\section{Ideal polyhedra and approximation of laminations.} 

In this section, we show how to approximate a pair of measured laminations that strongly fill in $\HH^2$ by a sequence of pairs of ``polyhedral'' laminations, that is, measured laminations on ideal polygons with support contained in a union of disjoint diagonals. We first define what we mean by a ``polyhedral'' measured lamination on $\HH^2$.

\begin{defi}
  A measured lamination on $\HH^2$ is {\em polyhedral} if its support is the disjoint union of a finite set of complete geodesics.
\end{defi}

\begin{defi}
  Let $P$ be an idea polyhedron in $AdS^3$, and let $P_-$ and $P_+$ be its past and future boundary components.\marginnote{JM: vérifier que c'est défini dans la section 2.}
  Each of those two pleated ideal polygons defines a polyhedral measured lamination. We denote these two laminations by $\lambda_-(P)$ and $\lambda_+(P)$ respectively.
  We denote by $V(P)$ the $n$-tuple of vertices of $P$, cyclically oriented in the order in which they occur on any acausal meridian containing them. 
\end{defi}

By construction, $\lambda_\pm(P)$ have a support which is a disjoint union of diagonals, that is, complete geodesics connecting two vertices. 

\begin{theorem}
Let $\lambda_-$, $\lambda_+\in \cML$ be a pair of laminations that strongly fill.
There exists a sequence $(P_n)_{n\in \N}$ of ideal polyhedra and a sequence $u_n$ of functions from $V(P_n)$ to $\RP^1$ such that $((u_{n})_*(\lambda_-(P_n)))_{n\in \N}$ and $((u_{n})_*(\lambda_+(P_n)))_{n\in \N}$ converge to $\lambda_-$ and $\lambda_+$, respectively.
\end{theorem}

The convergence considered here is in the sense of weak-$*$ convergence on measures in $\RP^1\times \RP^1\setminus \Delta$.\marginnote{JM: vérifier, c'est bien ça ?}


\begin{proof}
Let us first construct a weighted spherical graph out of which we will get a polyhedron using the main result of \cite{idealpolyhedra}. 

We fix $n\in\mathbb{N}$ and we consider the disk $D(o,n)$ of radius $n$ centered at a fixed point $o\in\mathbb{H}^2$ . We remove from $\lambda_-$ and $\lambda_+$ the leaves which does not intersect $D(o,n)$. We denote those truncated laminations by $\lambda_-^n$ and $\lambda_+^n$; our aim is to approximate them by polyhedral laminations. The total weight of those truncated lamination is finite, it equals twice the sum of the intersections of the circle $C(o,n)$ with the laminations $\lambda_+$ and $\lambda_-$. We denote this total weight by $a_n$.

We now arrange $k$ points on the circle $\mathbb{S}^1=\partial_\infty\mathbb{H}^2$ according to the following procedure. The number $k$ will be chosen later as a function of $n$. We use the Poincaré disk model of $\mathbb{H}^2$. We first set $a_1=i$. Assuming that $a_j$ has been constructed, we construct $a_{j+1}$ in such a way that the leaves of $\lambda_-^n$ and $\lambda_+^n$ limiting in the interval $\left[a_ja_{j+1}\right)$ have a total weight of $\frac{a_n}{k}$. If a single leaf carries too much weight, we divide it into parallel geodesics and we fairly split the weights. This defines $k$ points on $\mathbb{S}^1$.


We now describe a weighted graph $\Gamma$ which encodes the 1-skeleton and exterior dihedral angles of polyhedron we are looking for.
\begin{enumerate}
\item $\Gamma$ has a vertex $v_j$ in each segment $\left[a_ja_{j+1}\right)$.
\item Two vertices $v_j$ and $v_{j'}$ are related by an edge whose weight is the total weight of the leaves of $\lambda_-^n$ (resp. $\lambda_+^n$) from the interval $\left[a_ja_{j+1}\right)$ to $\left[a_{j'}a_{j'+1}\right)$, if there is any such leave.
\item The adjacent vertices $v_j$ and $v_{j+1}$ are related by an edge of weight $-\frac{a_n}{2k}$.
\end{enumerate}
Given an edge $e$ of this graph, we denote by $\theta(e)$ its weight.


\begin{lemma}
The weighted graph $\Gamma$ satisfies the hypothesis of \cite[Theorem 1.4]{idealpolyhedra} for $k$ large enough.
\end{lemma}

\begin{proof}
The fact that the graph has a Hamiltonian circle is obvious. The link condition is also trivially satisfied. We need to show is that if $e^*_1,e^*_2,\cdots, e^*_q$ is a closed path in the dual graph $\Gamma^*$ which does not bound a face of $\Gamma^*$ but intersect exactly two edges of the ``equator'' (edges of $\Gamma$ with negative weight), then\marginnote{JM: rewritten. No reason why $p=q$, right?}
\[\sum_{i=1}^q\theta(e^*_i)>0.\]
This issue reduces\marginnote{JM: ok but we need to explain why.} to the following statement: let $\gamma$ be a geodesic path in $\mathbb{H}^2$ connecting the interval $\left(v_1v_2\right)$ to $\left(v_{p}v_{p+1}\right)$. Then the total weight of the edges of $\lambda_-^n$ and $\lambda_+^n$ it crosses is bigger than $\frac{a_n}{k}$. We argue differently according to the relative position of $v_1$ and $v_p$.

\textbf{1st case:} The path $\gamma$ does not enter the disk $D(o,n)$. Since each edge of $\lambda_\pm$ enters $D(o,n)$, $\gamma$ must cross all the edges emanating from $v_2$ and $v_p$, for a total weight of $\frac{2a_n}{k}$.


