J_1 \cdot K_1 \cdot \dots \cdot J_m \cdot K_m$, so that it does not intersect $\beta$. Then, $\Gamma'$ is polygonal curve of combinatorial length 0.

In all cases, $\Gamma'$ is homotopic to $\Gamma$, and its combinatorial length now is $m - \#I$. The two notions of neutralization coincide when $I = \{j\}$, and the polygonal curve derived by neutralizing simultaneously all the $(K_j)_{j \in I}$ is the same as the one derived by successively neutralizing the cells $K_a, \dots, K_b$ (in whichever order).

**8.3. Geodesic representative and length inequalities.** We now endow $\mathbf{S}$ with a hyperbolic metric $Y \in \mathcal{T}_{g_S, n_S}^*$, and consider polygonal curves as geometric objects.

8.3.1. *Geodesic representative.* As suggested in Remark 2.8, we can assume by choosing the right representative in the Teichmüller space of S that β is a multi-geodesic, which then implies that the cells $K_j$ are geodesic segments. We replace each bridge $J_j$ by its orthogeodesic representative $\tilde{J}_j$. As the segments $J_j$ glide to their orthogeodesic representatives $\tilde{K}_j$, the segments $K_j$ glide to new segments $\tilde{K}_j$ whose endpoints match
those of the family $\bar{J}_j$. We call $(\bar{J}_j, \bar{K}_j)_{j \in \mathbb{Z}_m}$ the piecewise geodesic representative of $\Gamma$. This allows us to define the maps

$$ \begin{array}{l} \mathcal{T}_{g_S, n_S}^* \to \mathbb{R}_{>0} \\ Y \mapsto \ell_Y(J_j) := \ell(\bar{J}_j) \end{array} $$

$$ \begin{array}{l} \mathcal{T}_{g_S, n_S}^* \to \mathbb{R}_{>0} \\ Y \mapsto \ell_Y(K_j) := \ell(\bar{K}_j). \end{array} $$

Note that now $\bar{J}_j$ and $\bar{K}_j$ form a right angle at their meeting point, and so do $\bar{K}_j$ and $\bar{J}_{j+1}$.

In the following, unless specified otherwise, we always take geodesic representatives of polygonal curves, and we suppress all the bars $\bar{\phantom{x}}$ from the notation.

We notice that the discussion in §4.4 is valid for any polygonal curve, not only the family $(\Gamma_\lambda)_{\lambda \in \Lambda}$ of boundary components of our pair of pants decomposition. In particular, if we introduce the short-hand notations

$$ (8.2) \qquad L_j := \ell_Y(J_j) \quad \text{and} \quad \tau_j := \ell_Y(K_j) $$

then we can apply Proposition 4.11 to any polygonal curve $\Gamma$, and obtain that $\ell_Y(\Gamma) = Q_m(\vec{L}, \vec{\tau})$ for the length-vectors $\vec{L} = (L_j)_{j \in \mathbb{Z}_m}$ and $\vec{\tau} = (\tau_j)_{j \in \mathbb{Z}_m}$.

### 8.3.2. Neutralization and lengths.
Let us now examine the geometric impact of neutralizing cells in a polygonal curve $\Gamma$.

First, we observe that if $\Gamma'$ is obtained by neutralization of a cell $K_j$ in a polygonal curve $\Gamma$, then, due to taking geodesic representatives, the new cell $\tilde{K}_{j-1}$ is strictly included in $K_{j-1}$, and the new cell $\tilde{K}_{j+1}$ in $K_{j+1}$ (see Figure 29). This justifies the name “new cells” introduced in Definition 8.5. This is also the case when neutralizing several cells simultaneously.

We extend the notation (8.2) to the definitions introduced in §8.2.2 and 8.2.3 related to new cells and bridges, so that:

* $\tilde{L}_j, \tilde{\tau}_{j-1}$ and $\tilde{\tau}_{j+1}$ respectively denote the lengths of the new bridge $\tilde{J}_j$ and the new cells $\tilde{K}_{j-1}$ and $\tilde{K}_{j+1}$ obtained when we neutralize the cell $K_j$;

* $\tilde{L}_I, \tilde{\tau}_{a-1}$ and $\tilde{\tau}_{b+1}$ respectively denote the lengths of the new bridge $\tilde{J}_I$ and the new cells $\tilde{K}_{a-1}$ and $\tilde{K}_{b+1}$ obtained when we neutralize simultaneously $(K_j)_{j \in I}$ for $I = \{a, \dots, b\}$.

Importantly, if $\Gamma'$ is a polygonal curve derived from $\Gamma$, because $\Gamma'$ is homotopic to $\Gamma$, we have

$$ Q_{|\Gamma|}(\vec{L}, \vec{\tau}) = \ell_Y(\Gamma) = \ell_Y(\Gamma') = Q_{|\Gamma'|}(\vec{L}', \vec{\tau}') $$

where $\vec{L}', \vec{\tau}'$ are the length parameters of $\Gamma'$. As a consequence, we obtain multiple distinct rewritings of the length of the same closed geodesic, depending on which polygonal curve we use, $\Gamma$, or any derived polygonal curve $\Gamma'$. Our goal in the following will be to neutralize enough cells so that the expression of the length is well-behaved and can be easily understood.

We make the following observation.
<figure><img src="image_36.png" /><figcaption>FIGURE 29. Geodesic representative of a polygonal curve and of a neutralization.</figcaption></figure>

**Lemma 8.6.** Let $\Gamma$ be a polygonal curve of combinatorial length $m$, and $I = \{a, \dots, b\}$ be an interval of $\mathbb{Z}_m$. Then,

$$ (8.3) \qquad \begin{aligned} \tilde{L}_I &= \operatorname{fn}(L_a, \dots, L_{b+1}, \tau_a, \dots, \tau_b) \\ \tilde{\tau}_{a-1} &= \tau_{a-1} - \operatorname{fn}(L_a, \dots, L_{b+1}, \tau_a, \dots, \tau_b) \\ \tilde{\tau}_{b+1} &= \tau_{b+1} - \operatorname{fn}(L_a, \dots, L_{b+1}, \tau_a, \dots, \tau_b). \end{aligned} $$

This is a straightforward consequence of applying the following elementary lemma to the highlighted right-angled polygon in Figure 29.

**Lemma 8.7.** Let $m \ge 3$. In any right-angled convex polygon $(J_j, K_j)_{j \in \mathbb{Z}_m}$, any three consecutive lengths can be written as functions of the $2m-3$ others:

$$ (\ell(K_j), \ell(J_j), \ell(K_{j+1})) = \operatorname{fn}((\ell(J_k))_{k \in \mathbb{Z}_m \setminus \{j\}}, (\ell(K_k))_{k \in \mathbb{Z}_m \setminus \{j, j+1\}}). $$

*Proof of Lemma 8.7.* When $m = 3$, this is a direct consequence of the basic formulas for convex right-angled hexagons (see e.g. [5, Theorem 2.4.1 (i-ii)]). For $m > 3$, we obtain the result by induction, introducing a right-angled hexagon at each step. $\square$

**8.4. Favorable regions for studying the length of a polygonal curve.** We are now ready to study the variations of the function

$$ (8.4) \qquad F_m(\vec{L}, \vec{\tau}) = \frac{1}{2} \sum_{\alpha \in \{0,1\}^m} (-1)^{|\alpha|} \operatorname{hyp}_{-\partial\alpha} \left(\frac{\vec{L}}{2}\right) e^{-\alpha \cdot \vec{\tau}}. $$

We observe that the first term, corresponding to $\alpha = (0, \dots, 0)$, has the coefficient

$$ \operatorname{hyp}_{-1}\left(\frac{\vec{L}}{2}\right) = \prod_{j=1}^{m} \sinh\left(\frac{L_j}{2}\right). $$

We shall identify a “favourable” region of the Teichmüller space where this term dominates all others, so that $\cosh(\ell_Y(\Gamma)/2)$ is comparable to $\exp(\frac{1}{2}(\sum_{j=1}^m \tau_j)) \prod_{j=1}^m \sinh(L_j/2)$.
8.4.1. *Height of a cell.* First, we introduce a geometric quantity associated to each cell, called its *height*. We will see in §8.4.2 how this is the relevant quantity one needs to introduce to describe the variations of the function $F_m$.

**Definition 8.8.** Let $\Gamma$ be a polygonal curve of combinatorial length $m$, and $j \in \mathbb{Z}_m$. The height of the cell $K_j$ is defined as

$$ (8.5) \qquad Z_{K_j}^{\Gamma} := \tanh(L_j) \tanh(L_{j+1}) e^{\tau_j}. $$

We also use the short-hand notation $Z_j := Z_{K_j}^{\Gamma}$, when convenient.

Note that, if $m = 1$, the height is defined as $Z_1 = \tanh^2(L_1)e^{\tau_1}$. If $m = 0$, the polygonal curve has no cell, so there is no notion of height.

We prove the following bounds on the height $Z_j$ of a cell.

**Lemma 8.9.** For $m \ge 2$, $1 \le Z_j \le e^{\tau_j}$ and upon neutralizing the cell $K_j$, we have

$$ (8.6) \qquad \tilde{L}_j \le \log Z_j + L_j + L_{j+1}. $$

*Proof.* The upper bound $Z_j \le e^{\tau_j}$ is a direct consequence of the definition. The two other bounds can be obtained from the trigonometric formula for right-angled hexagons, see e.g. [5, Theorem 2.4.1 (i)], which yield

$$ (8.7) \qquad \cosh(\tilde{L}_j) = \sinh(L_j) \sinh(L_{j+1}) \cosh(\tau_j) - \cosh(L_j) \cosh(L_{j+1}). $$

Indeed, $\cosh(\tilde{L}_j) \ge 0$ together with (8.7) imply that

$$ \tanh(L_j) \tanh(L_{j+1}) \cosh(\tau_j) \ge 1 $$

and hence $Z_j \ge 1$. Then, we rewrite (8.7) as

$$ (\cosh(\tau_j) - 1) \sinh(L_j) \sinh(L_{j+1}) = \cosh(\tilde{L}_j) + \cosh(L_j - L_{j+1}) \ge \cosh(\tilde{L}_j). $$

Since $\cosh(\tau_j) - 1 = 2 \sinh^2(\tau_j/2) \le e^{\tau_j}/2$, this implies that

$$ 2 \cosh(\tilde{L}_j) \le e^{\tau_j} \sinh(L_j) \sinh(L_{j+1}) = Z_j \cosh(L_j) \cosh(L_{j+1}) $$

which yields the result since $e^x/2 \le \cosh(x) \le e^x$. $\square$

We notice that (8.6) means that the height $Z_j$ of a cell $K_j$ can be used to bound the length $\tilde{L}_j$ of the roof over it, under the additional assumption that the lengths of the adjacent bridges are bounded. More generally, the height of the cell $K_j$ can be used to bound the length of the orthogeodesic $W_j$ constructed below and represented in Figure 30.

**Definition 8.10.** For $j \in \mathbb{Z}_m$, we define the orthogeodesic $W_j$ by:

* $W_j = K_j$ if $L_j \ge 1$ and $L_{j+1} \ge 1$;

* $W_j$ is the orthogeodesic between $K_{j-1}$ and $J_{j+1}$ if $L_j < 1$ and $L_{j+1} \ge 1$;

* $W_j$ is the orthogeodesic between $J_j$ and $K_{j+1}$ if $L_j \ge 1$ and $L_{j+1} < 1$;

* $W_j = \tilde{J}_j$ otherwise, i.e. if $L_j < 1$ and $L_{j+1} < 1$.

We then have the following bound.

**Lemma 8.11.** For any $j$, $\ell(W_j) \le \log(Z_j) + 2$.
<figure><img src="image_37.png" /></figure>

<figure><img src="image_38.png" /></figure>

<figure><img src="image_39.png" /><figcaption>FIGURE 30. The orthogeodesic $W_j$ depending on the values of $L_j$ and $L_{j+1}$.</figcaption></figure>

*Proof.* If both $L_j$ and $L_{j+1}$ are smaller than one, then this follows from (8.6). When both $L_j$ and $L_{j+1}$ are greater or equal to 1, it is a straightforward consequence of the definition and the fact that $\log(1/\tanh(1)) < 1$. In the last cases, say $L_j < 1$ and $L_{j+1} \ge 1$, we use the hyperbolic formulas for right-angles pentagons (see [5, Theorem 2.3.4 (i)]) to write $\cosh \ell(W_j) = \sinh L_j \sinh \tau_j$, which once again yields the claim since $\log(2/\sinh(1)) < 1$. $\square$

8.4.2. Favourable regions and estimates on $F_m$. We are ready to use our notion of height of a cell to identify a condition upon which the leading term of $F_m$ is the term $\alpha = (0, \dots, 0)$.

**Lemma 8.12.** For any polygonal curve $\Gamma$ of combinatorial length $m \ge 1$, any $Y \in \mathcal{T}_{g_S, n_S}^*$, any multi-index $\alpha \in \{0, 1\}^m$ distinct from $(0, \dots, 0)$,

$$ (8.8) \qquad \frac{\operatorname{hyp}_{-\partial\alpha}\left(\frac{\vec{L}}{2}\right)}{\operatorname{hyp}_{-1}\left(\frac{\vec{L}}{2}\right)} e^{-\alpha \cdot \vec{\tau}} \le \prod_{j \in \alpha} \frac{4}{Z_{K_j}^{\Gamma}}. $$

In particular, if all the $Z_j$ go to $+\infty$, then for all $\alpha \ne (0, \dots, 0)$,

$$ \frac{\operatorname{hyp}_{-\partial\alpha}\left(\frac{\vec{L}}{2}\right)}{\operatorname{hyp}_{-1}\left(\frac{\vec{L}}{2}\right)} e^{-\alpha \cdot \vec{\tau}} = o(1). $$

We can then deduce that $F_m(\vec{L}, \vec{\tau}) \sim \operatorname{hyp}_{-1}(\vec{L}/2)$ and $\cosh(\ell_Y(\Gamma)/2) \sim \operatorname{hyp}_{-1}\left(\frac{\vec{L}}{2}\right) \exp\left(\frac{1}{2}\left(\sum_{j=1}^m \tau_j\right)\right)$, as quantified in the following statement.

**Corollary 8.13.** Let $\Gamma$ be a polygonal curve of combinatorial length $m \ge 1$. For a real number $a \ge (m+3)\log 2$, if we assume

$$ \forall j \in \mathbb{Z}_m, \quad Z_{K_j}^{\Gamma} \ge e^a $$

then we have

$$ (8.9) \qquad F_m(\vec{L}, \vec{\tau}) \ge \frac{1}{4} \operatorname{hyp}_{-1}\left(\frac{\vec{L}}{2}\right) \quad \text{and} \quad \cosh\left(\frac{\ell_Y(\Gamma)}{2}\right) \ge \frac{e^{ma/2}}{2^{m+2}}. $$
This motivates the following definitions.

**Notation 8.14.** In the following, we let $a = a(r) = \max((2r+3)\log(2), 2\log(8 \cosh(3/2)))$, where, recall, $r$ is the number of intersections of our generalized eight. We also fix two smooth functions $\chi_{\Gamma,0}, \chi_{\Gamma,\infty}$ on $\mathbb{R}$, with the property that $\chi_{\Gamma,0}$ is supported in $[-e^{a+1}, e^{a+1}]$, $\chi_{\Gamma,0} \equiv 1$ on $[-e^a, e^a]$, and $\chi_{\Gamma,0} + \chi_{\Gamma,\infty} \equiv 1$.

**Definition 8.15.** Let $\Gamma$ be a polygonal curve of combinatorial length $m \le 2r$. We call $\Gamma$-favourable region the region of Teichmüller space

$$ \{Y \in \mathcal{T}_{g_S, n_S}^* : \forall j \in \mathbb{Z}_m, Z_{K_j}^{\Gamma} > e^a\}. $$

**Remark 8.16.** In particular, using Corollary 8.13 and the fact that $e^{a/2} \ge 8 \cosh 3/2$, we obtain that, for any polygonal curve $\Gamma$, if $Y$ is in the $\Gamma$-favorable region, then $\ell_Y(\Gamma) \ge 3$.

*Proof of Lemma 8.12.* Let $m \ge 2$. For $\alpha \in \{0, 1\}^m$ such that $\alpha \ne (0, \dots, 0)$, let us write $\{j \in \mathbb{Z}_m : \alpha_j = 1\}$ as a disjoint union of intervals,

$$ \{j \in \mathbb{Z}_m : \alpha_j = 1\} = \bigsqcup_{i=1}^l \{k_i, \dots, l_i\}, $$

so that $\alpha_j = 1$ for $k_i \le j \le l_i$ but $\alpha_{k_i-1} = 0 = \alpha_{l_i+1}$. Then, by definition, $(\delta\alpha)_j = -1$ when $j = k_i$ or $l_i + 1$ and $+1$ otherwise. It follows that

$$ \frac{\operatorname{hyp}_{-\partial\alpha}\left(\frac{\vec{L}}{2}\right)}{\operatorname{hyp}_{-1}\left(\frac{\vec{L}}{2}\right)} e^{-\alpha \cdot \tau} = \prod_{i=1}^{l} \coth\left(\frac{L_{k_i}}{2}\right) \coth\left(\frac{L_{l_i+1}}{2}\right) \prod_{k_i \le j \le l_i} e^{-\tau_j}. $$

We note that, for all $x \ge 0$, we have $\coth(x/2) \le 2\coth(x)$. Therefore,

$$ \frac{\operatorname{hyp}_{-\partial\alpha}\left(\frac{\vec{L}}{2}\right)}{\operatorname{hyp}_{-1}\left(\frac{\vec{L}}{2}\right)} e^{-\alpha \cdot \tau} \le \prod_{i=1}^{l} 4 \coth(L_{k_i}) \coth(L_{l_i+1}) \prod_{k_i \le j \le l_i} e^{-\tau_j}. $$

We then inject $1 \le \prod_{k_i < j \le l_i} 4 \coth^2(L_j)$ in this equation, and obtain

$$ \frac{\operatorname{hyp}_{-\partial\alpha}\left(\frac{\vec{L}}{2}\right)}{\operatorname{hyp}_{-1}\left(\frac{\vec{L}}{2}\right)} e^{-\alpha \cdot \tau} \le \prod_{i=1}^{l} \prod_{k_i \le j \le l_i} 4 \coth(L_j) \coth(L_{j+1}) e^{-\tau_j} = \prod_{j \in \alpha} \frac{4}{Z_j} $$

which is our claim. $\square$

*Proof of Corollary 8.13.* First we observe that, by hypothesis, for any $j$, $4/Z_j \le 4e^{-a} \le 2^{-m-1}$ which is smaller than 1. As a consequence, for any $\alpha \ne (0, \dots, 0)$, we have $\prod_{j \in \alpha} 4/Z_j \le 2^{-m-1}$. To then deduce the bound (8.9), we write the triangle inequality:

$$ F_m(\vec{L}, \vec{\tau}) \ge \frac{1}{2} \operatorname{hyp}_{-1}\left(\frac{\vec{L}}{2}\right) \left[1 - \sum_{\substack{\alpha \in \{0,1\}^m \\ \alpha \ne (0, \dots, 0)}} \frac{\operatorname{hyp}_{-\partial\alpha}\left(\frac{\vec{L}}{2}\right)}{\operatorname{hyp}_{-1}\left(\frac{\vec{L}}{2}\right)} e^{-\alpha \cdot \tau}\right] \ge \frac{1}{4} \operatorname{hyp}_{-1}\left(\frac{\vec{L}}{2}\right) $$
because the sum above is bounded above by $\sum_{\alpha \in \{0,1\}^m} 2^{-m-1} = 1/2$. There only remains to prove the last claim, which can be obtained by writing

$$ \cosh\left(\frac{\ell_Y(\Gamma)}{2}\right) = e^{\sum_{j=1}^m \tau_j/2} F_m(\vec{L}, \vec{\tau}) \ge \frac{1}{4} e^{\sum_{j=1}^m \tau_j/2} \prod_{j=1}^m \sinh\left(\frac{L_j}{2}\right) $$

thanks to the bound on $F_m$. Then, since $\sinh(x/2) \ge \tanh(x)/2$ for all $x \ge 0$,

$$ \cosh\left(\frac{\ell_Y(\Gamma)}{2}\right) \ge \frac{1}{2^{m+2}} e^{\sum_{j=1}^m \tau_j/2} \prod_{j=1}^m \tanh(L_j) = \frac{1}{2^{m+2}} \prod_{j=1}^m Z_j^{1/2} \ge \frac{e^{ma/2}}{2^{m+2}}. $$

□

8.4.3. *Derivatives of $F_m$.* Let us now estimate the logarithmic derivatives of the function $F_m$ within the favorable region.

