Combinatorics on Words and the Theory of Markoff

  • Christophe ReutenauerEmail author
Conference paper
Part of the Abel Symposia book series (ABEL, volume 13)


This is a survey on the theory of Markoff, in its two aspects: quadratic forms (the original point of view of Markoff), approximation of reals. A link wih combinatorics on words is shown, through the notion of Christoffel words and special palindromes, called central words. Markoff triples may be characterized, by using some linear representation of the free monoid, restricted to these words, and Fricke relations. A double iterated palindromization allows to construct all Markoff numbers and to reformulate the Markoff numbers injectivity conjecture (Frobenius, Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin 26:458–487, 1913).


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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Département de mathématiquesUniversité du Québec à MontréalMontréalCanada

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