\definecolor{ttttff}{rgb}{0.2,0.2,1}
\begin{figure}[!h]
\begin{tikzpicture}[scale=0.8][line cap=round,line join=round,>=triangle 45,x=1.0cm,y=1.0cm]
\clip(-5.87,-13.42) rectangle (34.33,8.12);
\draw [shift={(10.38,-2.32)}] plot[domain=1.68:4.33,variable=\t]({1*9.01*cos(\t r)+0*9.01*sin(\t r)},{0*9.01*cos(\t r)+1*9.01*sin(\t r)});
\draw [shift={(10.7,-2.28)}] plot[domain=1.94:4.24,variable=\t]({1*7.88*cos(\t r)+0*7.88*sin(\t r)},{0*7.88*cos(\t r)+1*7.88*sin(\t r)});
\draw [shift={(-35.65,-237.6)}] plot[domain=1.37:1.41,variable=\t]({1*241.03*cos(\t r)+0*241.03*sin(\t r)},{0*241.03*cos(\t r)+1*241.03*sin(\t r)});
\draw [shift={(-5.95,-77.88)}] plot[domain=1.33:1.47,variable=\t]({1*76.84*cos(\t r)+0*76.84*sin(\t r)},{0*76.84*cos(\t r)+1*76.84*sin(\t r)});
\draw [shift={(2.7,3.55)}] plot[domain=-1.85:0.55,variable=\t]({1*3.18*cos(\t r)+0*3.18*sin(\t r)},{0*3.18*cos(\t r)+1*3.18*sin(\t r)});
\draw [shift={(7,6.27)}] plot[domain=-2.55:0.12,variable=\t]({1*1.91*cos(\t r)+0*1.91*sin(\t r)},{0*1.91*cos(\t r)+1*1.91*sin(\t r)});
\draw [shift={(1.37,-3.65)}] plot[domain=-1.26:1.55,variable=\t]({1*2.26*cos(\t r)+0*2.26*sin(\t r)},{0*2.26*cos(\t r)+1*2.26*sin(\t r)});
\draw [shift={(2.87,-7.77)}] plot[domain=-0.7:1.96,variable=\t]({1*2.13*cos(\t r)+0*2.13*sin(\t r)},{0*2.13*cos(\t r)+1*2.13*sin(\t r)});
\draw [line width=1.6pt,color=ttttff] (1.41,-1.39)-- (2.85,-1.54);
\draw (6.71,-8.26) node[anchor=north west] {$D(0,n)$};
\draw (4.27,-9.47) node[anchor=north west] {$ \partial_\infty \mathbb{H}^2 $};
\draw (2.28,-0.71) node[anchor=north west] {$ \lambda_0 $};
\draw (6.91,-1.39) node[anchor=north west] {$\partial_r b^+$};
\draw [shift={(1.35,-0.39)}] plot[domain=-1.51:1.09,variable=\t]({1*1*cos(\t r)+0*1*sin(\t r)},{0*1*cos(\t r)+1*1*sin(\t r)});
\draw (2.26,0.43) node[anchor=north west] {$\partial_0 b^+$};
\draw (7.73,0.28) node[anchor=north west] {$\partial_l b^+$};
\draw (5.6,2.75) node[anchor=north west] {$\partial B^+$};
\end{tikzpicture}
\label{fig:notation}
\caption{Notations used in the approximation procedure.}
\end{figure}


\textbf{2nd case:} The geodesic $\gamma$ does enter the disk $D(0,n)$. Note that the case where the path enter $D(0,n)$ for a length at least $c$ corresponding to $\varepsilon=\frac{2a_n}{k}$ in the definition of the strong filling condition is easily taking care of by the strong filling hypotheses. Nevertheless, the following argument also handles the case where the intersection of $\gamma$ and $D(0,n)$ is short. We will use the following notations (see figure \ref{fig:notation}). We denote by $B^+$ and $B^-$ the polygons corresponding to the union of the faces that $\gamma$ meets in the dual graph of $\lambda_n^+$ and $\lambda_n^-$ respectively. We also consider the quadrilateral $b$ which is the convex hull of the boundary edges crossed by $\gamma$, i.e in the above notations the edges $\left(v_1v_2\right)$ and $\left(v_{p}v_{p+1}\right)$ and we denote by $b^+$ and $b^-$ the two copies of $b$ in the upper and lower side. Then $\partial_r b^+$ and $\partial_l b^+$ are the remaining sides of $b^+$ (same for $b^-$). Finally $\lambda_0$ is the part of $\partial_r b^+$ lying outside $D(0,n)$.
\end{proof}

The fact that $\lambda_+(P_n)$ converge to $\lambda_+$ and $\lambda_-(P_n)$ converge to $\lambda_-$ follows directly from the definition of the topology on $\cML$. \note{good ref}

\end{proof}

We now associate to those truncated laminations, an approximation of the quasi-circle we are looking for. Indeed, let $P_n$ be the ideal polyhedra we constructed in the previous theorem. With its Hamiltonian path, we connect all of its vertices by a path $v^n:\mathbb{RP}^1\rightarrow \partial_\infty$AdS$^3$ which is piecewise $\mathcal{C}^1$ with breaking points only at the vertices and with left and right derivatives everywhere. Moreover we take $v^n$ uniformly acausal in the sense that its (left and right) tangent are uniformly far from a lightlike line. This later condition rephrases as $v^n$ being quasi-symmetric. \marginnote{JM: je ne vois pas bien pourquoi, il me semble qu'il manque une preuve.}
{"config": "arxiv", "file": "2006.13470/laminations3old.tex"}