**Proposition 8.17.** Let $\Gamma = (J_j, K_j)_{j \in \mathbb{Z}_m}$ be a polygonal curve of combinatorial length $m \ge 1$. If the metric $Y$ is in the $\Gamma$-favourable region, then for any subset $\pi \subseteq \mathbb{Z}_m$,

$$ \left| \prod_{j \in \pi} \frac{\partial}{\partial \tau_j} (\log F_m)(\vec{L}, \vec{\tau}) \right| \le \mathrm{fn}(m) \prod_{j \in \pi} \frac{1}{Z_j}. $$

*Proof.* Let $k = \#\pi$. The partial derivative $\partial^\pi \log F_m$ is a linear combination (with universal coefficients depending only on $k$) of fractions of the form $\prod_{j=1}^k (\partial^{\pi_j} F_m/F_m)$ where the non-empty subsets $\pi_1, \dots, \pi_k$ of $\mathbb{Z}_m$ form a partition of $\pi$. We then write for each $j \in \{1, \dots, k\}$

$$ \frac{\partial^{\pi_j} F_m}{F_m} = \sum_{\alpha \in \{0,1\}^m} (-1)^{|\alpha|} \frac{\mathrm{hyp}_{-\partial\alpha}\left(\frac{\vec{L}}{2}\right) \partial^{\pi_j}(e^{-\alpha \cdot \vec{\tau}})}{F_m}. $$

To bound the denominator, we use (8.9), which states that $F_m(\vec{L}, \vec{\tau}) \ge \frac{1}{4} \mathrm{hyp}_{-1}(\vec{L}/2)$. For the numerator, we note that $\partial^{\pi_j}(e^{-\alpha \cdot \vec{\tau}}) = 0$ unless $\alpha_i = 1$ for all $i \in \pi_j$, and in the latter case we have $\partial^{\pi_j}(e^{-\alpha \cdot \vec{\tau}}) = (-1)^{\#\pi_j} e^{-\alpha \cdot \vec{\tau}}$. In particular, the term $\alpha = (0, \dots, 0)$ does not contribute to the sum. Then,

$$ \left| \frac{\partial^{\pi_j} F_m}{F_m} \right| \le 4 \sum_{\substack{\alpha \in \{0,1\}^m \\ \alpha \ne (0, \dots, 0)}} \frac{\mathrm{hyp}_{-\partial\alpha}\left(\frac{\vec{L}}{2}\right) e^{-\alpha \cdot \vec{\tau}}}{\mathrm{hyp}_{-1}\left(\frac{\vec{L}}{2}\right)} \le 2^{3m+2} \prod_{i \in \pi_j} \frac{1}{Z_i} $$

by Lemma 8.12, using the fact that there are at most $2^m$ terms in the sum. Taking the product over $j = 1, \dots, k$ yields

$$ \prod_{j=1}^{k} \left| \frac{\partial^{\pi_j} F_m}{F_m} \right| \le 2^{(3m+2)k} \prod_{i \in \pi} \frac{1}{Z_i}. $$

Summing over all partitions $\pi_1, \dots, \pi_k$ of $\pi$, and noticing that $k \le m$, yields the announced bound. □
8.4.4. *Decoration for one polygonal curve.* We recall that our aim in this section is to provide a well-described formula for the length of a polygonal curve $\Gamma$. Proposition 8.17 tells us that this is achieved as soon as our metric $Y$ lies in the $\Gamma$-favorable region. If this is not the case, then we shall replace $\Gamma$ by a derived polygonal curve $\Gamma'$ obtained by neutralizing a cell $K$ such that $Z_K^\Gamma \le e^a$, and hope that $Y$ now falls in the $\Gamma'$-favourable region. Since the combinatorial length strictly decreases under derivation, we can iterate this process finitely many times until we reach a favourable region for a polygonal curve $\Gamma''$ derived from $\Gamma$. Since $\Gamma''$ is homotopic to $\Gamma$, this yields a well-behaved formula for the length $\ell_Y(\Gamma)$ in terms of the lengths of the cells and bridges of $\Gamma''$.

The *decorations*, defined below, will serve to enumerate all possible cases in this recursive process. Later on (§8.6), decorations will be used to construct a partition of unity on the Teichmüller space: a decoration $\infty$ on a cell $K$ will mean that we consider the region of Teichmüller space where the height $Z_K^\Gamma$ is high, and on the opposite the decoration 0 will mean that we consider the region where the height $Z_K^\Gamma$ is low.

Let us provide the general definition of a decoration for a polygonal curve $\Gamma$.

**Definition 8.18.** A decorated polygonal curve $(\Gamma, d)$ is a polygonal curve $\Gamma$ of combinatorial length $|\Gamma| \ge 1$, together with a map $d : \text{Cell}(\Gamma) \to \{0, \infty\}$. A cell $K \in \text{Cell}(\Gamma)$ is called *favourable* if $d(K) = \infty$ and *unfavourable* if $d(K) = 0$. A decorated polygonal curve $(\Gamma, d)$ is called *favourable* if all cells are favourable.

A decorated polygonal curve $(\Gamma', d')$ is said to be *derived from* $(\Gamma, d)$ if $\Gamma'$ is derived from $\Gamma$ by neutralizing a cell $K \in \text{Cell}(\Gamma)$ such that $d(K) = 0$. In addition, we ask that the map $d'$ coincides with $d$ on $\text{Cell}(\Gamma') \cap \text{Cell}(\Gamma)$.

The following definition allows us to consider the iterative derivation process described above.

**Definition 8.19.** Let $\Gamma$ be a polygonal curve. An *exhaustion* for $\Gamma$ is a sequence of decorated polygonal curves $(\Gamma, \mathbf{d}) = (\Gamma^k, d^k)_{1 \le k \le t}$, such that

* $\Gamma^1 = \Gamma$;

* for every $k < t$, $(\Gamma^k, d^k)$ is not favourable, and $(\Gamma^{k+1}, d^{k+1})$ is derived from $(\Gamma^k, d^k)$;

* for $k = t$, $(\Gamma^t, d^t)$ is either favourable or of combinatorial length = 0.

**Remark 8.20.** In particular, any polygonal curve of combinatorial length 0 is left untouched by the exhaustion process. It follows that, if we apply this process to a polygonal curve $\beta_\lambda$ with $\lambda \in \Lambda^\beta$ (i.e. a component of $\beta$), then nothing happens.

**Notation 8.21.** For an exhaustion $(\Gamma, \mathbf{d}) = (\Gamma^k, d^k)_{1 \le k \le t}$ of a polygonal curve $\Gamma$, we denote as $(\Gamma^\infty, d^\infty) = (\Gamma^t, d^t)$ the decorated polygonal curve at the terminal step of the exhaustion.

**8.5. Families of polygonal curves and simultanous decorations.** The integral (5.13), our main focus, simultaneously involves all the lengths $(\ell_Y(\Gamma_\lambda))_{\lambda \in \Lambda}$ through the function $1_\mathcal{D}$. We will therefore need to consider simultaneously all the polygonal curves $(\Gamma_\lambda)_{\lambda \in \Lambda}$. This motivates the need to study families of polygonal curves, which we do in this section.

In the following, our families are indexed by our set $\Lambda$ introduced in §3.3 to index our pair of pants decomposition, but the precise structure of $\Lambda$ does not matter. The
one thing that is important is that we have ordered the set $\Lambda$ with the order relation $\lambda_1 < \lambda_2$ if $\text{step}(\lambda_1) < \text{step}(\lambda_2)$, where step($\lambda$) is the step of the construction of the pair of pants decomposition at which $\Gamma_\lambda$ appears. The considerations below can be applied to any partially ordered set $\Lambda$.

**8.5.1. Compatibility of polygonal curves.** When considering families of polygonal curves, we will need to understand how the different polygonal curves in our family intersect one another. Without any hypothesis, two polygonal curves can have quite wild intersections and combinatorics. The following notion of compatibility will be key to restrict the possible behaviours and describe our families of polygonal curves.

**Definition 8.22 (Compatibility of two polygonal curves).** Let $\Gamma_1, \Gamma_2$ be two polygonal curves based on $\beta$. We say that $\Gamma_1$ and $\Gamma_2$ are compatible if:

* if two cells $K^1 \in \text{Cell}(\Gamma_1)$, $K^2 \in \text{Cell}(\Gamma_2)$ lie in the same component of $\beta$ and have the same orientation, then they are either disjoint or one is included in the other;

* two bridges $J^1 \in \text{Br}(\Gamma_1)$, $J^2 \in \text{Br}(\Gamma_2)$ are either equal or disjoint (in particular, they cannot have an endpoint in common unless they are equal).

As a consequence of our definition of polygonal curves, for any cells $K^1, K^2$ within one component $\beta$ and oriented the same way, the bridges adjacent to $K^1$ and $K^2$ all lie on the same side of that component of $\beta$. The compatibility condition then implies that any two compatible polygonal curves $\Gamma_1$ and $\Gamma_2$ are homotopic to simple curves not intersecting one another.

**Definition 8.23 (Inclusion of cells).** Let $\Gamma_1, \Gamma_2$ be two compatible polygonal curves. Let $K^1 \in \text{Cell}(\Gamma_1)$ and $K^2 \in \text{Cell}(\Gamma_2)$ be two cells of $\Gamma_1, \Gamma_2$ respectively. We say that the cell $K^1$ is *included* in the cell $K^2$, and write $K^1 \subseteq K^2$, if $K^1$ is a subset of $K^2$ and their orientations are the same. We will say that the two cells *coincide*, and write $K^1 = K^2$, if $K^1 \subseteq K^2$ and $K^2 \subseteq K^1$. In this case, we say that $\Gamma_1$ and $\Gamma_2$ have a cell in common. We say that $K^1$ is strictly included in $K^2$, and write $K^1 \subsetneq K^2$, if $K^1 \subseteq K^2$ and $K_1 \neq K_2$.

Because of the second item in Definition 8.22, if $K^1 = K^2$, then the bridges adjacent to this common cell are equal.

**Remark 8.24.** The reason we insist on only comparing cells with the same orientation is that we want the notion of inclusion to be purely topological, that is, independent on the choice of a hyperbolic metric. If $K^1 \subseteq K^2$ for topological polygonal curves $\Gamma_1, \Gamma_2$, then, by the Useful Remark 2.11, we have that the geodesic representatives $\bar{K}^1$ and $\bar{K}^2$ satisfy $\bar{K}^1 \subseteq \bar{K}^2$ in the sense of the inclusion of sets.

**Definition 8.25.** We say a family of polygonal curves $(\Gamma_\lambda)_{\lambda \in \Lambda}$ form a compatible family if

* these polygonal curves are pairwise compatible;

* for $K^1 \in \text{Cell}(\Gamma_{\lambda_1})$ and $K^2 \in \text{Cell}(\Gamma_{\lambda_2})$, if $K^1 \subsetneq K^2$, then $\lambda_2 < \lambda_1$.

**Example 8.26.** By construction, the family $(\Gamma_\lambda)_{\lambda \in \Lambda}$ of boundary components of our pair of pants decomposition is a compatible family of polygonal curves.
**Notation 8.27.** For $\Gamma = (\Gamma_\lambda)_{\lambda \in \Lambda}$ a compatible family of polygonal curves, we let

$$ \mathrm{Cell}(\Gamma) = \bigcup_{\lambda \in \Lambda} \mathrm{Cell}(\Gamma_\lambda), $$

where the union runs over the $\lambda \in \Lambda$ such that $|\Gamma_\lambda| \ge 1$. For $K \in \mathrm{Cell}(\Gamma)$, if $\lambda \in \Lambda$ is such that $K \in \mathrm{Cell}(\Gamma_\lambda)$, then we denote by $\tilde{J}(K)$ the roof over $K$ in $\Gamma_\lambda$ and $Z_K^\Gamma := Z_K^{\Gamma_\lambda}$ the height of $K$. We call *complexity* of $\Gamma$, and denote $|\Gamma|$, its number of cells.

Thanks to the compatibility condition, $\tilde{J}(K)$ and $Z_K^\Gamma$ do not depend on the choice of a $\lambda \in \Lambda$ for which $K \in \mathrm{Cell}(\Gamma_\lambda)$. The following property will allow us to compare heights within a compatible family. It says that the height is (almost) increasing for the inclusion relation.

**Proposition 8.28.** *Consider a compatible family* $\Gamma = (\Gamma_\lambda)_{\lambda \in \Lambda}$. *Let $K, K' \in \mathrm{Cell}(\Gamma)$, such that $K' \subseteq K$. Then*

$$ (8.10) \qquad Z_K^\Gamma \ge e^{-2} Z_{K'}^\Gamma. $$

*Proof.* Let us lift the polygonal curve to the hyperbolic plane. Let $K$ and $K'$ denote two lifts of our cells, such that $K' \subseteq K$. Let $J_-$, $J_+$ denote lifts of the bridges arriving on $K$ and coming out of $K$, and similarly $J'_-$, $J'_+$ for $K'$. By definition of the inclusion of cells, all of those bridges lie on the same side of $K$. We denote as $L_\pm$ and $L'_\pm$ the lengths of these bridges, and $\tau, \tau'$ the lengths of the cells $K, K'$.

Let $t_\pm$ denote the distance between the origins of $J_\pm$ and $J'_\pm$ along $K$. We shall only prove the property when $t_+, t_- > 0$, i.e. the bridges $J_\pm$ and $J'_\pm$ are disjoint – the proof is simpler when one or both of these lengths is equal to 0.

Let $\tilde{\beta}_+, \tilde{\beta}'_+$ denote the lifts of the multicurve $\beta$ on which the bridges $J_+, J'_+$ terminate. Since there is no right-angled quadrilateral in $\mathbb{H}$, these two lifts must differ. Then, completing the picture with the orthogeodesic between $\beta_+$ and $\beta'_+$, we obtain a convex right-angled hexagon, see Figure 31.

<figure><img src="image_40.png" /><figcaption>FIGURE 31. Illustration of the proof of Proposition 8.28.</figcaption></figure>

In a convex right-angled hexagon, the lengths $\ell_1, \ell_2$ of any two consecutive edges must satisfy $\sinh(\ell_1)\sinh(\ell_2) \ge 1$. Indeed, one can cut the hexagon along one of its
orthogeodesics to obtain a right-angled pentagon with two consecutive edges of lengths $\ell_1, \ell_2$, which then must satisfy the aforementioned inequality by [5, Therorem 2.3.4(i)].

It then follows that $\sinh(t_+)\sinh(L_+) \ge 1$. If $L_+ \le 1$, then $\sinh(L_+) \ge \tanh(L_+)/e$ and we hence obtain

$$ (8.11) \qquad e^{t_+} \tanh(L_+) \ge 1/e. $$

Otherwise, $e^{t_+} \ge 1$ and $\tanh(L_+) \ge \tanh(1) \ge 1/e$, so the inequality (8.11) also holds. By symmetry, (8.11) remains true replacing all + by $-$.

Then, we write $\tau = \tau' + t_+ + t_-$, and hence

$$ Z_K^\Gamma = \tanh(L_+) \tanh(L_-) e^{\tau'+t_++t_-} \ge \frac{e^{\tau'}}{e^2} \ge e^{-2} \tanh(L'_+) \tanh(L'_-) e^{\tau'} = e^{-2} Z_{K'}^\Gamma $$

which was our claim. $\square$

**8.5.2. Neutralization in a compatible family.** In §8.2.2, we defined a notion of neutralization of a cell $K$ for a single polygonal curve. Now, we define and discuss the notion of neutralization of a cell $K$ in a compatible family of polygonal curves $\Gamma = (\Gamma_\lambda)_{\lambda \in \Lambda}$.

**Definition 8.29.** Let $\Gamma = (\Gamma_\lambda)_{\lambda \in \Lambda}$ be a family of polygonal curves. Let $K \in \text{Cell}(\Gamma)$.

For every $\lambda \in \Lambda$, we let $\Gamma'_\lambda$ be the polygonal curve derived from $\Gamma_\lambda$ by simultaneously neutralizing all cells $K' \in \text{Cell}(\Gamma_\lambda)$ such that $K' \subseteq K$. We say that the family $\Gamma' = (\Gamma'_\lambda)_{\lambda \in \Lambda}$ is derived from $\Gamma$ by neutralizing $K$.

We show, using the Useful Remark, that the notion of compatibility is stable by derivation.

**Proposition 8.30.** If $\Gamma$ is a compatible family and $\Gamma'$ is derived from $\Gamma$ by neutralizing a cell, then $\Gamma'$ is a compatible family.

*Proof.* Let $\tilde{K}^1, \tilde{K}^2 \in \text{Cell}(\Gamma')$ be two cells with non-empty intersection and oriented the same way. We wish to prove that either $\tilde{K}^1 \subseteq \tilde{K}^2$ or $\tilde{K}^2 \subseteq \tilde{K}^1$, and that the bridges adjacent to these cells are equal or disjoint.

For $i = 1, 2$, let $\lambda_i \in \Lambda$ denote an index for which $\tilde{K}^i \in \Gamma'_{\lambda_i}$. By definition, $\Gamma'_{\lambda_i}$ is derived from $\Gamma_{\lambda_i}$ by neutralizing some cells, and therefore the cell $\tilde{K}^i$ is contained in a cell $K^i$ of $\Gamma_{\lambda_i}$.

