International Journal of Mathematics and Mathematical Sciences
Volume 32 (2002), Issue 4, Pages 193-262
doi:10.1155/S0161171202012875
L'interprétation matricielle de la théorie de Markoff classique
5 rue de Bon Pasteur, Metz 57070, France
Received 22 April 2001; Revised 5 October 2001
Copyright © 2002 Serge Perrine. 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.
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,ℤ)
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 (F 2)des automorphismes du groupe libre engendré par deux
éléments F 2.
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,ℤ) 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 (F 2) of automorphisms of the free group F 2 generated by two elements.