Éléments de comptage sur les générateurs du groupe modulaire et les $\lambda$-quiddités

Abstract

The aim of this article is to count the $n$-tuples of positive integers $(a_1,\dots ,a_n)$ solutions of the equation $\left(\begin{array}{cc}{a}_n& -1\\ 1& 0\end{array}\right)\left(\begin{array}{cc}{a}_{n-1}& -1\\ 1& 0\end{array}\right)\cdots \left(\begin{array}{cc}{a}_1& -1\\ 1& 0\end{array}\right)=\pm M$ when $M$ is equal to the generators of the modular group $S=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$ and $T=\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$. To count these elements, we will study the $\lambda$-quiddities, which are the solutions of the equation in the case $M=Id$ (related to Coxeter's friezes), whose last component is fixed.

in French language