By the compatibility condition, two lifts of $\Gamma_{\lambda_1}$ and $\Gamma_{\lambda_2}$ to $\mathbb{H}^2$ cannot intersect transversally. Consider two such lifts, say $(J_j^1, K_j^1)_{j \in \mathbb{Z}}$ and $(J_j^2, K_j^2)_{j \in \mathbb{Z}}$, chosen so that $K_0^1$ and $K_0^2$ are respective lifts of $K^1$ and $K^2$ and $K_0^1 \cap K_0^2 \neq \emptyset$.

Let us consider the index

$$ m = \inf\{j \ge 0 : t(K_j^1) \neq t(K_j^2)\}, $$

i.e. the first place where the two polygonal curves $\Gamma_{\lambda_1}$ and $\Gamma_{\lambda_2}$ start disagreeing after meeting along $K_0^1$ and $K_0^2$ (see Figure 32). First, we observe that if $m = +\infty$, then $\Gamma_{\lambda_1} = \Gamma_{\lambda_2}$ and thus $\Gamma'_{\lambda_1} = \Gamma'_{\lambda_2}$, which is enough to conclude.

If now $m < \infty$, then $J_j^1 = J_j^2$ for all $j \in \{1, \dots, m\}$. Thus, if $m > 0$, then $K_m^1$ and $K_m^2$ have the same origin and, in particular, $K_m^1$ and $K_m^2$ are two distinct cells, with an intersection and oriented the same way. This is obvisouly also true by hypothesis if $m = 0$. Now, because $\Gamma_{\lambda_1}$ and $\Gamma_{\lambda_2}$ are compatible, we either have $K_m^2 \subsetneq K_m^1$ and
$\lambda_1 < \lambda_2$, or $K_m^1 \subsetneq K_m^2$ and $\lambda_2 < \lambda_1$ – let us assume w.l.o.g. that we are in the first situation.

If we now consider the index

$$n = \sup\{j \le 0, o(K_j^1) \ne o(K_j^2)\},$$

since we assumed that $m < +\infty$, $\Gamma_{\lambda_1} \ne \Gamma_{\lambda_2}$, and hence $n > -\infty$. By the same reasoning as above, the cells $K_n^1$ and $K_n^2$ are distinct, with an intersection, and oriented the same way. Since we have now specified that $\lambda_1 < \lambda_2$, we must have $K_n^2 \subsetneq K_n^1$.

<figure><img src="image_41.png" /><figcaption>FIGURE 32. Example of two compatible polygonal curves under a neutralization procedure. The full lines are the initial polygonal curves. The dotted lines are the new cells $\tilde{K}^1$ and $\tilde{K}^2$ that we need to compare, and the dashed lines the new bridges. On this example, we have $m = 0$ and $n = -1$.</figcaption></figure>

Because two lifts of $\Gamma_{\lambda_1}$ and $\Gamma_{\lambda_2}$ cannot cross transversally, our lift of $\Gamma_{\lambda_2}$ then lies entirely on the right of our lift of $\Gamma_{\lambda_1}$, as represented in Figure 32.

Now, for $i = 1, 2$, let $\tilde{J}^i$ denote the bridge coming out of $\tilde{K}^i$ in $\Gamma'_{\lambda_i}$. By definition of the neutralization procedure, the bridge $\tilde{J}^i$ is the orthogeodesic between the cell $K_0^i$ and a cell $K_{j_i}^i$ for an index $j_i \ge 0$, and the cells $K_j^i$ for $0 < j < j_i$ are neutralized. There are now two cases to consider.

* If $j_2 \ge m$, we can use the Useful Remark 2.11 to deduce that the origin of $\tilde{J}^2$ lies on the right of the origin of $\tilde{J}^1$ on $K_0^1$, and the bridges $\tilde{J}^1$ and $\tilde{J}^2$ are equal or do not cross, which is exactly our claim.

* Otherwise, $j_2 < m$, or, in other words, the two polygonal curves $\Gamma_{\lambda_1}$ and $\Gamma_{\lambda_2}$ coincide up until the index $j_2$. By definition of the neutralization for a family of curve, if a cell $K_j^2$ of $\Gamma_{\lambda_2}$ is contained in a cell $K_j^1$ of $\Gamma_{\lambda_1}$ and if $K_j^1$ is neutralized, then $K_j^2$ must also be neutralized. Therefore, $j_1 \le j_2$, and we can once again use the Useful Remark to conclude.
The same reasoning also allows to prove that the origin $o(\tilde{K}^2)$ lies on the left of $o(\tilde{K}^1)$, and the bridges arriving on $\tilde{K}^1$ and $\tilde{K}^2$ are equal or disjoint. $\square$

**8.5.3. Simultaneous decoration.** We are now ready to extend the notion of decoration introduced in §8.4.4 to families of polygonal curves.

**Definition 8.31.** Let $\Gamma = (\Gamma_\lambda)_{\lambda \in \Lambda}$ be a compatible family of polygonal curves. A (simultaneous) decoration of $\Gamma$ is a map $d : \text{Cell}(\Gamma) \to \{0, \infty\}$ such that any cell common to several $\Gamma_\lambda$ bears only one decoration.

**Definition 8.32.** Let $(\Gamma, d)$ be a decorated family. A cell $K \in \text{Cell}(\Gamma)$ is called unfavourable (for the decoration $d$) if $d(K) = 0$ or if $K \subseteq K'$ with $d(K') = 0$. It is called favourable otherwise. A maximal unfavourable cell is a cell which is unfavourable and maximal for inclusion.

A simultaneous decoration $d$ of the family $\Gamma$ is called favourable if all its cells are favourable.

By convention, recall that polygonal curves of combinatorial length 0 have no cells. As a consequence, if $\Gamma = (\Gamma_\lambda)_{\lambda \in \Lambda}$ contains only polygonal curves such that $|\Gamma_\lambda| = 0$, then $\text{Cell}(\Gamma) = \emptyset$ and in particular $\Gamma$ is favorable.

**Definition 8.33.** A decorated family $(\Gamma', d')$ is said to be derived from $(\Gamma, d)$ if $\Gamma'$ is derived from $\Gamma$ by neutralizing a maximal unfavourable cell and the map $d'$ coincides with $d$ on all cells in common between $\Gamma$ and $\Gamma'$.

**Definition 8.34.** Let $\Gamma = (\Gamma_\lambda)_{\lambda \in \Lambda}$ be a compatible family of polygonal curves. An exhaustion of $\Gamma$ is a sequence of decorated families $(\Gamma, d) = (\Gamma^k, d^k)_{1 \le k \le t}$, where:

* $\Gamma^1 = \Gamma$;

* for every $1 \le k \le t$, $\Gamma^k = (\Gamma_\lambda^k)_{\lambda \in \Lambda}$ is a compatible family of polygonal curves;

* for every $1 \le k < t$, $(\Gamma^k, d^k)$ is not favourable, and $(\Gamma^{k+1}, d^{k+1})$ is derived from $(\Gamma^k, d^k)$;

* for $k = t$, $(\Gamma^t, d^t)$ is favourable.

We let $\text{Cell}(\Gamma) = \bigcup_{k=1}^t \text{Cell}(\Gamma^k)$. We denote as $|\Gamma| = \#\text{Cell}(\Gamma)$ the complexity of the exhaustion $\Gamma$. If $K \in \text{Cell}(\Gamma)$, we denote $d(K) = d^k(K)$ for any $k$ such that $K \in \text{Cell}(\Gamma^k)$.

Note that, by construction, the value of the decoration $d(K) = d^k(K)$ is independent from the choice of an index $k$ such that $K \in \text{Cell}(\Gamma^k)$. This is also true of the roof over a cell $\tilde{J}(K)$ and height of a cell $Z_K^\Gamma$ introduced in Notation 8.27.

**Notation 8.35 (Terminal cells).** Let $(\Gamma, d) = (\Gamma^k, d^k)_{1 \le k \le t}$ be a sequence of decorated families of polygonal curves. We denote as $(\Gamma^\infty, d^\infty)$ the decorated polygonal curve $(\Gamma^t, d^t)$. The cells of the last family $\Gamma^\infty$ are called the terminal cells of $(\Gamma, d)$. The set of terminal cells associated with $(\Gamma, d)$ is denoted $\text{Term}(\Gamma, d)$, in other words

$\text{Term}(\Gamma, d) := \text{Cell}(\Gamma^\infty)$.
**8.6. Another partition of unity on Teichmüller space.** Let us now focus on our compatible family of polygonal curves $\Gamma = (\Gamma_\lambda)_{\lambda \in \Lambda}$ defined as the boundary components of the pair of pants decomposition introduced in §3.2.

We recall that we have fixed a cutoff value $a = a(r)$ and associated cutoff functions $\chi_{\Gamma,0}, \chi_{\Gamma,\infty}$ such that $\chi_{\Gamma,0} + \chi_{\Gamma,\infty} = 1$ in Notation 8.14.

**Notation 8.36.** Let $(\Gamma, d) = (\Gamma^k, d^k)_{1\le k\le t}$ be an exhaustion of the family $(\Gamma_\lambda)_{\lambda\in\Lambda}$. For a hyperbolic metric $Y \in \mathcal{T}_{g_S,n_S}^*$, we let

$$ (8.12) \qquad \Psi_{\Gamma,d}(Y) := \prod_{K \in \text{Cell}(\Gamma)} \chi_{\Gamma,d(K)}(Z_K^{\Gamma}). $$

The function $\Psi_{\Gamma,d}$ is a cut-off function on the Teichmüller space $\mathcal{T}_{g_S,n_S}^*$. It achieves our purpose described through this section. Indeed, if $Y \in \mathcal{T}_{g_S,n_S}^*$ lies in the support of $\Psi_{\Gamma,d}$, then for any cell $K \in \text{Cell}(\Gamma)$,

* if $d(K) = 0$, then $Z_K^\Gamma \le e^{a+1}$;

* if $d(K) = \infty$, then $Z_K^\Gamma \ge e^a$.

We can extract from the family of functions $\Psi_{\Gamma,d}$ with $(\Gamma, d)$ an exhaustion a partition of unity on Teichmüller space :

**Proposition 8.37.** There exists a family of exhaustions $\mathfrak{C}$ of the family $(\Gamma_\lambda)_{\lambda \in \Lambda}$ such that

$$ (8.13) \qquad \forall Y \in \mathcal{T}_{g_S,n_S}^*, \quad \sum_{(\Gamma,d) \in \mathfrak{C}} \Psi_{\Gamma,d}(Y) = 1. $$

We omit the proof, a tedious but surpriseless recursive procedure on the number of cells of the family of polygonal curves $(\Gamma_\lambda)_{\lambda \in \Lambda}$.

**8.7. Neutral variables associated with $(\Gamma, d)$.** Let us fix an exhaustion $(\Gamma, d) = (\Gamma^k, d^k)_{1\le k\le t}$ of the family $(\Gamma_\lambda)_{\lambda \in \Lambda}$ of boundary components of our pair of pants decomposition of $S$. On the support of $\Psi_{\Gamma,d}$, we shall enlarge the set of neutral variables originally defined in §5.3, and hence diminish the number of variables with respect to which we need to apply the operator $\mathcal{L}$ under the integral (5.13).

Amongst the terminal cells of $(\Gamma, d)$, some are particularly important.

**Notation 8.38.** We denote as $\text{Term}_{\text{BC}}(\Gamma, d) \subseteq \text{Term}(\Gamma, d)$ the set of cells of $(\Gamma_\lambda^\infty)_{\lambda \in \Lambda_{\text{BC}}^\Gamma}$. We call these cells *boundary terminal cells*.

We observe that, by definition of the neutralization procedure, any $K \in \text{Term}_{\text{BC}}(\Gamma, d)$ is included in a cell of the initial polygonal curve $(\Gamma_\lambda^1)_{\lambda \in \Lambda_{\text{BC}}^\Gamma}$. The cells of this family are amongst the segments $(K_q)_{q \in \Theta}$ defined in §4.3. We also note that each segment $(K_q)_{q \in \Theta}$ can contain at most one boundary terminal cell.

We now recall that, in §4.3, we have defined for each $q \in \Theta$ a set $\Theta_\tau(q) \subseteq \Theta$ such that we can write the length $\tau_q$ of $K_q$ as $\tau_q = \sum_{q' \in \Theta_\tau(q)} \theta_{q'}$.

**Definition 8.39.** Let $q \in \Theta$. We call $K_q$ a *surviving boundary segment* if $K_q$ contains a boundary terminal cell. In that case, we let

$$ (8.14) \qquad Z_{\Gamma,d}(K_q) := Z_K^\Gamma $$
where $K$ is the unique boundary terminal cell contained in $K_q$. We denote as $\Theta_{\text{surv}}$ the set of indices $q \in \Theta$ such that $K_q$ is a surviving boundary segment and $\vec{\tau}_{\text{surv}} = (\tau_q)_{q \in \Theta_{\text{surv}}}$.

**Notation 8.40.** We define additional *neutral variables* associated to the exhaustion $(\Gamma, \mathbf{d})$,

$$ (8.15) \qquad \Theta_{\text{nt}}(\Gamma, \mathbf{d}) := \bigcup_{q \in \Theta \setminus \Theta_{\text{surv}}} \Theta_{\tau}(q). $$

For a vector $\vec{\theta} \in \mathbb{R}^{\Theta}$, we denote as $\vec{\theta}_{\text{nt}}(\Gamma, \mathbf{d}) = (\theta_q)_{q \in \Theta_{\text{nt}}(\Gamma, \mathbf{d})} \in \mathbb{R}^{\Theta_{\text{nt}}(\Gamma, \mathbf{d})}$ the restriction of the vector $\vec{\theta}$ to the indices in $\Theta_{\text{nt}}(\Gamma, \mathbf{d})$.

**Remark 8.41.** If a boundary component $\lambda \in \Lambda_{\text{BC}}$ is reduced in the exhaustion procedure to a component of combinatorial length 0, i.e. if $|\Gamma_{\lambda}^{\infty}| = 0$, then all variables $\theta_q$ with $\lambda(q) = \lambda$ are neutral variables.

Lemma 8.6 allows us to draw the following observations as to the variables appearing in a terminal polygonal curve.

**Lemma 8.42.** Let $(\Gamma, \mathbf{d})$ be an exhaustion of $(\Gamma_{\lambda})_{\lambda \in \Lambda}$. Then,

* for any $q \in \Theta_{\text{surv}}$, if $K$ is the unique boundary terminal cell contained in $K_q$,

$$ (8.16) \qquad \tau_q = \ell(K) + \text{fn}(\vec{L}, \vec{\theta}_{\text{nt}}(\Gamma, \mathbf{d})); $$

* any bridge $J \in \text{Br}(\Gamma^{\infty})$ of the terminal family of polygonal curves $\Gamma^{\infty}$ satisfies

$$ (8.17) \qquad \ell(J) = \text{fn}(\vec{L}, \vec{\theta}_{\text{nt}}(\Gamma, \mathbf{d})). $$

**8.8. Structure and derivatives of the function $\Psi_{\Gamma,d}(Y)$**. Later on, we will need to be able to estimate the derivatives of the function $\Psi_{\Gamma,d}(Y)$ with respect to certain variables. We prove the following.

**Proposition 8.43.** We can rewrite $\Psi_{\Gamma,d}(Y)$ under the form

$$ \Psi_{\Gamma,d}(Y) = \tilde{\Psi}_{\Gamma,d}(\vec{\tau}, \vec{L}, \vec{\theta}_{\text{nt}}(\Gamma, \mathbf{d})) $$

for a smooth function $\tilde{\Psi}_{\Gamma,d} : \mathbb{R}_{>0}^{\Theta} \times \mathbb{R}_{>0}^{r} \times \mathbb{R}^{\Theta_{\text{nt}}(\Gamma, \mathbf{d})} \to \mathbb{R}$ satisfying, for any set $\tilde{\pi} \subseteq \Theta$,

$$ \left| \frac{\partial^{|\tilde{\pi}|} \tilde{\Psi}_{\Gamma, \mathbf{d}}}{\prod_{q \in \tilde{\pi}} \partial \tau_q} \right| \le \text{fn}(|\Gamma|) \left( \max_{0 \le k \le |\tilde{\pi}|} \sup_{x \ge 0} \left( |\bar{\chi}_{\Gamma, 0}^{(k)}(x)| + |\bar{\chi}_{\Gamma, \infty}^{(k)}(x)| \right) \right)^{|\Gamma|} $$

where $\bar{\chi}_{\Gamma,d}(x) := \chi_{\Gamma,d}(e^x)$ for $d \in \{0, \infty\}$.

Recall that the quantity $|\Gamma|$ is the complexity of $\Gamma$, i.e. its number of cells, which is bounded above by $2r\#\Lambda \le 6r^2$.

**Proof.** Let us first make a useful observation. For any cell $K \in \text{Cell}(\Gamma)$, there exists $K^1 \in \text{Cell}(\Gamma^1)$ containing $K$. As a consequence, $K = K^1 \setminus I_K$ for a union of intervals $I_K$, corresponding to some intervals that were removed in the neutralization procedure. Recalling that the cells of $\Gamma^1$ are amongst the collection of segments $(K_{qq'})_{q,q' \in \Theta}$ introduced in §4.3, we obtain that there exists a set of indices $\Theta_{\Gamma}(K) \subseteq \Theta$ such that

$$ (8.18) \qquad \ell_Y(K) = \sum_{q \in \Theta_{\Gamma}(K)} \tau_q - \ell_Y(I_K). $$
We now write

$$ (8.19) \qquad \Psi_{\Gamma,d}(Y) = \prod_{K \in \text{Cell}(\Gamma)} \bar{\chi}_{\Gamma,d(K)}(\ell_Y(K) + T_K) $$

where for $K \in \text{Cell}(\Gamma)$,

* $\ell_Y(K)$ is the length of the cell $K$;

* if $J_{\pm}(K)$ denote the two bridges adjacent to $K$, then

$$ T_K = \log \tanh(\ell_Y(J_-(K))) + \log \tanh(\ell_Y(J_+(K))). $$

By (8.18) this is equal to

$$ (8.20) \qquad \Psi_{\Gamma,d}(Y) = \prod_{K \in \text{Cell}(\Gamma)} \bar{\chi}_{\Gamma,d(K)} \left( \sum_{q \in \Theta_{\Gamma}(K)} \tau_q - \ell_Y(I_K) + T_K \right). $$

We prove, by the same method as Lemma 8.42, that for all $K \in \text{Cell}(\Gamma)$,

$$ T_K - \ell_Y(I_K) = \text{fn}(\vec{L}, \vec{\theta}_{\text{nt}}(\Gamma, d)). $$

Calling $f_K$ this function, define

$$ \tilde{\Psi}_{\Gamma,d}(\vec{\tau}, \vec{L}, \vec{\theta}_{\text{nt}}(\Gamma, d)) := \prod_{K \in \text{Cell}(\Gamma)} \bar{\chi}_{\Gamma,d} \left( \sum_{q \in \Theta_{\Gamma}(K)} \tau_q + f_K(\vec{L}, \vec{\theta}_{\text{nt}}(\Gamma, d)) \right). $$

The upper bound on the derivatives is a simple application of the chain rule. $\square$

## 9. PROOF OF THE MAIN THEOREM FOR GENERALIZED EIGHTS

We are now ready to prove the main result, Theorem 1.22, for generalized eights.

