International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2002 / Article

Open Access

Volume 32 |Article ID 854739 | 70 pages | https://doi.org/10.1155/S0161171202012875

L'interprétation matricielle de la théorie de Markoff classique

Received22 Apr 2001
Revised05 Oct 2001

Abstract

On explicite l'approche de Cohn (1955) de la théorie de Markoff. On montre en particulier comment l'arbre complet des solutions de l'équation diophantienne associée apparasît comme quotient du groupe GL(2,) des matrices 2×2 à coefficients entiers et de déterminant ±1 par un sous-groupe diédral D6 à 12 éléments. Différents développements intermédiaires sont faits autour du groupe Aut(F2)des automorphismes du groupe libre engendré par deux éléments F2.We detail the approach followed by Cohn for the Markoff theory. We show particularly how appears the whole tree of solutions for the associated Diophantine equation as a quotient of the group GL(2,) of matrices 2×2 with integer coefficients and determinant ±1 by its dihedral subgroup D6 with 12 elements. Some developments are made with the group Aut(F2) of automorphisms of the free group F2 generated by two elements.

Copyright © 2002 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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