arXiv subjects
Publications and source records attributed to Flavien Mabilat.
We characterize sequences of positive integers (c 1 , c 2 , ..., cn) for which the (2 x 2)-matrix c 1 --1 1 0 $\times$ $\times$ $\times$ cn --1 1 0 belongs to the principal congruence subgroup of level 2 in SL(2, Z). The answer is given in terms of dissections of a convex n-gon into a mixture of triangles and quadrilaterals.
In this article we study the combinatorics of congruence subgroups of the modular group. We consider the notion of minimal monomial solutions. These are the solutions of a matrix equation (also appearing in the study of Coxeter friezes), modulo an integer $N$, whose components are identical and minimal for this property. Our objective is to study the integers $N$ for which all these solutions have a certain irreducibility property.
In this paper, we study the combinatorics of congruence subgroups of the modular group by generalizing results obtained in the non-modular case. For this, we define a notion of irreducible solutions from which we can build all the solutions. In particular, we give a particular solution, irreducible for any $N$, and the list of irreducible solutions for $N \leq 6$.
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 $M=\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$.
As part of the study of Coxeter's friezes, M. Cuntz introduced the notion of irreducible $λ$-quiddity cycle. The objective of this note is to list all the irreducible $λ$-quiddity cycles on the ring $\mathbb{Z}[α]$ with $α$ a transcendent complex number. ----- Dans le cadre de l'étude des frises de Coxeter, M.Cuntz a introduit la notion de $λ$-quiddité irréductible. L'objectif de cette note est de lister toutes les $λ$-quiddités irréductibles sur l'anneau $\mathbb{Z}[α]$ dans le cas où $α$ est un nombre complexe transcendant.