### 9.1. Writing as a pseudo-convolution.
We recall that, in §5, we have reduced the problem to the study of the integral (5.13)

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

with the function $h$ equal to the length of $c$ in the coordinates $(\vec{L}, \vec{\theta})$, see (5.16), and the density $\varphi_0$ given in (5.15). Under this form, the integral $\text{Int}(\ell)$ is already written as a pseudo-convolution of Friedman-Ramanujan functions. However, as seen in §6.5, we cannot directly apply Theorem 6.28 (or its variants) to it to conclude to Theorem 1.22.

We shall therefore input the considerations of §7 and 8. More precisely, let us use the partitions of unity $(\Psi_{\xi,Q})_{\xi \in \Xi, Q \subseteq I(\xi)}$ and $(\Psi_{\Gamma,d})_{(\Gamma,d) \in \mathfrak{C}}$ defined in §7.6 and 8.6 to subdivide the integral into

$$ \text{Int}(\ell) = \sum_{(\xi, Q), (\Gamma, d)} \text{Int}_{(\xi, Q), (\Gamma, d)}(\ell), $$

where $\xi$ ranges over $\Xi$, $Q$ over the subsets of $I(\xi)$, $(\Gamma, d)$ over the set of exhaustions $\mathfrak{C}$ constructed in Proposition 8.37, and

$$ \text{Int}_{(\xi, Q), (\Gamma, d)}(\ell) = \int_{h(\vec{\theta}_{\text{ac}})=\ell} \varphi_0(\vec{\theta}_{\text{ac}}) \Psi_{\xi, Q}(\vec{L}, \vec{\theta}) \Psi_{\Gamma, d}(\vec{L}, \vec{\theta}) \prod_{q \in \Theta_{\text{ac}}} \theta_q^{K_q} e^{\theta_q} \frac{d\vec{\theta}_{\text{ac}}}{d\ell}. $$

We now further enlarge the set of neutral parameters.
**Notation 9.1.** We define the full set of neutral parameters as

$$ \Theta'_{\text{nt}} = \Theta_{\text{nt}} \cup \Theta_{\text{nt}}(\xi, \mathcal{Q}) \cup \Theta_{\text{nt}}(\Gamma, \mathbf{d}), $$

where $\Theta_{\text{nt}}(\xi, \mathcal{Q})$ and $\Theta_{\text{nt}}(\Gamma, \mathbf{d})$ are defined in (7.23) and (8.15) respectively. The active variables are $\tilde{\theta}'_{\text{ac}} := (\theta_q)_{q \in \Theta'_{\text{ac}}}$ with $\Theta'_{\text{ac}} = \Theta \setminus \Theta'_{\text{nt}}$. We denote $\tilde{\theta}_{\text{ac}}^{\sim} = (\tilde{\theta}_q)_{q \in \Theta'_{\text{ac}}}$ the set of modified $\theta$ parameters introduced in Notation 7.15.

From now on, all the neutral variables $\tilde{\theta}'_{\text{nt}} := (\theta_q)_{q \in \Theta'_{\text{nt}}}$ will be frozen, and any function which depends only on the neutral variables may be treated as a constant. Our variables will be the $\tilde{\theta}_{\text{ac}}^{\sim}$, which are always $\ge \log(2)$, and hence circumvent the problems raised by crossings. We saw in Lemma 7.20 that $\tilde{\theta}_q$ and $\theta_q$ just differ by a constant (that is, a function of neutral variables).

We shall therefore consider the restriction of the integral $\text{Int}_{(\xi, \mathcal{Q}), (\Gamma, \mathbf{d})}$ defined as

$$ (9.1) \quad \text{Int}'_{(\xi, \mathcal{Q}), (\Gamma, \mathbf{d})}(\ell) = \int_{h(\tilde{\theta}_{\text{ac}}^{\sim}) = \ell} \varphi_0(\tilde{\theta}_{\text{ac}}) \Psi_{\xi, \mathcal{Q}}(\tilde{L}, \tilde{\theta}) \Psi_{\Gamma, \mathbf{d}}(\tilde{L}, \tilde{\theta}) \prod_{q \in \Theta'_{\text{ac}}} \theta_q^{K_q} e^{\theta_q} \frac{d\tilde{\theta}_{\text{ac}}^{\sim}}{d\ell}. $$

Note that the only modification is that we have removed the functions $\theta_q^{K_q} e^{\theta_q}$ for $q \in \Theta'_{\text{nt}} \setminus \Theta_{\text{nt}}$ (now constant), and restricted the integral to the variables $\tilde{\theta}_{\text{ac}}^{\sim}$, which are identical to $\tilde{\theta}'_{\text{ac}}$, up to an additive function of the neutral parameters.

We can now see the integral $\text{Int}'_{(\xi, \mathcal{Q}), (\Gamma, \mathbf{d})}(\ell)$ as the $(h, \varphi)$-convolutions of the functions $\theta_q^{K_q} e^{\theta_q}$, indexed by $q \in \Theta'_{\text{ac}}$ of cardinal $n_{\text{ac}} = \#\Theta'_{\text{ac}}$, where

* the function $h$ is now given by (7.20), which reads

$$ h(\tilde{\theta}_{\text{ac}}^{\sim}) = \ell_Y(\mathbf{c}) = \sum_{i=1}^{n} M_{n'_i}((\epsilon \tilde{L}_q)_{q \in I(\xi^i)}, (\tilde{\theta}_q)_{q \in I(\xi^i)}) $$

where $n'_i := \#I(\xi^i)$ is the number of non-erased $\theta$ parameters in the $i$-th component $\mathbf{c}_i$, and $\tilde{L}_q, \tilde{\theta}_q$ are the modified parameters introduced in §7.4 after erasing;

* the function $\varphi$ given by

$$ (9.2) \quad \varphi(\tilde{\theta}_{\text{ac}}^{\sim}) = \Psi_{\xi, \mathcal{Q}}(\tilde{L}, \tilde{\theta}) \Psi_{\Gamma, \mathbf{d}}(\tilde{L}, \tilde{\theta}) \mathbb{1}_{\mathcal{D}}(\tilde{L}, \tilde{\theta}) \prod_{\lambda \in V_+^\beta} \chi_z(y_\lambda)^2 \prod_{\lambda \in V^\Gamma \setminus W^\Gamma} \left( \log F_{m_\lambda}(\tilde{L}^\lambda, \tilde{\tau}^\lambda) \right)^{d_\lambda}. $$

In the next two sections, we show that the functions $h$ and $\varphi$ have the structure described in the second variant of Theorem 6.28, presented in §6.6.2.

## 9.2. Properties of the function h.
By definition, on the support of the function $\Psi_{\xi, \mathcal{Q}}$, we have $\tilde{\theta}_q \ge \log(2)$ for every $q \in \Theta'_{\text{ac}}$. We can therefore apply Proposition 6.24, Corollary 6.25 and Remark 6.26 and prove that the function $h$, seen as a function of the variables $\tilde{\theta}_{\text{ac}}^{\sim}$, belongs in the class $\mathcal{E}_{n_{\text{ac}}}^{(\log 2)}$ introduced in Definition 6.22. We can then deduce that the function $h$ is of class $\mathcal{C}^\infty$ on the support of $\Psi_{\xi, \mathcal{Q}}$, and that on this set,

* for every $q \in \Theta'_{\text{ac}}$,

$$ |\partial_q h(\tilde{\theta}_{\text{ac}}^{\sim}) - 1| \le C e^{-\tilde{\theta}_q}; $$
• for every $\pi \subseteq \Theta'_{ac}$ such that $|\pi| \ge 2$,

$$|\partial^\pi h(\vec{\theta}_{ac})| \le C e^{-\sum_{q \in \pi} \tilde{\theta}_q},$$

where $C = C(\log 2, n)$ is the constant from Proposition 6.24. As a consequence, the function $h$ satisfies the hypothesis (h) defined in Notation 6.29 with respect to the variables $\vec{\theta}_{ac}$, with the constant $C$.

**9.3. Derivatives of the function $\varphi$.** We shall rewrite the function $\varphi$ under the special form required in (6.30), i.e. split it into a product $\varphi = \phi \times \psi$ where

$$\psi(\vec{\theta}_{ac}) = \Psi_{\xi, \mathcal{Q}}(\vec{L}, \vec{\theta}) \mathbb{1}_{\mathcal{D}}(\vec{L}, \vec{\theta}) \prod_{\lambda \in V_+^\beta} \chi_z(y_\lambda)^2$$

and

$$\phi(\vec{\tau}_{\text{surv}}) = \Psi_{\Gamma, \mathbf{d}}(\vec{L}, \vec{\theta}) \prod_{\lambda \in V^\Gamma \setminus W^\Gamma} \left( \log F_{m_\lambda}(\vec{L}^\lambda, \vec{\tau}^\lambda) \right)^{d_\lambda}$$

where we recall that $\vec{\tau}_{\text{surv}}$ is defined in Definition 8.39. To see that $\phi$ may be expressed as a function of the parameters $\vec{\tau}_{\text{surv}}$ (once we have set the neutral parameters as constants), we refer to Lemma 8.42. We prove the following.

**Proposition 9.2.** The functions $\psi$ and $\phi$ are $C^\infty$ on the support of $\Psi_{\Gamma, \mathbf{d}}$. Furthermore,

• for any $\pi \subseteq \Theta'_{ac}$, $|\partial^\pi \psi| \le \text{fn}(r)e^{-\sum_{q \in \pi} \tilde{\theta}_q}$;

• for any $\tilde{\pi} \subseteq \Theta_{\text{surv}}$, if $d := \sum_{\lambda \in V^\Gamma \setminus W^\Gamma} d_\lambda$,

$$\left| \frac{\partial^{|\tilde{\pi}|} \phi}{\prod_{q \in \tilde{\pi}} \partial \tau_q} \right| \le \text{fn}(r, d) \ell_Y(\mathbf{c})^d \prod_{\mathcal{K} \in \tilde{\pi}} \frac{1}{Z_{\Gamma, \mathbf{d}}^{1/2}(\mathcal{K})}.$$

These inequalities play the role of the derivative controls in Assumption (φ-CE-form), see Notation 6.43.

*Proof.* Let us first establish the smoothness of $\psi$ and $\phi$.

We study the discontinuity of the indicator function $\mathbb{1}_{\mathcal{D}}$. By Remark 8.41, if $\lambda \in \Lambda$ satisfies $|\Gamma_\lambda^\infty| = 0$, then $\Theta_\ell(\lambda) \subseteq \Theta'_{nt}$. In other words, if $\lambda \in \Lambda$ satisfies $q \in \Theta_\ell(\lambda)$ for an active parameter $q \in \Theta'_{ac}$, then we must have $|\Gamma_\lambda^\infty| \ge 1$. In that scenario, by definition of the exhaustion, all the cells $K \in \text{Cell}(\Gamma_\lambda^\infty)$ must be favorable. Then, on the support of $\Psi_{\Gamma, \mathbf{d}}$, by definition, we must have $Z_K^\Gamma \ge e^a$ for all cells in $\Gamma_\lambda^\infty$; in other words, $Y$ is in the $\Gamma_\lambda^\infty$-favorable region. By Remark 8.16, we obtain that $\ell_Y(\Gamma_\lambda) \ge 3$. Combined with Proposition 3.21, this yields that $\Psi_{\Gamma, \mathbf{d}}$ vanishes near the discontinuities of the indicator function $\mathbb{1}_{\mathcal{D}}$. In other words, the derivatives of $\mathbb{1}_{\mathcal{D}}$ with respect to $\vec{\theta}_{ac}^\sim$ are identically equal to zero on the support of $\Psi_{\Gamma, \mathbf{d}}$.

Let us now examine the regularity of $\chi_z(y_\lambda)$ for $\lambda \in V_+^\beta$. Let $q \in \Theta'_{ac}$ with $q \in \Theta_\ell(\lambda)$ (otherwise $\chi_z(y_\lambda)$ does not depend on $\theta_q$). Then there also exists $q'$ such that $q \in \Theta_\tau(q')$: otherwise, $\beta_\lambda$ would not be reached by any of the bars $B_1, \dots, B_r$, contradicting the fact that the multi-loop $\mathbf{c}$ was assumed to be connected, and not a simple curve. Necessarily, the corresponding $\mathcal{K}_{q'}$ must be a surviving boundary segment (otherwise, we would have $q \in \Theta_{\text{nt}}(\Gamma, \mathbf{d})$ by definition of the set of neutral variables). This implies that $e^{\tau_{q'}} \ge$
$Z_{\Gamma,d}(K_{q'}) \ge e^a$. Because $\beta_\lambda$ contains the segment $K_{q'}$, we then have $y_j \ge \tau_{q'} \ge a > 1$. But the function $\chi_z$ is assumed to be constant on $[1, +\infty)$. Thus, we obtain that the derivatives of $\chi_z(y_\lambda)$ with respect to $\theta_{ac}^\sim$ are identically equal to zero on the support of $\Psi_{\Gamma,d}$.

The smoothness of the functions $\Psi_{\Gamma,d}$, $\Psi_\xi, \mathcal{Q}$ and $\log(F_{m_\lambda}(\vec{L}^\lambda, \vec{\tau}^\lambda))$ are immediate consequences of their definitions. To estimate the derivatives $\partial^{\tilde{\pi}}\phi$ we use the Leibniz rules. The derivatives of $\phi$ will contain derivatives of $\log F_{m_\lambda}(\vec{L}^\lambda, \tau^\lambda)$ and derivatives of $\Psi_{\Gamma,d}$. We have all tools to estimate both.

By Lemma 8.42, the lengths of terminal cells of $(\Gamma, d)$ coincide (up to additive “constants”) with the components of $\vec{\tau}_{\text{surv}} = (\tau_q)_{q \in \Theta_{\text{surv}}}$. We may therefore reformulate Proposition 8.17 as follows: on the support of the test function $\Psi_{\Gamma,d}$, for every $\lambda \in \Lambda_{\text{BC}}^{\Gamma}$, for every set $\tilde{\pi}$ of surviving boundary segments with $|\tilde{\pi}| \ge 1$, we have

$$ \left| \frac{\partial^{\tilde{\pi}} \log F_{m_\lambda}(\vec{L}^\lambda, \vec{\tau}^\lambda)}{\prod_{q \in \tilde{\pi}} \partial \tau_q} \right| \le \text{fn}(m_\lambda) \prod_{q \in \tilde{\pi}} \frac{1}{Z_{\Gamma,d}(K_q)}. $$

As a consequence,

$$ \left| \frac{\partial^{\tilde{\pi}} \left( \left( \log F_{m_\lambda}(\vec{L}^\lambda, \vec{\tau}^\lambda) \right)^{d_\lambda} \right)}{\prod_{q \in \tilde{\pi}} \partial \tau_q} \right| \le \text{fn}(m_\lambda) d_\lambda^{|\tilde{\pi}|} |\log F_{m_\lambda}(\vec{L}^\lambda, \vec{\tau}^\lambda)|^{d_\lambda - |\tilde{\pi}|} \prod_{q \in \tilde{\pi}} \frac{1}{Z_{\Gamma,d}(K_q)}. $$

Since $m_\lambda$ is always less than $2r$, we can write $\max_{m_\lambda \le 2r} \text{fn}(m_\lambda) = \text{fn}(r)$. By definition of $F_{m_\lambda}$,

$$ |\log F_{m_\lambda}(\vec{L}^\lambda, \vec{\tau}^\lambda)| \le Q_{m_\lambda}(\vec{L}^\lambda, \vec{\tau}^\lambda) = \ell_Y(\Gamma_\lambda) \le \ell_Y(\mathbf{c}) $$

by Proposition 3.22. We then deduce that

$$ (9.3) \qquad \left| \frac{\partial^{\tilde{\pi}} \left( \left( \log F_{m_\lambda}(\vec{L}^\lambda, \vec{\tau}^\lambda) \right)^{d_\lambda} \right)}{\prod_{q \in \tilde{\pi}} \partial \tau_q} \right| \le \text{fn}(r, d_\lambda) \ell_Y(\mathbf{c})^{d_\lambda} \prod_{q \in \tilde{\pi}} \frac{1}{Z_{\Gamma,d}^{1/2}(K_q)} $$

using the trivial but later useful observation that $1/Z_{\Gamma,d}(K_q) \le 1/Z_{\Gamma,d}^{1/2}(K_q)$.

Let us turn to the derivatives of $\Psi_{\Gamma,d}$. It follows from Proposition 8.43, the bound $|\Gamma| \le 6r^2$ and the fact that our partition functions $\chi_{\Gamma,0}$, $\chi_{\Gamma,\infty}$ are fixed by Notation 8.14, that

$$ \left| \frac{\partial^{\tilde{\pi}} \Psi_{\Gamma,d}}{\prod_{q \in \tilde{\pi}} \partial \tau_q} \right| \le \text{fn}(r). $$

We then harmonize this upper bound with (9.3) by artificially introducing a product on the right-hand side. In order to do so, we note that, if $q \in \tilde{\pi}$ is a surviving boundary segment, then it contains a terminal cell $K$. On the support of the derivative above, $Z_K^{\Gamma_\infty} \le e^{a+1}$ which, by Proposition 8.28, implies that $Z_{\Gamma,d}(K_q) \le e^{a+3}$. We obtain

$$ (9.4) \qquad \left| \frac{\partial^{\tilde{\pi}} \Psi_{\Gamma,d}}{\prod_{q \in \tilde{\pi}} \partial \tau_q} \right| \le \text{fn}(r) \prod_{K \in \tilde{\pi}} \frac{1}{Z_{\Gamma,d}^{1/2}(K)}. $$

The bounds on the derivatives of $\psi$ are trivial consequences of the Leibniz rules together with the previous observations on the derivatives of $\mathbb{1}_{\mathcal{D}}$ and $\chi_z(y_\lambda)$. The only
derivatives appearing are the derivatives $\partial^\pi \Psi_{\xi,Q}$, which are bounded by $\text{fn}(\#Q, |\pi|) \le \text{fn}(r)$, and vanish for $\tilde{\theta}_q \ge 2 \log(2)$, and hence satisfy

$$|\partial^\pi \Psi_{\xi,Q}| \le \text{fn}(r) 4^{|\pi|} \prod_{q \in \pi} e^{-\tilde{\theta}_q} \le \text{fn}(r) \prod_{q \in \pi} e^{-\tilde{\theta}_q}$$

which leads to the claim.

<figure><img src="image_42.png" /></figure>

**9.4. Comparison estimate.** Now, the last thing we need to check is the hypothesis (6.38) in Assumption (φ-CE-form), i.e., the comparison estimate. It will take the form of the following inequality.

