Quelques éléments de combinatoire des matrices de $SL_{2}(\mathbb{Z})$

Abstract: A Theorem of V.Ovsienko characterizes sequences of positive integers $(a_1,a_2,\ldots,a_n)$ such that the $(2\times2)$-matrix $\begin{pmatrix} a_n & -1 \ 1 & 0 \end{pmatrix}\cdots \begin{pmatrix} a_1 & -1 \ 1 & 0 \end{pmatrix}$ is equal to $\pm Id$. In this paper, we study this equation when we replace $\pm Id$ by $\pm M$. In particular, we give a combinatorial description of the solutions of this equation in terms of dissections of convex polygons in the cases $M=\begin{pmatrix} 0 & -1 \ 1 & 0 \end{pmatrix}$ and $M=\begin{pmatrix} 1 & 1 \ 0 & 1 \end{pmatrix}$.