Quelques Éléments de Combinatoire des Matrices de $SL_2(\mathbb{Z})$

Flavien Mabilat

Published 2020-04-29Version 1

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 matrices $M$ such that some properties verified by the previous equation are still true when we replace $\pm Id$ by $\pm M$. We also give a combinatorial description of the solutions of this equation when $M=\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ in terms of dissections of convex polygons.