**Proposition 9.3.** On the support of $\Psi_{\Gamma,d}$, we have

$$\sum_{\lambda \in \Lambda_{BC} \setminus W} \ell_Y(\Gamma_\lambda) \le \ell_Y(\mathbf{c}) + \sum_{q \notin \Theta_\tau(\tilde{\pi})} \tilde{\theta}_q + \sum_{q \in \tilde{\pi}} (\log Z_{\Gamma,d}(K_q) + 2) + \text{fn}(r).$$

We can rewrite this inequality under the form (6.38) in terms of the variables $\tilde{\theta}_{ac}^\sim$ with the comparison functions

$$\log Z_q := \log Z_{\Gamma,d}(K_q) + 2$$

$$\mathcal{H}(\tilde{\theta}_{ac}^\sim) := \frac{1}{2} \left( \sum_{\lambda \in \Lambda_{BC} \setminus W} \ell_Y(\Gamma_\lambda) - \text{fn}(r) \right).$$

The proof of the proposition can be found in §10, see Proposition 10.19. We first show how it allows to end the proof of our main theorem.

**9.5. Applying the operator $\mathcal{L}$**. We are now ready to apply the operator $\mathcal{L}$ to the integral (9.1), in order to prove that the result is a Friedman-Ramanujan remainder (an element of $\mathcal{R}_w$). By Proposition 1.26, this implies that the desired function is a Friedman-Ramanujan function, which is our claim. In order to write down the terms we obtain when we apply the operator $\mathcal{L}$, we will use Proposition 6.45, the second variant of our stability result.

The integration variable $x_1, \dots, x_n$ are the active parameters $\tilde{\theta}_q$ with $q \in \Theta'_{ac}$ (and in particular $n = n_{ac} = \#\Theta'_{ac}$), and the linear forms $\tau_1, \dots, \tau_m$ are the components of the vector $\vec{\tau}_{\text{surv}} = (\tau_q)_{q \in \Theta_{\text{surv}}}$, given by (4.14). The functions $h, \phi$ and $\psi$ are those described in §9.2 and §9.3, and the functions $f_1, \dots, f_n$ are the functions $\tilde{\theta}_q \mapsto \tilde{\theta}_q^{K_q} e^{\theta_q}$ for $q \in \Theta'_{ac}$.

The previous discussion proves that, as a function of the variables $\tilde{\theta}_{ac}^\sim$, the functions $h$, $f_1, \dots, f_n$ and $\varphi$ satisfy Assumptions (h), (f) and (φ-CE-form) from Notation 6.29 and 6.43 (provided we prove the comparison estimate). The constant K from Assumption (f) is

$$(9.5) \qquad K = \sum_{q \in \Theta'_{ac}} (K_q + 1)$$

which, by (5.11), is equal to

$$(9.6) \qquad K = \sum_{q \in \Theta'_{ac}} K_q + n_{ac} \le \sum_{j \in V} (K_j - 1) + n_{ac} \le \sum_{j \in V} (K_j - 1) + 2r.$$
We note that the functions $\theta_q^{K_q} e^{\theta_q}$ only have a principal term here (so we could take $N_q = 0$), but because we assume that $K_q \le N_q$ in Assumption (f), we take $N_q = K_q$ for all $q \in \Theta'_{ac}$.

We can then apply Proposition 6.45. We obtain that applying $\mathcal{L}^K$ to (9.1) yields a sum of terms bounded by

$$ (9.7) \qquad \mathrm{fn}(r, \mathrm{K}) \prod_{j \in V} \|\tilde{g}_j\|_{\mathcal{F}^{K_j, K_j}} (1+\ell)^K e^{\frac{\ell}{2}} \int \mathbb{1}_{\mathcal{D}}(\vec{L}, \vec{\theta}) \prod_{\lambda \in V_+^\beta} \chi_z(y_\lambda)^2 \prod_{q \in \Theta'_{ac}} e^{\theta_q} \prod_{\lambda \in \Lambda_{BC} \setminus W} e^{-y_\lambda/2} \frac{d\vec{\theta}_{ac}^\sim}{d\ell}. $$

As a consequence, our goal will be achieved if we prove that the integral in (9.7) is polynomially bounded, after integration with respect to all the frozen variables $\vec{\theta}'_{nt}$.

All the variables that were frozen are recorded in (5.17) and just after (9.1). Re-integrating them in the calculation, we are lead to consider the product of $\mathrm{fn}(r, \mathrm{K}) \prod_{j \in V} \|\tilde{g}_j\|_{\mathcal{F}^{K_j, K_j}}$ with

$$ (1 + \ell)^K e^{\frac{\ell}{2}} \int_{h(\vec{L}, \vec{\theta}) = \ell} \mathbb{1}_{\mathcal{D}}(\vec{L}, \vec{\theta}) \prod_{\lambda \in V_+^\beta} \chi_z(y_\lambda)^2 P(\vec{L}, \vec{\theta}) T_{V,m}(\vec{L}, \vec{\theta}) \prod_{i=1}^r \sinh(L_i) \frac{d^r \vec{L} \, d^{2r} \vec{\theta}}{d\ell}. $$

where

$$ P(\vec{L}, \vec{\theta}) = \prod_{q \in \Theta_{\ell}(V_+) \cap \Theta_{nt}} \theta_q^{K_q} \prod_{q \in \Theta'_{nt} \setminus \Theta_{nt}} \theta_q^{K_q} \prod_{q \in \Theta_{ac}} e^{\theta_q} \prod_{\lambda \in W} r_{\lambda}(\vec{L}, \vec{\theta}) \prod_{\lambda \in \Lambda_{BC} \setminus W} e^{-y_{\lambda}/2}. $$

Recall that we have assumed that we have the decay

$$ |r_{\lambda}(\vec{L}, \vec{\theta})| \le \begin{cases} (1 + y_{\lambda})^{N_{\lambda} - 1} e^{y_{\lambda}/2} & (\lambda \in W^{\beta}) \\ (1 + y_{\lambda})^{N_{\lambda} - 1} e^{-y_{\lambda}/2} & (\lambda \in W^{\Gamma}). \end{cases} $$

We also remember that $\sum_{q \in \Theta_{ac}} \theta_q \le \sum_{\lambda \in V_+^\beta \setminus W^\beta} y_\lambda = \sum_{\lambda \in V^\beta \setminus W^\beta} y_\lambda + \sum_{\lambda \in \Lambda_{in}^\beta} y_\lambda$. It follows that

$$ P(\vec{L}, \vec{\theta}) \le (1 + \ell)^{N'} \prod_{\lambda \in V^\beta \setminus W^\beta} e^{y_\lambda} \prod_{\lambda \in \Lambda_{in}^\beta} e^{y_\lambda} \prod_{\lambda \in W^\beta} e^{y_\lambda/2} \prod_{\lambda \in W^\Gamma} e^{-y_\lambda/2} \prod_{\lambda \in \Lambda_{BC} \setminus W} e^{-y_\lambda/2} $$

for $N' \le \sum_{\lambda \in \Theta'_{nt}} N_\lambda$. Therefore, since $(V^\beta \setminus W^\beta) \cup (V^\Gamma \setminus W^\Gamma) \subseteq \Lambda_{BC} \setminus W$,

$$ P(\vec{L}, \vec{\theta}) \le (1 + \ell)^{N'} \prod_{\lambda \in V^\beta} e^{y_\lambda/2} \prod_{\lambda \in \Lambda_{in}^\beta} e^{y_\lambda} \prod_{\lambda \in V^\Gamma} e^{-y_\lambda/2}. $$

Putting everything together, we have proved that the integral (9.1) is bounded by a sum of terms of the form

$$ \int_{h(\vec{L}, \vec{\theta}) = \ell} \mathbb{1}_{\mathcal{D}}(\vec{L}, \vec{\theta}) \frac{\prod_{\lambda \in V^\beta} \chi_z(y_\lambda) e^{y_\lambda/2} \prod_{\lambda \in \Lambda_{in}^\beta} \chi_z^2(y_\lambda) e^{y_\lambda}}{\prod_{\lambda \in V^\Gamma} e^{y_\lambda/2}} T_{V,m}(\vec{L}, \vec{\theta}) \prod_{i=1}^r \sinh(L_i) \frac{d^r \vec{L} \, d^{2r} \vec{\theta}}{d\ell}, $$

multiplied by $\mathrm{fn}(r, \mathrm{K}) \prod_{j \in V} \|\tilde{g}_j\|_{\mathcal{F}^{K_j, K_j}} (1 + \ell)^{K+N'} e^{\ell/2}$. We shall now bound this by a quantity for which we can use our change of variables, Corollary 3.13, backwards. Indeed,
we note that $\chi_z(y)e^{y/2} \le c \sinh(y/2)$ for a universal constant $c$, and hence the quantity above is smaller than

$$ 
\begin{aligned}
& \mathrm{fn}(r) \int_{h(\vec{L}, \vec{\theta})=\ell} \mathbb{1}_{\mathcal{D}}(\vec{L}, \vec{\theta}) \frac{\prod_{\lambda \in V^{\beta}} \sinh\left(\frac{y_{\lambda}}{2}\right) \prod_{\lambda \in \Lambda_{\mathrm{in}}^{\beta}} \sinh^2\left(\frac{y_{\lambda}}{2}\right)}{\prod_{\lambda \in V^{\Gamma}} \sinh\left(\frac{y_{\lambda}}{2}\right)} T_{V,m}(\vec{L}, \vec{\theta}) \prod_{i=1}^{r} \sinh(L_i) \frac{d^r \vec{L} \, d^{2r} \vec{\theta}}{d\ell} \\
& = \mathrm{fn}(r) \int_{h(\vec{L}, \vec{\theta})=\ell} \prod_{j=1}^{k} x_{i_j} \delta(x_{i_j} - x_{i'_j}) \frac{\mathrm{dVol}_{g_S, n_S}^{\mathrm{WP}}(Y)}{d\ell}.
\end{aligned}
$$

We have shown that this quantity is polynomially bounded in the weak sense in Corollary 3.25, giving a bound

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

As a conclusion, for $K = \sum_{j \in V} (K_j - 1) + 2r$ we have shown that

$$ \mathcal{L}^{\mathcal{K}} \mathcal{J} \in \mathcal{R}^{\mathcal{N}} $$

where $N = K + N' + 3\chi(\mathbf{S}) + 1 \le \sum_{j \in V} N_j + 3\chi(\mathbf{S}) + 1$, implying that $\mathcal{J}$ belongs to $\mathcal{F}_w^{K,N}$. Furthermore we have shown that

$$ (9.8) \qquad \| \mathcal{J} \|_{\mathcal{F}_w^{K,N}} \le \mathrm{fn}(g_S, n_S, K, N) \prod_{j \in V} \| \tilde{g}_j \|_{\mathcal{F}_{K_j, N_j}}. $$

This proves Theorem 1.22.

9.6. **Theorem 9.4.** As a step towards the version of §12, we add a small modification to Theorem 1.22. The only difference is that we impose the lengths of some components $c_j$ to stay bounded.

**Theorem 9.4.** With the notations of Theorem 1.22, let $W$ be a subset of $\{1, \dots, n\}$ and $\kappa = (\kappa_j)_{j \in W}$ be positive real numbers. Then, the function $\mathcal{J} = \mathcal{J}_{T,V,m,f,W,\kappa}$ defined by

$$ (9.9) \qquad \mathcal{J}(\ell) = \int_{\ell_Y(\mathbf{c})=\ell} \prod_{j \in W} \mathbb{1}_{[0, \kappa_j]}(\ell_Y(\mathbf{c}_j)) \prod_{j \in V} f_j(x_j) \prod_{j=1}^{k} x_{i_j} \delta(x_{i_j} - x_{i'_j}) \frac{\mathrm{dVol}_{g_S, n_S}^{\mathrm{WP}}(\mathbf{x}, Y)}{d\ell} $$

is a Friedman-Ramanujan function in the weak sense. More precisely, let $\rho[\mathbf{T}, W]$ be the number of unshielded simple portions of the multi-loop $\mathbf{c}$ that are not in $(\mathbf{c}_j)_{j \in W}$,

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

Then $\mathcal{J} \in \mathcal{F}_w^{K,N}$, and

$$ (9.10) \qquad \| \mathcal{J} \|_{\mathcal{F}_w^{K,N}} \le \mathrm{fn}(g_S, n_S, K, N) e^{\frac{1}{2} \sum_{j \in W} \kappa_j} \prod_{j \in V} \| \tilde{g}_j \|_{\mathcal{F}_{K_j, N_j}}. $$

*Proof.* The proof of Theorem 9.4 follows exactly the same lines as the proof of Theorem 1.22, except that the set of neutral variables is enlarged to $\Theta'_{\mathrm{nt}}(W) := \Theta'_{\mathrm{nt}} \cup \{q \in$
$\Theta, \mathrm{Comp}(q) \in W$} (with the notation $\mathrm{Comp}(q)$ from §2.5.2, indicating to which component $c_i$ the index $q$ belongs). The set of active variables is therefore reduced to $\Theta \setminus \Theta'_{\mathrm{nt}}(W)$.

Proposition 9.3 is modified as follows:

$$ \sum_{\lambda \in \Lambda_{BC} \setminus W} \ell_Y(\Gamma_\lambda) \le \ell_Y(\mathbf{c}) + \sum_{q \notin \Theta_\tau(\bar{\pi})} \tilde{\theta}_q + \sum_{q \in \bar{\pi}} (\log Z_{\Gamma,d}(K_q) + 2) + \sum_{j \in W} \kappa_j + \mathrm{fn}(r) $$

and this last term is responsible for the extra factor $e^{\frac{1}{2} \sum_{j \in W} \kappa_j}$ in the quantitative statement of Theorem 9.4. $\square$

**Remark 9.5.** Note that $g$, the genus of the large surface in which $S$ is embedded at the beginning of the paper, does not appear in Theorems 1.22 and 9.4. However, a notable point is that the numbers $(\kappa_j)_{j \in W}$ may be taken to be functions of $g$. This will happen in the application to Theorem 1.6, where we have constraints such as $\ell_Y(\mathbf{c}_j) \le \kappa_j$ with $\kappa_j = \kappa \log g$, with $\kappa > 0$ arbitrarily small.

**Remark 9.6.** Another point is that the new exponents K, N are bounded above by the ones obtained in the initial result, Theorem 1.22. In particular, the property also holds with the exponents from Theorem 1.22.

## 10. PROOF OF THE CRUCIAL COMPARISON ESTIMATE (PROPOSITION 9.3)

The proof of the comparison estimate, Proposition 9.3, is divided in two parts: a lower bound on $\ell_Y(\mathbf{c})$ (presented in §10.1) and an upper bound on $\sum_{\lambda \in \Lambda_{BC} \setminus W^\Gamma} \ell_Y(\Gamma_\lambda)$ (in §10.2).

10.1. **Lower bound on $\ell_Y(\mathbf{c})$.** Let us first prove a lower bound on the length of $\mathbf{c}$. We first prove a general lemma, and then proceed to the proof in §10.1.2.

10.1.1. A lemma. Let us first prove a comparison inequality in a simple case. Let $\gamma_1, \dots, \gamma_M$ be $M$ complete oriented geodesics in the hyperbolic plane, assumed to be pairwise aligned, with $\gamma_{k+1}$ on the right of $\gamma_k$ for all $k$. Let us define the following notations.

* For $1 \le k \le M-1$, $H_{k,k+1}$ is the orthogeodesic segment from $\gamma_k$ to $\gamma_{k+1}$, of length denoted as $\theta_k$, with endpoints $G_k \in \gamma_k$ and $D_k \in \gamma_{k+1}$.

* $D_0$ and $G_M$ are two arbitrary points on $\gamma_1$ and $\gamma_M$ respectively.

* For $1 \le k \le M$, we denote as $L_k$ the distance between $D_{k-1}$ and $G_k$.

We then prove the following.

**Lemma 10.1.** We have

$$ (10.1) \qquad e^{\mathrm{dist}(D_0, G_M)} \ge \frac{1}{4^{M-1}} \exp \left( \sum_{k=1}^{M} L_k \right) \prod_{k=1}^{M-1} E_k(\theta_k) $$

where the functions $E_k$ are defined by

$$ E_k(\theta) = \begin{cases} \frac{1}{2} \exp(\theta) & \text{if the geodesic from } D_0 \text{ to } G_M \text{ crosses } H_{k,k+1} \\ \cosh \theta - 1 = 2 \sinh^2(\frac{\theta}{2}) & \text{otherwise.} \end{cases} $$
*Proof.* The proof relies on the following classic formula for hyperbolic birectangles, see [5, Equation (2.3.2)]. For a geodesic quadrilateral with consecutives sides $ρ_1, t, ρ_2, d$ with right angles between $ρ_1, t$ and between $t, ρ_2$, we have :

$$ (10.2) \qquad \cosh d = \cosh \rho_1 \cosh \rho_2 \cosh t - \sinh \rho_1 \sinh \rho_2. $$

We then obtain that:

* if $ρ_1ρ_2 ≥ 0$,

$$ (10.3) \qquad e^d \ge \cosh d \ge \cosh \rho_1 \cosh \rho_2 (\cosh t - 1) \ge \frac{1}{4} e^{\rho_1 + \rho_2} (\cosh t - 1); $$

* and if $ρ_1ρ_2 ≤ 0$,

$$ (10.4) \qquad e^{|d|} \ge \cosh d \ge \cosh \rho_1 \cosh \rho_2 \cosh t \ge \frac{1}{2^3} e^{|\rho_1|+|\rho_2|+|t|}. $$

Let us now prove the result by induction on $M ≥ 2$. For $M = 2$, this is a direct consequence of (10.3) and (10.4) applied to the quadrilateral of vertices $D_0, G_1, D_1, G_2$.

For $M > 2$, assume the result known for $M - 1$ geodesics, and let us pass to rank $M$. We observe that the geodesic from $D_0$ to $G_M$ has to cross successively $γ_2, \dots, γ_{M-1}$. Call $\tilde{G}_{M-1}$ its intersection point with $γ_{M-1}$. The induction assumption at rank $M - 1$ tells us that

$$ e^{\text{dist}(D_0, \tilde{G}_{M-1})} \ge \frac{1}{4^{M-2}} \exp \left( \sum_{k=1}^{M-2} L_k + \text{dist}(D_{M-2}, \tilde{G}_{M-1}) \right) \prod_{k=1}^{M-2} E_k(\theta_k). $$

By the case $M = 2$,

$$ e^{\text{dist}(\tilde{G}_{M-1}, G_M)} \ge \frac{1}{4} \exp(\text{dist}(\tilde{G}_{M-1}, G_{M-1}) + L_M) E_{M-1}(\theta_{M-1}). $$

The result at rank $M$ follows by putting together the two previous inequalities, given that we have the triangular inequality

$$ L_{M-1} = \text{dist}(D_{M-2}, G_{M-1}) \le \text{dist}(D_{M-2}, \tilde{G}_{M-1}) + \text{dist}(\tilde{G}_{M-1}, G_{M-1}), $$

and the equality

$$ \text{dist}(D_0, G_M) = \text{dist}(D_0, \tilde{G}_{M-1}) + \text{dist}(\tilde{G}_{M-1}, G_M). $$

□

We will actually use the following variant.

