# Markov's Theorem and 100 Years of the Uniqueness Conjecture

## A Mathematical Journey from Irrational Numbers to Perfect Matchings

• Martin Aigner
Book

1. Front Matter
Pages i-x
2. ### Numbers

1. Front Matter
Pages 1-1
2. Martin Aigner
Pages 3-29
3. Martin Aigner
Pages 31-41
3. ### Trees

1. Front Matter
Pages 43-43
2. Martin Aigner
Pages 45-62
3. Martin Aigner
Pages 63-77
4. ### Groups

1. Front Matter
Pages 79-79
2. Martin Aigner
Pages 81-111
3. Martin Aigner
Pages 113-131
5. ### Words

1. Front Matter
Pages 133-133
2. Martin Aigner
Pages 135-157
3. Martin Aigner
Pages 159-182
6. ### Finale

1. Front Matter
Pages 183-183
2. Martin Aigner
Pages 185-206
3. Martin Aigner
Pages 207-248
7. Back Matter
Pages 249-257

### Introduction

This book takes the reader on a mathematical journey, from a number-theoretic point of view, to the realm of Markov’s theorem and the uniqueness conjecture, gradually unfolding many beautiful connections until everything falls into place in the proof of Markov’s theorem. What makes the Markov theme so attractive is that it appears in an astounding variety of different fields, from number theory to combinatorics, from classical groups and geometry to the world of graphs and words.

On the way, there are also introductory forays into some fascinating topics that do not belong to the standard curriculum, such as Farey fractions, modular and free groups, hyperbolic planes, and algebraic words. The book closes with a discussion of the current state of knowledge about the uniqueness conjecture, which remains an open challenge to this day.

All the material should be accessible to upper-level undergraduates with some background in number theory, and anything beyond this level is fully explained in the text.

This is not a monograph in the usual sense concentrating on a specific topic. Instead, it narrates in five parts – Numbers, Trees, Groups, Words, Finale – the story of a discovery in one field and its many manifestations in others, as a tribute to a great mathematical achievement and as an intellectual pleasure, contemplating the marvellous unity of all mathematics.

### Keywords

Continued Fractions Lagrange Spectrum Markov Theorem Modular Group Uniqueness Conjecture of Markov Numbers

#### Authors and affiliations

• Martin Aigner
• 1
1. 1.Fachbereich Mathematik und Informatik, Institut für MathematikFreie Universität BerlinBerlinGermany