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L'interprétation matricielle de la théorie de Markoff classique

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  • Serge Perrine

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 D 6 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 D 6 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.

Suggested Citation

  • Serge Perrine, 2002. "L'interprétation matricielle de la théorie de Markoff classique," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 32, pages 1-70, January.
    • Handle: RePEc:hin:jijmms:854739
    DOI: 10.1155/S0161171202012875
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