**Lemma 10.2.** Let $\mathcal{Q} = \{\psi_1 < \psi_2 < \dots < \psi_j\} \subseteq \{1, \dots, M-1\}$. If we assume that $\theta_k \ge \log 2$ for all $k \in \mathcal{Q}$, then if we define $\psi_0 := 0$ and $\psi_{j+1} := M$,

$$ (10.5) \qquad \text{dist}(D_0, G_M) \ge \sum_{k \in \mathcal{Q}} \theta_k + \sum_{k=0}^{j} \text{dist}(D_{\psi_k}, G_{\psi_{k+1}}) - 5(M-1) \log 2. $$

*Proof.* The proof is the same as above, now noting that the hypothesis on $\mathcal{Q}$ implies that $E_k(\theta_k) \ge \frac{1}{8}e^{\theta_k}$ for all $k \in \mathcal{Q}$. □
**10.1.2. The lower bound.** Recall the partition of unity $\Psi_{\xi,Q}$ defined in Definition 7.23, indexed by elements $\xi \in \Xi$ and subsets $Q$ of the set of parameters $I(\xi) \subseteq \Theta$ (which intuitively corresponds to the set of “nice” variables $\theta_q$, i.e. the ones we do not erase). We have the decomposition $I(\xi) = \bigsqcup_{i=1}^n I(\xi^i)$ depending on the index of the component $c$ we consider. Consider $\tilde{I}(\xi^i) \subseteq \mathbb{Z}$ the periodic extension of $I(\xi^i)$, and let $\varphi^i : \mathbb{Z} \to \mathbb{Z}$ be an increasing numbering of it:

$$ \tilde{I}(\xi^i) = \{\varphi_k^i, k \in \mathbb{Z}\}, \quad \varphi_k^i < \varphi_{k+1}^i. $$

Denoting $n_{\xi^i} = \#I(\xi^i)$, we have the periodicity property that $\varphi_{k+n_{\xi^i}}^i = \varphi_k^i + n_i$.

We also recall some notation related to the lifts of the geometric situation to the hyperbolic plane introduced in §4.2.2. For a component $i \in \{1, \dots, n\}$, $(\tilde{\beta}(q))_{q \in \Theta_i}$ is a sequence of lifts of geodesics in the family $\beta$, which satisfies the following periodicity property: $\tilde{\beta}(q + n_i)$ is the image of $\tilde{\beta}(q)$ by the hyperbolic translation of axis $\tilde{c}_i$ and of length $\ell_Y(c_i)$.

Last, recall the intervals $\mathcal{I}'_q = [G_q, D_q]$ introduced in Notation 7.13.

**Notation 10.3.** Throughout this section, we fix $\xi \in \Xi$ and $Q \subseteq I(\xi)$. For $i \in \{1, \dots, n\}$, let $Q_i := Q \cap I(\xi^i) \subseteq \Theta^i \simeq \mathbb{Z}_{n_i}$. We pick a numbering of $Q_i$

$$ Q_i = \{\psi_1^i < \dots < \psi_{j_i}^i\} \subseteq \{1, \dots, n_i\}. $$

Let $\tilde{Q}^i = \{\psi_k^i\}_{k \in \mathbb{Z}} \subseteq \Theta^i \simeq \mathbb{Z}$ denote its cyclic extension.

Note that, since $Q^i \subseteq I(\xi^i)$, the sequence $(\psi_k^i)_{k \in \mathbb{Z}}$ is an extraction of $(\varphi_k^i)_{k \in \mathbb{Z}}$.

**Definition 10.4.** We call *junction* any geodesic segment $[D_{\psi_k^i}, G_{\psi_{k+1}^i}]$, with $i \in \{1, \dots, n\}$ and $k \in \{1, \dots, j_i\}$. We denote $\text{Junc}(Q)$ the set of all junctions. Two segments of $\text{Junc}(Q)$ will be said to be *disjoint* if they intersect only transversally.

Remark that certain junctions are bars (if $\psi_{k+1}^i = \psi_k^i + 1$).

**Proposition 10.5.** For any $Y$ in the support of $\Psi_{\xi,Q}$,

$$ (10.6) \qquad \ell_Y(c) \ge \sum_{\lambda \in \Lambda^\beta} \ell_Y(\beta_\lambda) + \sum_{p \in \text{Junc}(Q)} \ell(p) - 12r \log 2. $$

**Remark 10.6.** The proof consists, for each $i \in \{1, \dots, n\}$, in constructing a representative $c_i^f$ of the homotopy class of $c_i$, made of the concatenation of the intervals $\mathcal{I}'_q$ and of junctions. Inequality (10.6) means that $c = (c_1^f, \dots, c_n^f)$ almost minimizes the length in its homotopy class.

*Proof.* Let $Y$ be a metric on the support of $\Psi_{\xi,Q}$. We prove this property component by component and then sum the different bounds. Let us therefore fix $i \in \{1, \dots, n\}$.

Denote by $\gamma_k$ the complete orthogeodesic between $\tilde{\beta}_{\varphi_{k-1}^i}$ and $\tilde{\beta}_{\varphi_k^i}$: it contains the geodesic segment $\tilde{B}_{\varphi_{k-1}^i \varphi_k^i}$. By construction of the set $I(\xi^i)$, all these geodesics are parallel, and can be oriented so that $\gamma_{k+1}$ is on the right of $\gamma_k$. We are in the geometric situation described in §10.1.1, with $M = n_{\xi^i} + 1$. The distance between $\gamma_k$ and $\gamma_{k+1}$ is $\tilde{\theta}_{\varphi_k^i}$. The orthogeodesic segment denoted by $H_{k,k+1}$ in §10.1.1 is $[G_{\varphi_k^i}, D_{\varphi_k^i}]$.
The infinite geodesic $\tilde{c}_i$ intersects successively all geodesics $\gamma_k$. Calling $D_0$ its intersection point with $\gamma_1$ and $G_{n_{\xi^i+1}}$ its intersection with $\gamma_{n_{\xi^i+1}}$, we have $\ell_Y(\mathbf{c}_i) = \text{dist}(D_0, G_{n_{\xi^i+1}})$. We can apply inequality (10.5) to the set of indices $\psi_k^i$, which applies here because all the $\tilde{\theta}_{\psi_k^i}$ are greater than $\log 2$ on the support of $\Psi_{\xi, Q}$. We obtain that

$$ (10.7) \qquad \ell_Y(\mathbf{c}_i) \ge \sum_{q \in \mathcal{Q}^i} \tilde{\theta}_q + \sum_{k=1}^{j_i} \text{dist}(D_{\psi_k^i}, G_{\psi_{k+1}^i}) - 5n_i \log 2. $$

By summing over $i \in \{1, \dots, n\}$ and using $\ell_Y(\mathbf{c}) = \sum_{i=1}^n \ell_Y(\mathbf{c}_i)$, we obtain:

$$ \ell_Y(\mathbf{c}) \ge \sum_{q \in \mathcal{Q}} \tilde{\theta}_q + \sum_{i=1}^{n} \sum_{k=1}^{j_i} \text{dist}(D_{\psi_k^i}, G_{\psi_{k+1}^i}) - 10r \log 2. $$

Because the $\tilde{\theta}_q$ are all smaller than $2 \log 2$ for $q \notin \mathcal{Q}$, we may write

$$ \begin{aligned} \sum_{q \in \mathcal{Q}} \tilde{\theta}_q &\ge \sum_{q \in I(\xi)} \tilde{\theta}_q - 2r \log 2 \\ &\ge \sum_{q \in \Theta} \theta_q - 2r \log 2 = \sum_{\lambda \in \Lambda^\beta} \ell_Y(\beta_\lambda) - 2r \log 2 \end{aligned} $$

where we used (7.16) in the second line. This is the claim. Note that the right-hand side represents the total length of the representative $\mathbf{c}^f = (\mathbf{c}_1^f, \dots, \mathbf{c}_n^f)$ mentioned in Remark 10.6. $\square$

10.1.3. The polygons $\text{Pol}(q)$. We observe that, for any $1 \le i \le n$ and $1 \le k \le j_i$, the junction $J_k^i = [D_{\psi_k^i}, G_{\psi_{k+1}^i}]$ leaves from $\tilde{\beta}_{\psi_k^i}$ and arrives at $\tilde{\beta}_{\psi_{k+1}^i}$. It is homotopic with gliding endpoints to the concatenation

$$ \left( \cdot^{\psi_{k+1}^i - 1}_{q=\psi_k^i+1} \tilde{B}_{q-1,q} \cdot \tilde{\mathcal{I}}_q \right) \cdot \tilde{B}_{\psi_{k+1}^i - 1, \psi_{k+1}^i} $$

Examining the points at infinity of the infinite geodesic containing a lift of $J_k^i$, and using the fact that it has to cross both $\tilde{\beta}_{\psi_k^i}$ and $\tilde{\beta}_{\psi_{k+1}^i}$, we make the following two observations.

* The junction $J_k^i$ crosses all the geodesics $\tilde{\beta}_q$ with $q \in \{\psi_k^i + 1, \dots, \psi_{k+1}^i\}$, assuming $q$ is a crossing parameter.

* The junction $J_k^i$ crosses all the bars $\tilde{B}_q$ with $q \in \{\psi_k^i + 1, \dots, \psi_{k+1}^i\}$, assuming both $q-1$ and $q$ are U-turn parameters.

We deduce the following.

**Lemma 10.7.** Let $q \in \Theta$, and let $1 \le i \le n$ and $1 \le k \le j_i$ for which $q \in \{\psi_k^i + 1, \dots, \psi_{k+1}^i\}$. There exists a contractible geodesic polygon $\text{Pol}(q)$, which is bordered, on one side by the full bar $\tilde{B}_q$, on other sides, by subsegments of $J_k^i$, $\tilde{\beta}_{q-1}$, $\tilde{\beta}_q$ and possibly $\tilde{\beta}_{q-2}$, $\tilde{\beta}_{q+1}$, $\tilde{B}_{q-1}$ and $\tilde{B}_{q+1}$, and so that none of the geodesics $\tilde{\beta}_{q'}$ or the bars $\tilde{B}_{q'}$ meet the interior of $\text{Pol}(q)$.

*Proof.* The construction is made by exhaustion of all cases, depending on whether $q-1$ and $q$ are U-turn parameters or crossing parameters.
* If $q-1$ and $q$ are crossing parameters, we simply consider the polygon with sides delimited by $\tilde{B}_q$, and subsegments of $\tilde{\beta}_q$, $\tilde{\beta}_{q-1}$ and $J_k^i$, as represented by the shaded area in Figure 33.

<figure><img src="image_43.png" /><figcaption>FIGURE 33. The polygon $\text{Pol}(q)$ when $q-1$ and $q$ are crossing parameters.</figcaption></figure>

* If $q-1$ is a crossing parameter and $q$ is a U-turn, then there are four cases to consider, depending on whether $J_k^i$ intersects $\tilde{B}_q$ or not, and whether it first intersects $\tilde{B}_{q+1}$ or $\tilde{\beta}_{q+1}$. See Figure 34.

<figure><img src="image_44.png" /><figcaption>FIGURE 34. The polygon $\text{Pol}(q)$ in the mixed case.</figcaption></figure>

* The symmetric case, when $q-1$ is a U-turn parameter and $q$ a crossing parameter.

* In the case when both $q-1$ and $1$ are U-turn parameters, there are also four cases to consider, depending on whether $J_k^i$ first intersects $\tilde{\beta}_{q-2}$ or $\tilde{B}_{q-1}$, and $\tilde{\beta}_{q+1}$ or $\tilde{B}_{q+1}$. See Figure 35.

<figure><img src="image_45.png" /></figure>

**Remark 10.8.** In the case the junction $J_k^i$ intersects the bar $\tilde{B}_q$, the polygon $\text{Pol}(q)$ is not convex but rather made of two disjoint true polygons $\text{Pol}(q) = \text{Pol}^1(q) \sqcup \text{Pol}^2(q)$, one at the left of the bar $\tilde{B}_q$, the other one at its right. It can also happen that the polygon
<figure><img src="image_46.png" /><figcaption>FIGURE 35. The polygon $\text{Pol}(q)$ when $q-1$ and $q$ are U-turn parameters.</figcaption></figure>

$\text{Pol}(q)$ is degenerate, which happens if and only if the junction $J_k^i$ coincides with the bar $\tilde{B}_q$.

Because of the presence of these contractible polygons $\text{Pol}(q)$, all the bars $\tilde{B}_q$ (or their projections $\bar{B}_q$ to $S$), for $q \in \{\psi_k^i+1, \dots, \psi_{k+1}^i\}$, are said to be *superseded* by the polygon $\text{Pol}(q)$. Each bar in the family $\bar{B}$ is superseded twice: if $\iota: \Theta \to \Theta$ is the involution switching signs, then $\bar{B}_q$ is superseded by $\text{Pol}(q)$ and $\bar{B}_{\iota q}$ by $\text{Pol}(\iota q)$.

**Remark 10.9.** In the special case when $q \in \{\psi_k^i+1, \dots, \psi_{k+1}^i-1\}$ is a U-turn parameter, then the junction $J_k^i$ does not intersect $\tilde{\beta}_q$, and the entirety of the simple portion $\tilde{\mathcal{I}}_q$ is included in the boundary of $\text{Pol}(q)$. We say that $\tilde{\mathcal{I}}_q$ is *shielded* by the junction $J_k^i$: there is a contractible component of $S \setminus \mathbf{c}^f$ containing $\tilde{\mathcal{I}}_q$ in its boundary.

**10.2. Upper bound on** $\ell_Y(\Gamma_\lambda)$. The upper bound will be related to neutralizations within the family of polygonal curves $(\Gamma_\lambda)_{\lambda \in \Lambda}$. We invite the reader to remind themselves of the notations introduced in §8, and in particular the notion of height of a cell $Z_K^\Gamma$ from Notation 8.27, the conventions related to simultaneous decorations from §8.5.3, and the notion of surviving boundary segment $\mathcal{K}_q$, $q \in \Theta_{\text{surv}}$, and associated height $Z_{\Gamma,d}(\mathcal{K}_q)$, introduced in Definition 8.39.

**10.2.1. A useful lemma.** The following lemma, illustrated in Figure 36, is an obvious consequence of the variational characterization of geodesics and orthogeodesics. Here we use Lemma 8.11, which allows us to compare certain orthogedesics with the height function introduced in Definition 8.5.

**Proposition 10.10.** Let $\Gamma = (J_j, K_j)_{j \in \mathbb{Z}_m}$ be a polygonal curve. Let $I = \{a, \dots, b\}$ be an interval of $\mathbb{Z}_m$ of length $\ge 2$. For $x, y$ be two points in the roof $\tilde{J}_I$ over $(K_i)_{i \in I}$,
denote as $\gamma_{xy}$ the subsegment of $\tilde{J}_I$ between $x$ and $y$. Let $\gamma = \gamma_1 \cdot \kappa_1 \cdot \gamma_2 \cdot \cdots \cdot \kappa_n \cdot \gamma_{n+1}$ (with $n \ge 1$) be a path joining $x$ and $y$, homotopic to $\gamma_{xy}$, such that:

• for $1 \le i \le n+1$, $\gamma_i$ is a piecewise geodesic path;

• for $1 \le i \le n$, $\kappa_i$ is a geodesic segment contained in a segment $K_{m_i}$ for a $m_i \in I$.

Then, for any arbitrary subset $\pi$ of $\mathbb{Z}_m$,

$$ (10.8) \qquad \ell(\gamma_{xy}) \le \sum_{\substack{1 \le i \le n \\ \kappa_i \notin \pi}} \ell(\kappa_i) + \sum_{i=1}^{n+1} \ell(\gamma_i) + \sum_{j \in I \cap \pi} (\log Z_{K_j}^{\Gamma} + 2), $$

where $\kappa_i \notin \pi$ means that the segment $K_{m_i}$ containing $\kappa_i$ has index $m_i \notin \pi$.

<figure><img src="image_47.png" /><figcaption>FIGURE 36. Illustration of the length inequality in a neutralized polygonal curve.</figcaption></figure>

In practice, $\gamma$ will be simple (which implies that the indices $m_i$ are monotonically ordered), and the paths $\gamma_i$ will not intersect the cells $(K_j)_{j \in \mathbb{Z}_m}$ except at their endpoints.

**Remark 10.11.** If $x$ is the origin of $\tilde{J}_I$, then we can replace in this upper bound $\gamma_1$ by any piecewise geodesic path $\gamma'_1$, starting anywhere on $K_{a-1}$ and ending at the origin of $\kappa_1$. A similar remark holds if $y$ is the endpoint of $\tilde{J}_I$ lying in $K_{b+1}$.

The same result also holds with $I = \mathbb{Z}_m$, $\tilde{J}_I$ replaced by $\Gamma$, and $\gamma = \gamma_1 \cdot \kappa_1 \cdot \gamma_2 \cdot \cdots \cdot \kappa_n$ a piecewise geodesic curve (hence closed), freely homotopic to $\Gamma$ (the only difference being that there is the same number of $\kappa_i$ and $\gamma_i$, i.e. (10.8) now runs for $1 \le i \le n$ for both sums).

*Proof.* First we observe that, under the hypotheses of the statement, by the variational definition of the geodesic,

$$ (10.9) \qquad \ell(\gamma_{xy}) \le \sum_{i=1}^{n} \ell(\kappa_i) + \sum_{i=1}^{n+1} \ell(\gamma_i). $$

We then observe that the path $\gamma$ has to intersect each of the segments $W_{j_i}$ defined in Definition 8.10. More precisely, there is a portion of $\gamma$ containing $\kappa_i$ and homotopic to a
portion of $W_{j_i}$. Thus, we replace $\gamma$ by a piecewise geodesic path in the same homotopy class, where each segment $\kappa_i$ such that $m_i \in \pi$ is suppressed and replaced by a portion of $W_{m_i}$. The result then follows from the minimizing property of $\gamma_{xy}$ together with Lemma 8.11. □

**10.2.2.** *New representatives of $\Gamma_\lambda$. In the spirit of the previous length inequality, we construct a new representative to each free homotopy class $\Gamma_\lambda$, which approximates its length well.*

**Notation 10.12.** Throughout the rest of this section, we fix an exhaustion $(\Gamma, d) = (\Gamma^k, d^k)_{1\le k\le t}$ of the family of polygonal curves $\Gamma = (\Gamma_\lambda)_{\lambda\in\Lambda}$. For $\lambda \in \Lambda$, write

$$ \Gamma_{\lambda}^{k} = (J_{j}^{\lambda,k}, K_{j}^{\lambda,k})_{j \in \mathbb{Z}_{n_{\lambda}}^{k}}. $$

For $k=t$ the terminal polygonal curve, we simply denote $\Gamma_{\lambda}^{\infty} = (J_{j}^{\lambda}, K_{j}^{\lambda})_{j \in \mathbb{Z}_{n_{\lambda}}^{\infty}}$.

