Markov's Theorem and 100 Years of the Uniqueness Conjecture

A Mathematical Journey from Irrational Numbers to Perfect Matchings

  • Martin Aigner

Table of contents

  1. Front Matter
    Pages i-x
  2. Numbers

    1. Front Matter
      Pages 1-1
    2. Martin Aigner
      Pages 3-29
  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

About this book


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.


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

Bibliographic information

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