From Christoffel Words to Markoff Numbers

In 1875, Elwin Bruno Christoffel introduced a special class of words on a binary alphabet linked to continued fractions which would go on to be known as Christoffel words. Some years later, Andrey Markoff published his famous theory, now called the Markoff theory. Markoff's theory characterized certain quadratic forms and real numbers by extremal inequalities. Both classes are constructed using certain natural numbers, known as Markoff numbers, and they form part of a solution to the Markoff Diophantine equation. More basically, they are constructed using certain words, essentially Christoffel words. The link between Christoffel words and Markoff's theory was noted by Ferdinand Frobenius in 1913, but has been neglected in recent times. Motivated by this overlooked connection, this book looks to expand on the relationship between these two areas. The first part of the book focuses on the classical theory of Markoff, while Part II explores the more advanced and recent results of the theory of Christoffel words. This new edition includes many additional exercises and solutions, as well as expanded sections on quadratic forms and quadratic numbers, two new chapters on standard words and the commutator subgroup, and revised results in combinatorics on words.

1 - Basics
2 - Words
3 - Markoff numbers
4 - The Markoff property
5 - Continued fractions
6 - Binary quadratic forms
7 - Lagrange number
8 - Words and quadratic numbers
9 - Lagrange numbers less than three
10 - Markoff's Theorem for approximations
11 - Markoff's theorem for quadratic forms
12 - Numerology
13 - Historical notes
14 - Palindromes and periods
15 - Conjugates and factors
16 - Standard words
17 - Lyndon words and Christoffel words
18 - Sturmian words
19 - Stern-Brocot Tree
20 - Free group on two generators
21 - The commutator subgroup SL2(Z)'
22 - Solution to selected exercises

Christophe Reutenauer was educated at École Normale Supérieure in Paris, and subsequently undertook his postgraduate studies at the Université de Paris 6 in 1977, before going on to complete his doctorate thesis at the same institution in 1980. He was a former researcher at the CNRS (Centre National de la Recherche Scientifique) in Paris, and LITP (Laboratoire d'Informatique Théorique et de Programmation) from 1976 to 1990. Reutenauer has been a professor at UQAM (Université du Québec à Montréal) since 1985, and was also a professor at the University of Strasbourg between 1999 and 2001. Since then, he has been an invited professor or researcher at several universities, including Saarbrücken, Darmstadt, Roma, Napoli, Palermo, UQAM, San Diego (UCSD), Strasbourg, Montpelier, Bordeaux, Paris-Est, Nice, and the Mittag-Leffler Institute. He was also the Canadian Research Chair for Algebra, Combinatorics and mathematical Informatics between 2001 and 2015.