Remember that we also fixed a $\xi \in \Xi$ and $Q \subseteq I(\xi)$, see Notation 10.3.

In the following we focus on the elements $\lambda \in \Lambda_{BC}^{\Gamma}$, that is to say, boundary curves. Let us call $\gamma_\lambda$ the closed geodesic homotopic to $\Gamma_\lambda^1$. For $k \in \{1, \dots, t\}$ there exists a closed annular region $A_\lambda^k$ bordered, on one side by $\gamma_\lambda$, on the other side by the polygonal curve $\Gamma_\lambda^k$. Note that we have the inclusion $A_\lambda^k \subseteq A_\lambda^{k-1}$ for $k \ge 2$.

The following lemma is an immediate consequence of our construction.

**Lemma 10.13.** For $k \in \{1, \dots, t\}$, we can find a simple, piecewise geodesic representative $\tilde{\Gamma}_{\lambda}^k$ of the free homotopy class $\Gamma_{\lambda}$, contained in the annulus $A_{\lambda}^k$, written as a cyclic concatenation

$$ (10.10) \qquad \tilde{\Gamma}_{\lambda}^{k} = \gamma_{1}^{\lambda,k} \cdot \kappa_{1}^{\lambda,k} \cdot \dots \cdot \gamma_{s_{\lambda}^{k}}^{\lambda,k} \cdot \kappa_{s_{\lambda}^{k}}^{\lambda,k} $$

where:

* $\gamma_j^{\lambda,k}$ is piecewise geodesic, made of portions of junctions and bridges of $\Gamma_\lambda^k$;

* $\kappa_j^{\lambda,k}$ is a geodesic segment contained in one of the cells of $\Gamma_\lambda^k$.

The representative $\tilde{\Gamma}_{\lambda}^k$ is unique if we ask that the annulus $\tilde{A}_{\lambda}^k$ bordered by $\gamma_\lambda$ and by $\tilde{\Gamma}_{\lambda}^k$ does not contain any junction in its interior.

In particular, $\gamma_j^{\lambda,k}$ does not intersect the cells of $\Gamma_\lambda^k$ except at its endpoints.

For $k=t$, we write $\gamma_j^\lambda = \gamma_j^{\lambda,t}$ and $\kappa_j^\lambda = \kappa_j^{\lambda,t}$ for $1 \le j \le s_\lambda = s_\lambda^t$.

**10.2.3.** *Length inequalities: junctions.* We prove the following comparison between the lengths of the paths $(\gamma_j^{\lambda,1})_{1\le j \le s_\lambda^1}$ and the total length of all junctions.

**Lemma 10.14.** We have that

$$ \sum_{\lambda \in \Lambda_{BC}^{\Gamma}} \sum_{j=1}^{s_{\lambda}^{1}} \ell(\gamma_{j}^{\lambda,1}) \le \sum_{p \in \text{Junc}(Q)} \ell(p). $$

*Proof.* We observe that, for $\lambda \in \Lambda_{BC}^{\Gamma}$, the union of paths $\bigcup_{1\le j \le s_\lambda^1} \gamma_j^{\lambda,1}$ is composed of disjoint subsegments of junctions, that we call $(j_l^\lambda)_{1\le l \le M_\lambda}$, and subsegments of the
bridges of $\Gamma^1$, which we denote as $(b_l^\lambda)_{1\le l\le m_\lambda}$. Since the bridges of $\Gamma^1$ are bars $\bar{B}_q$, the paths $(b_l^\lambda)_{1\le l\le m_\lambda}$ are subsegments of bars.

We claim that to each segment $b_l^\lambda$ we can associate a subset $P(b_l^\lambda)$ satisfying the following properties:

* $P(b_l^\lambda)$ is included in $\bigcup_{p \in \text{Junc}(Q)} p$ and disjoint from all $(j_{l'}^{\lambda'})_{\lambda', 1 \le l' \le M_\lambda}$;

* $\ell(b_l^\lambda) \le \ell(P(b_l^\lambda))$;

* the $P(b_l^\lambda)$ are mutually disjoint for distinct $(l, \lambda)$.

Indeed, $b_l^\lambda$ is included in the bar $\bar{B}_q$ for an index $q$. This means that $b_l^\lambda$ is one of the boundary segments of the polygon $\text{Pol}(q) \cup \text{Pol}(\iota q)$. The two polygons $\text{Pol}(q)$ and $\text{Pol}(\iota q)$ share a side, namely $\bar{B}_q$. Because their union still contains the subset $b_l^\lambda$ of $\bar{B}_q$ in its boundary, their intersection $\text{Pol}(q) \cap \text{Pol}(\iota q)$ must be a polygon, the boundary of which consists of the union of:

* the entire segment $b_l^\lambda$ itself;

* a subset of $\bigcup_{p \in \text{Junc}(Q)} p$ made of pieces of boundary of $\text{Pol}(q)$ in the interior of $\text{Pol}(\iota q)$, or pieces of boundary of $\text{Pol}(\iota q)$ in the interior of $\text{Pol}(q)$;

* and, perhaps, some pieces of $\beta_{\lambda(q)}$ or $\beta_{\lambda(q-1)}$, orthogonal to $b_l^\lambda$.

We define $P(b_l^\lambda)$ to be the set of points in the boundary of $\text{Pol}(q) \cap \text{Pol}(\iota q)$ which belong to $\bigcup_{p \in \text{Junc}(Q)} p$. By an enumeration of all cases, a few of which are illustrated in Figure 37, we check that this set satisfies the length inequality, using either the variational characterization of geodesics or orthogeodesics.

<figure><img src="image_48.png" /><figcaption>FIGURE 37. The construction of $P(b_l^\lambda)$ in all cases where $q$ and $\iota(q)$ are crossing parameters. The overall shaded area is the union of $\text{Pol}(q)$ and $\text{Pol}(\iota(q))$, with the intersection represented in green. The dotted lines are the irrelevant portions of the junctions and bars, and the dashed lines the multi-curve $\beta$.</figcaption></figure>
We can then write

$$ 
\begin{aligned}
\sum_{\lambda \in \Lambda_{\text{BC}}^{\Gamma}} \sum_{j=1}^{s_{\lambda}^{1}} \ell(\gamma_j^{\lambda, 1}) &= \sum_{\substack{\lambda \in \Lambda_{\text{BC}}^{\Gamma} \\ 1 \le l \le M_{\lambda}}} \ell(j_l^{\lambda}) + \sum_{\substack{\lambda \in \Lambda_{\text{BC}}^{\Gamma} \\ 1 \le l \le m_{\lambda}}} \ell(b_l^{\lambda}) \\
&\le \ell\left(\bigcup_{\substack{\lambda \in \Lambda_{\text{BC}}^{\Gamma} \\ 1 \le l \le M_{\lambda}}} j_l^{\lambda} \sqcup \bigcup_{\substack{\lambda \in \Lambda_{\text{BC}}^{\Gamma} \\ 1 \le l \le m_{\lambda}}} P(b_l^{\lambda})\right) \le \sum_{p \in \text{Junc}(Q)} \ell(p)
\end{aligned}
$$

because the $j_l^{\lambda}$ and $P(b_l^{\lambda})$ are disjoint subsegments of the set of junctions. $\square$

<figure><img src="image_49.png" /></figure>

**10.2.4. Length inequalities and double points.** Let us now prove some length inequalities related to the portions of cells $(\kappa_j^{\lambda})_{\lambda,j}$. We observe that the sets $(\beta_{\lambda})_{\lambda \in \Lambda_{\text{BC}}^{\beta}}$ and $(\kappa_j^{\lambda})_{\lambda \in \Lambda_{\text{BC}}^{\Gamma}, 1 \le j \le s_{\lambda}}$ are included in $\beta$, and that each point of $\beta$ belongs to at most two of them. In particular,

$$ (10.11) \qquad \sum_{\lambda \in \Lambda_{\text{BC}}^{\beta}} \ell_Y(\beta_{\lambda}) + \sum_{\lambda \in \Lambda_{\text{BC}}^{\Gamma}} \sum_{1 \le j \le s_{\lambda}} \ell(\kappa_j^{\lambda}) \le 2\ell_Y(\beta). $$

This is also true if we restrict the left-hand-side to smaller sets.

**Notation 10.15.** In the rest of this section, we shall consider arbitrary subsets $V^{\Gamma} \subseteq \Lambda_{\text{BC}}^{\Gamma}$, $V^{\beta} \subseteq \Lambda_{\text{BC}}^{\beta}$, $\pi \subseteq \Theta_{\text{surv}}$. We let $V = V^{\beta} \sqcup V^{\Gamma}$. We write $\kappa_j^{\lambda} \notin \pi$ iff the cell containing $\kappa_j^{\lambda}$ is not in $\pi$. We denote $\Theta_{\tau}(\pi) := \cup_{q \in \pi} \Theta_{\tau}(q)$ the set of indices corresponding to the segments $K_q$ that are in $\pi$.

Plugging in these subsets, (10.11) directly implies

$$ (10.12) \qquad \sum_{\lambda \in V^{\beta}} \ell_Y(\beta_{\lambda}) + \sum_{\lambda \in V^{\Gamma}} \sum_{\substack{1 \le j \le s_{\lambda} \\ \kappa_j^{\lambda} \notin \pi}} \ell(\kappa_j^{\lambda}) \le 2\ell_Y(\beta). $$

The aim of this subsection is to improve this inequality by determining the points which are counted twice.

**Definition 10.16.** We call a point in $\beta$ simple if it satisfies either of these conditions:

* it belongs to $\cup_{\lambda \in \Lambda_{\text{in}}^{\beta}} \beta_{\lambda}$ and to at most one interval $\kappa_j^{\lambda}$ with $\lambda \in V^{\Gamma}$ and $\kappa_j^{\lambda} \notin \pi$;

* it belongs to $\cup_{\lambda \in V^{\beta}} \beta_{\lambda}$ and to none of the intervals $\kappa_j^{\lambda}$ with $\lambda \in V^{\Gamma}$ and $\kappa_j^{\lambda} \notin \pi$;

* or it belongs to $\cup_{\lambda \in \Lambda_{\text{BC}}^{\beta} \setminus V^{\beta}} \beta_{\lambda}$.

Other points are called *double*. We call $D$ the set of double points.

Obviously, if $|D|$ stands for the 1-dimensional Lebesgue measure of $D$ as a subset of $\beta$,

$$ (10.13) \qquad \sum_{\lambda \in V^{\beta}} \ell_Y(\beta_{\lambda}) + \sum_{\lambda \in V^{\Gamma}} \sum_{\substack{1 \le j \le s_{\lambda} \\ \kappa_j^{\lambda} \notin \pi}} \ell(\kappa_j^{\lambda}) \le \ell_Y(\beta) + |D|. $$

We now prove a bound on $|D|$.
**Proposition 10.17.** We have that

$$|D| \leq \sum_{\substack{q \in \mathbb{Q} \cap \Theta_{\ell}(V) \\ q \notin \Theta_{\tau}(\pi)}} \tilde{\theta}_q.$$

The proof essentially revolves around the following lemma.

**Lemma 10.18.** All the following points are simple:

(a) points in $K_q$ where $K_q$ is a cell of $\Gamma_\lambda$ with $\lambda \notin V^\Gamma$ or $\kappa_l^\lambda \in \pi$;

(a') points in $\bigcup_{\lambda \notin V^\beta} \beta_\lambda$;

(b) for $q \in I(\xi)$, points in $\bar{I}_q \setminus [G_q, D_q]$;

(c) points in $\bar{I}_q$ where $q \notin I(\xi)$;

(d) for $q \in I(\xi) \setminus \mathcal{Q}$, all points in $\bar{I}_q$;

*Proof of Lemma 10.18.* Items (a), (a') are obvious.

Item (b) is also straightforward, because there are two junctions arriving at $G_q$ and $D_q$, which are responsible for the points in $\bar{I}_q \setminus [G_q, D_q]$ being simple.

For item (c), if $q \notin I(\xi)$ then there is a junction that crosses $\beta_{\lambda(q)}$ transversally, say at a point $x_q$, which is a vertex both of $\mathrm{Pol}(q)$ and of $\mathrm{Pol}(q+1)$. The two intervals $[x_q, o(\bar{I}_{q+1})]$ and $[x_q, t(\bar{I}_q)]$ consist of simple points (see Figure 38), and their union contains $\bar{I}_q$.

<figure><img src="image_50.png" /><figcaption>FIGURE 38. Illustration of case (c) in the proof of Lemma 10.18. The two highlighted polygonals are $\mathrm{Pol}(q)$ and $\mathrm{Pol}(q+1)$. The dashed path is $\bar{I}_q$.</figcaption></figure>

Regarding item (d), if $q \in (I(\xi) \setminus \mathcal{Q}) \cap \mathcal{C}$, there is again a junction that crosses $\beta_{\lambda(q)}$ transversally, and we can argue as for Item (c). If $q \in (I(\xi) \setminus \mathcal{Q}) \cap \mathcal{U}$ we use Remark 10.9, showing that the points in $\bar{I}_q$ are simple. $\square$

*Proof of Proposition 10.17.* We write $\beta = \bigcup_{q \in \Theta} \bar{I}_q$, where the geodesic segments are just defined as subsets of $\beta$, that is to say, we forget about their orientations. These intervals may overlap: this is the case if some of the $\theta_q$ were negative, or were positive but very large. By Lemma 10.18, the only possibly double points are contained in segments $[G_q, D_q]$ where $q \in \mathcal{Q} \cap \Theta_\ell(V)$ with $q \notin \Theta_\tau(\pi)$. This implies the claim if we recall that $\tilde{\theta}_q$ is the length of $[G_q, D_q]$. $\square$
10.2.5. The upper bound. We are now ready to state and prove our upper bound.

**Proposition 10.19.** Let $\xi \in \Xi$, $Q \subseteq I(\xi)$, and $(\Gamma, \mathbf{d})$ be an exhaustion of $(\Gamma_\lambda)_{\lambda \in \Lambda}$. For any arbitrary sets $V \subseteq \Lambda_{BC}$ and $\pi \subseteq \Theta_{\text{surv}}$, on the support of $\Psi_{\Gamma, \mathbf{d}}$, we have

$$ \sum_{\lambda \in V} \ell_Y(\Gamma_\lambda) \le \ell_Y(\mathbf{c}) + \sum_{\substack{q \in Q \cap \Theta_\ell(V) \\ q \notin \Theta_\tau(\pi)}} \tilde{\theta}_q + \sum_{q \in \pi} (\log Z_{\Gamma, \mathbf{d}}(K_q) + 2) + \text{fn}(r) $$

where $\Theta_\tau(\pi) = \bigcup_{q \in \pi} \Theta_\tau(q)$.

*Proof.* Denote $V^\beta = V \cap \Lambda^\beta$ and $V^\Gamma = V \cap \Lambda^\Gamma$. For $\lambda \in \Lambda_{BC}^\Gamma$ and $k \in \{1, \dots, t-1\}$, we define

$$ S_{\lambda}^{k} = \{K \in \text{Cell}(\Gamma_{\lambda}^{k}) : \exists K' \in \text{Cell}(\Gamma^{k+1}), K' \subseteq K\}, $$

the set of cells of $\Gamma_\lambda^k$ that “survive” at step $k+1$ of the exhaustion. Denote $Z_\lambda^k = \text{Cell}(\Gamma_\lambda^k) \setminus S_\lambda^k$ the set of cells of $\Gamma_\lambda^k$ that are neutralized at step $k+1$. By repeated application of Proposition 10.10, we obtain that, for any $Y$ in the support of $\Psi_{\Gamma, \mathbf{d}}$,

$$ \ell_Y(\Gamma_\lambda) = \ell(\gamma_\lambda) \le \sum_{j=1}^{s_\lambda} \ell(\gamma_j^\lambda) + \sum_{\substack{1 \le j \le s_\lambda \\ \kappa_j^\lambda \notin \pi}} \ell(\kappa_j^\lambda) + \sum_{q \in \pi \cap \Theta_\ell(\lambda)} (\log Z_{\Gamma, \mathbf{d}}(K_q) + 2) $$

and, for $k \in \{2, \dots, t\}$,

$$ \sum_{j=1}^{s_{\lambda}^{k}} \ell(\gamma_{j}^{\lambda,k}) \leq \sum_{j=1}^{s_{\lambda}^{k-1}} \ell(\gamma_{j}^{\lambda,k-1}) + \sum_{K \in \mathcal{Z}_{\lambda}^{k-1}} (\log Z_{K}^{\Gamma} + 2). $$

Put together, this yields

$$ \ell_Y(\Gamma_\lambda) \le \sum_{j=1}^{s_\lambda^1} \ell(\gamma_j^{\lambda,1}) + \sum_{\substack{1 \le j \le s_\lambda \\ \kappa_j^\lambda \notin \pi}} \ell(\kappa_j^\lambda) \\ + \sum_{q \in \pi \cap \Theta_\ell(\lambda)} (\log Z_{\Gamma, \mathbf{d}}(K_q) + 2) + \sum_{k=1}^{t-1} \sum_{K \in \mathcal{Z}_\lambda^k} (\log Z_K^\Gamma + 2). $$

Let us now use the decoration to further estimate this length. If $k \le t-1$, then for any cell $K \in \mathcal{Z}_\lambda^k$, there is a cell $K'$ of $\Gamma^k$, containing $K$, and bearing the decoration 0 for the exhaustion $(\Gamma, \mathbf{d})$. As a result, by definition of the test function $\Psi_{\Gamma, \mathbf{d}}$, on its support, we have $\log Z_{K'}^\Gamma \le a+1$. Proposition 8.28 then implies $\log Z_K^\Gamma \le a+3$. We can then deduce from the previous bound, after summation over $\lambda \in V^\Gamma$, that

$$ \sum_{\lambda \in V^\Gamma} \ell_Y(\Gamma_\lambda) \le \sum_{\lambda \in V^\Gamma} \ell(\gamma_j^{\lambda,1}) + \sum_{\lambda \in V^\Gamma} \sum_{\substack{1 \le j \le s_\lambda \\ \kappa_j^\lambda \notin \pi}} \ell(\kappa_j^\lambda) \\ + \sum_{q \in \pi} (\log Z_{\Gamma, \mathbf{d}}(K_q) + 2) + \sum_{\substack{1 \le k \le t-1 \\ \lambda \in V^\Gamma \\ K \in \mathcal{Z}_\lambda^k}} (a+9). $$
In the last sum, the number of terms in the triple sum $\sum_{k=1}^{t-1} \sum_{\lambda \in V^\Gamma} \sum_{K \in \mathbb{Z}_\lambda^k}$ is less than the total number of cells of all $\Gamma_\lambda$ with $\lambda \in V^\Gamma$; this is less than $4r$.

