Quelques éléments de combinatoire des matrices de $S{L}_2(\mathbb{Z})$

Abstract

A Theorem of V.Ovsienko characterizes sequences of positive integers $(a_1,a_2,\dots ,a_n)$ such that the $(2\times 2)$-matrix $\left(\begin{array}{cc}{a}_n& -1\\ 1& 0\end{array}\right)\cdots \left(\begin{array}{cc}{a}_1& -1\\ 1& 0\end{array}\right)$ 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 $M=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$ and $M=\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$.

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