We now use Lemma 10.14, equation (10.13) and Proposition 10.17 to bound the sum over $\gamma_j^\lambda$ and $\kappa_j^\lambda$. We obtain that

$$ \sum_{\lambda \in V} \ell_Y(\Gamma_\lambda) = \sum_{\lambda \in V^\Gamma} \ell_Y(\Gamma_\lambda) + \sum_{\lambda \in V^\beta} \ell_Y(\beta_\lambda) \\ \le \sum_{p \in \text{Junc}(Q)} \ell(p) + \sum_{\lambda \in \Lambda^\beta} \ell_Y(\beta_\lambda) + \sum_{\substack{q \in Q \cap \Theta_\ell(V) \\ q \notin \Theta_\tau(\pi)}} \tilde{\theta}_q + \sum_{q \in \pi} (\log Z_{\Gamma,d}(K_q) + 2) + \text{fn}(r). $$

To conclude, we use the lower bound proven at the beginning of this section, (10.6). $\square$

<figure><img src="image_51.png" /></figure>

## 11. ARBITRARY LOOP TOPOLOGIES

We now indicate how to prove Theorem 1.22 when the multi-loop c is not a generalized eight. In this case, the representatives of the homotopy class of c in minimal position may not be isotopic to one another. We choose arbitrarily one such representative – for instance, by putting an auxiliary hyperbolic metric on S and choosing the geodesic representative of c (possibly perturbing it to get rid of multiple intersections).

Let us denote by $\mathcal{N} \subseteq S$ a regular neighbourhood of c. The pair $(\mathcal{N}, c)$ is a generalized eight, and we may apply the construction of the pairs of pants decomposition of §3.2. We keep the same notation $(\Gamma_\lambda)_{\lambda \in \Lambda}$ for the pair of pants decomposition of $\mathcal{N}$, with the indices $\Lambda_{BC} = \Lambda_{BC}^\beta \sqcup \Lambda_{BC}^\Gamma$ corresponding to the boundary components of $\mathcal{N}$, split into the ones which are, or not, components of the simple multi-loop $\beta$.

In S, some of the curves $(\Gamma_\lambda)_{\lambda \in \Lambda_{BC}}$ may be contractible. Proposition 1.29 tells us that the existence of contractible $\beta_\lambda$ and $\Gamma_\lambda$ should help to prove Theorem 1.22, by making comparison estimates easier. On the other hand, contractible boundary components cause two new difficulties.

* While it is relatively easy to enumerate topological types of generalized eights, it seems daunting to make a list of all topological types of multi-loops. We sidestep this difficulty by only ever working with a fixed topological type, in this article. In the applications to spectral gaps of random hyperbolic surfaces [4], the discussion involves all possible topological types appearing in the Selberg trace formula, but we condition the Weil-Petersson measure on the set of tangle-free surfaces, in order to drastically reduce the number of local topological types to consider.

* When we vary the metric Y on S and consider the piecewise geodesic representative of c, as done in §4.2.1, the orthogeodesics $(\bar{B}_q)_{q \in \Theta}$ may be ill-defined. This happens if $B_q$ is a *problematic bar* in the sense of Definition 11.4.

### 11.1. Diagrams and the reconstitution procedure.
We recall the notion of diagram D representing the multi-loop c introduced in Definition 2.19. A diagram is the data of the simple multi-loop $\beta$ and the bars $B = (B_1, \dots, B_r)$.
11.1.1. *Simple portions of diagrams.* We extend the notions of simple portions, introduced in Definition 1.20 for multi-loops, to diagrams as follows: a *simple portion* of **D** is a maximal subsegment of **β** that does not meet the endpoints of the bars **B**. These simple portions may be *shielded* or *unshielded*, following once again Definition 1.20. The (un)shielded simple portion of **c** are in natural bijection with (un)shielded simple portion of **D**.

We do not apply this terminology to the bars $B_k$, because they do not play the same role as the curves $\beta_i$: indeed, when we describe the isotopy class of **c** by following its diagram, all the bars are traversed twice, whilst each simple portion of $\beta$ only once.

11.1.2. *Reconstitution procedure.* Given a diagram **D** = (β, **B**), we may reconstitute a multi-loop **c** stemming from **D** by the following procedure, guided by the *orientation rules* in §2.2.2.

The endpoints of the bars cut $\beta$ into $2r$ non-oriented segments $I_j$. Pick a starting segment $I_{j_0}$ and pick arbitrarily one of the two possible orientations of this segment. There a unique closed path starting with $I_{j_0}$, obtained by concatenating alternatively bars $B_k$ and segments $I_j$, and respecting the orientation rules. If this path goes through all segments $I_j$, then we obtain a loop (with one component) stemming from **D**. If not, select another segment $I_j$ that has not been traversed yet, and repeat the procedure until exhaustion of all segments $I_j$.

Starting from a diagram **D** with *r* bars, and letting $\Theta_{\mathbf{D}} := \{1, \dots, r\} \times \{\pm\}$, applying the reconstitution procedure gives us an orientation of each segment $I_j$ and their labelling by elements of $\Theta_{\mathbf{D}}$, as well as two oriented copies $(B_1^{\pm}, \dots, B_r^{\pm})$ of each bar, labelled by $\Theta_{\mathbf{D}}$.

If we start with a multi-loop **c** and obtain a diagram **D** by opening the intersections (either way), the reconstitution procedure will give us back a multi-loop isotopic to **c** (modulo orientation of each component).

The reader may check that a different choice of the original orientation of $I_{j_0}$ will just reverse the orientation of the corresponding component of **c**. This means that if a multi-loop stemming from **D** has *n* components, then the reconstitution procedure can lead to $2^n$ multi-loops stemming from **D**, all having the same geometric image, and differing only by orientation of each component.

It seems difficult to determine when two diagrams lead to homotopic multi-loops; we do not address this problem in this paper. In the case of generalized eights, this problem does not occur and we can state the following:

**Remark 11.1.** Let **D** be a diagram. Assume that $S \setminus D$ has no connected component which is a disk. Then a multi-loop stemming from **D** is a generalized eight and, as a consequence, **D** represents a generalized eight; all generalized eights leading to the diagram **D** are isotopic (modulo orientation of each component).

Note that each component of a multi-loop **c** obtained by the reconstitution procedure may be written by a concatenation such as the one we have obtained in §2.5.1, see (2.6).

11.1.3. *Extended diagrams.* We can extend the notation of diagram, discarding the condition that the bars do not self-intersect:
**Definition 11.2.** An extended diagram **D** in **S** is the data (modulo isotopy) of a collection of simple disjoint loops $\beta$, and of a collection of bars $B = (B_1, \dots, B_r)$, defined as closed segments with distinct endpoints lying in $\beta$, and disjoint from $\beta$ otherwise.

The reconstitution procedure of §11.1.2 also applies to an extended diagram **D**. *Simple portions* of an extended diagram are defined as for usual diagrams. A simple portion of **D** is said to be shielded (in **S**) if it lies in the boundary of a contractible component of $S \setminus D$.

**Definition 11.3.** Let $\mathbf{D} = (\beta, B)$ be an extended diagram in $S$, and let $\mathbf{c} = (c_1, \dots, c_n)$ be a multi-loop obtained from $\mathbf{D}$ by the reconstitution procedure. We say that $\mathbf{D}$ *satisfies the minimality condition* if:

* the components $c_i$ are non-contractible;

* if $\tilde{c}_i$ is a lift of a component of $\mathbf{c}$ in the universal cover of $S$, then $\tilde{c}_i$ has no self-intersection;

* any two lifts $\tilde{c}_i$, $\tilde{c}_j$ of components of $\mathbf{c}$ in the universal cover of $S$ can intersect at most once (with the convention that a shared bar counts for 1 intersection point).

Remark that if we start with a multi-loop $\mathbf{c}$ in minimal position, as we always do, then any diagram $\mathbf{D}$ representing it automatically satisfies the minimality condition; so this is a natural condition to ask for.

**Definition 11.4.** A bar of an extended diagram is *problematic* if it satisfies either of the following two conditions:

* one of its endpoints lies on a contractible loop $\beta_i$;

* its two endpoints lie on one same loop $\beta_i$, and the bar is homotopically trivial with endpoints gliding along $\beta_i$.

**Definition 11.5.** A bar of an extended diagram is *cylindrical* if it runs between $\beta_i$ and $\beta_j$ with $i \neq j$ and $\beta_i, \beta_j$ are non-contractible, in the same homotopy class.

Cylindrical bars count among the non-problematic bars.

**Definition 11.6.** An extended diagram $\mathbf{D}$ is said to be *good* if it has no problematic bar and if it satisfies the minimality condition.

**11.2. An auxiliary generalized eight: the skeleton.** The forthcoming Proposition 11.13 says that, by opening intersections from a multi-loop $\mathbf{c}$, we can create a generalized eight that has length *smaller than* $\mathbf{c}$ for any hyperbolic metric, and *still fills S* (or a surface obtained by removing cylinders from $S$). Such an auxilliary loop is called a *skeleton*; simple examples are represented in Figure 39.

11.2.1. *Removing one disk.* We first prove the following lemma.

**Lemma 11.7.** Let $\mathbf{D}$ be a diagram filling $S$. Assume that at least one of the connected components of $S \setminus \mathbf{D}$ is homeomorphic to a disk. Then we can remove one of the bars of $\mathbf{D}$ so that the resulting diagram $\mathbf{D}'$ fills a surface $S'$ obtained by removing cylinders from $S$.
We recall that we have defined the ways to remove a bar in a diagram in §2.4. By construction, $S \setminus D'$ has exactly one less embedded disk than $S \setminus D$ among its connected components.

*Proof.* If one of the loops $\beta_i$ are contractible, it borders a disk $D_i$. We remove a bar $B_k$ touching $\beta_i$ (and another components $\beta_j$) in the second way, to obtain $D'$. The filled surface $S'$ has same Euler characteristic as $S$. If $B_k$ lies inside $D_i$, then $\beta_j$ is also contractible and obviously $S' = S$. Otherwise, the bar $B_k$ is outside the disk $D_i$. We have $S' = S$ except in one particular case, which occurs when $\beta_i = \beta_j$. In this case, $S'$ is obtained from $S$ by removing a cylinder.

Assume now that none of the $\beta_i$ are contractible. Consider a contractible connected component $D$ of $S \setminus D$: pick a bar $B_k$ in the boundary of this contractible component and remove it the first way. Either $S' = S$, or $S'$ is obtained from $S$ by removing a cylinder (this can only happen if the two sides of $B_k$ touch the open disk $D$). $\square$

Examples of both cases appearing in this lemma can be found in Figure 39.

<figure><img src="image_52.png" /><figcaption>FIGURE 39. Three examples of (a) loops, (b) diagrams $D^{(1)}$ and (c) skeletons (shown in solid lines). The diagram $D^{(2)}$ is obtained by re incorporating the dashed bar to the skeleton. The two first examples are homotopic, non-isotopic loops.</figcaption></figure>

11.2.2. *Skeleton D'.* By iteration of the process, and using Remark 11.1, we obtain:

**Corollary 11.8.** For any diagram $D$ filling $S$, we can remove bars so that the resulting diagram $D'$ fills a surface $S'$ obtained by removing cylinders from $S$, and so that no
connected components of $S' \setminus D'$ is contractible. The diagram $D'$ represents a generalized eight $c'$.

**Definition 11.9.** Let $c$ be a multi-loop with $r$ self-intersections, and $D^{(1)} = (\beta, B)$ be the diagram obtained by opening all the intersections of $c$ the first way. We call skeleton of $c$ any diagram $D' = (\beta', B')$ obtained from $D^{(1)}$ by application of Corollary 11.8. We denote as $r' \le r$ its the number of bars and $\Theta' = \{1, \dots, r'\} \times \{\pm\}$. We also refer to the generalized eight $c'$ stemming from $D'$, obtained by removing intersections from $c$, as a skeleton of $c$.

**Remark 11.10.** We have $\chi(S) = \chi(S') = r'$.

**Remark 11.11.** A skeleton is not uniquely defined, because it depends on the order in which we choose the intersection points to remove when iterating Lemma 11.7. It suffices for us to know that a skeleton exists. The forthcoming discussion is completely dependent on the choice of a skeleton.

**Remark 11.12.** Even when $S$ is connected, the surface $S'$ may be disconnected, and some connected components may be cylinders containing a component of $c'$ which is a simple curve. This happens, in particular, when $\beta_i$ and $\beta_j^{\pm 1}$ are non-contractible curves bordering a cylinder containing a third curve $\beta_k$, in which case $\beta_k$ becomes a separate connected component of $S'$ once all its adjacent bars are removed the first way.

Note however that if a bar of $D$ joins two non-contractible curves $\beta_i$, and is not in the boundary of a contractible component of $S \setminus D$, then this bar stays in each skeleton $D'$. A skeleton $D'$ never contains the cylindrical bars and $S'$ may have more connected components than $S$, including cylinders (filled by a simple curve).

The multi-curve $\beta'$ corresponding to a skeleton $c'$ is homotopic to the collection of non-contractible components of $\beta$ in $D^{(1)}$. In particular, its number of components is smaller.

11.2.3. *Length inequality and skeleton.* We are now ready to prove our length inequality.

**Proposition 11.13.** Let $c$ be a multi-loop in minimal position, filling $S$. Let $c'$ be a skeleton of $c$. Then, for any hyperbolic Riemannian metric $Y$ on $S$,

$$ (11.1) \qquad \ell_Y(c') \le \ell_Y(c). $$

The proof relies on the following lemma.

**Lemma 11.14.** Let $c_1$ and $c_2$ be two multi-loops differing by one third Reidemeister move. Let $c'_1$ be obtained from $c_1$ by removing some intersections (in the first or the second way). Then there exists $c'_2$, homotopic to $c'_1$, obtained from $c_2$ by removing some intersections.

*Proof of Lemma 11.14.* This can be checked by simple inspection of a finite number of cases. Two cases are shown in Figure 40. $\square$

*Proof of Proposition 11.13.* A skeleton $c'$ is obtained from $c$ by removing some intersections. Let $Y$ be a hyperbolic Riemannian metric $Y$ on $S$ and $\gamma$ be the geodesic representative of $c$. By definition, $\ell_Y(c) = \ell(\gamma)$. Since $\gamma$ is a geodesic, $\gamma$ is in minimal
<figure><img src="image_53.png" /><figcaption>FIGURE 40. Illustration of two cases arising in the proof of Lemma 11.14.</figcaption></figure>

position: by the results of [7] it is isotopic to **c** up to a finite number of third Reidemeister moves.

For each Reidemeister move, we shall use Lemma 11.14. We obtain a multi-loop $\gamma'$ homotopic to the skeleton $\mathbf{c}'$, obtained from $\gamma$ by opening some intersections. In the homotopy class of $\gamma'$, there is a representative $\gamma_{\varepsilon}$ which just differs from $\gamma$ in an $\varepsilon$-neighbourhood of the intersections (with $\varepsilon > 0$ arbitrarily small) and such that $\ell(\gamma_{\varepsilon}) < \ell(\gamma)$. More precisely, $\gamma_{\varepsilon}$ may be obtained from local “switchings”, described as follows: take embedded $\varepsilon$-disks $D_1, \dots, D_r$ around each of the intersection points $a_1, \dots, a_r$, such that $D_i \cap \gamma$ is the union of two simple geodesic segments intersecting once, say $[x_1, x_3] \cup [x_2, x_4]$. The boundary $\partial D_i \cap \gamma$ consists of the four cyclically ordered points $x_1, x_2, x_3, x_4$, and the multi-loop $\gamma_{\varepsilon}$ with removed intersections is such that $D_i \cap \gamma_{\varepsilon} = [x_1, x_2] \cup [x_2, x_3]$. Outside the disks $D_i$, we have $\gamma_{\varepsilon} = \gamma$. By the triangle inequality, $\ell([x_1, x_2] \cup [x_2, x_3]) < \ell([x_1, x_3] \cup [x_2, x_4])$. Equivalently, $\ell(\gamma_{\varepsilon}) < \ell(\gamma)$. To prove the proposition, we combine the two inequalities $\ell_Y(\mathbf{c}') \le \ell(\gamma_{\varepsilon})$, and $\ell(\gamma_{\varepsilon}) < \ell(\gamma) = \ell_Y(\mathbf{c})$. $\square$

**11.3. Good representative of c.** The goal of this section is to construct a new representative $\mathbf{D}^{\text{opt}}$ of $\mathbf{c}$ which is a good extended diagram: recall that this means a diagram without problematic bars (Definition 11.4) and satisfying the minimality condition (Definition 11.3).

**11.3.1. Diagrams $\mathbf{D}^{(2)}$ and $\mathbf{D}''$.** From the construction of $\mathbf{c}'$, we see that there exists a diagram $\mathbf{D}^{(2)}$ on $\mathbf{S}$, representing $\mathbf{c}$, obtained by:

* gluing to some cylinders to $\mathbf{S}'$ to obtain $\mathbf{S}$;

* adding a collection of bars to the diagram $\mathbf{D}'$ in $\mathbf{S}$.

**Remark 11.15.** The diagram $\mathbf{D}^{(2)}$ is in general different from $\mathbf{D}^{(1)}$, because $\mathbf{D}^{(1)}$ was obtained by opening all intersections the first way, whereas $\mathbf{D}^{(2)}$ is obtained from $\mathbf{c}$ by
opening the second way certain intersections (running between contractible components of $\beta$).

The added bars may be problematic or non-problematic (cylindrical or not). Since the components of $\beta'$ in $D^{(2)}$ are non-contractible, the problematic bars of $D^{(2)}$ are only of one kind: they are homotopically trivial bars, running from a component of $\beta'$ to itself.

Recall that the (un)shielded simple portions of $c$ are in natural bijection with (un)shielded simple portions of $D^{(1)}$, and of $D^{(2)}$. A simple portion of $D'$ can be written as a disjoint union of simple portion of $D^{(2)}$, only one of which may be unshielded. Since $D'$ is a generalized eight, it has exactly $2\chi(S)$ simple portions. We deduce the following.

**Lemma 11.16.** $D^{(1)}$ and $D^{(2)}$ have at most $2\chi(S)$ unshielded simple portions.

**Definition 11.17.** A simple portion of $D'$ which contains only shielded simple portions of $D^{(2)}$ will be called a *priori neutral*.

We introduce a last auxilliary diagram.

**Notation 11.18.** Let $D''$ be the diagram obtained by removing from $D^{(2)}$ its problematic bars.

**Lemma 11.19.** $D''$ is a good diagram filling $S$.

11.3.2. *Diagram $D^{\text{opt}}$. We now prove the following.*

**Theorem 11.20.** Let $c$ be a multi-loop in minimal position. Assume that $c$ is connected and not double-filling. Then there exists good extended diagram $D^{\text{opt}}$, containing all the non-problematic bars of $D^{(2)}$, such that $c$ is homotopic to a multi-loop stemming from $D^{\text{opt}}$.

**Remark 11.21.** We may assume w.l.o.g. that $c$ is connected. If the surface $S$ filled by $c$ is not connected, the previous theorem applies in each connected component of $S$.

An example of a loop which is not a generalized eight, and a good extended diagram for this loop, is represented in Figure 41.

<figure><img src="image_54.png" /><figcaption>FIGURE 41. Example of Theorem 11.20.</figcaption></figure>