About this Title
Volodymyr Nekrashevych, International University Bremen, Bremen, Germany
Publication: Mathematical Surveys and Monographs
Publication Year 2005: Volume 117
ISBNs: 978-0-8218-3831-0 (print); 978-1-4704-1344-6 (online)
MathSciNet review: MR2162164
MSC (2000): Primary 20E08; Secondary 20F65, 37B15, 37F10
Self-similar groups (groups generated by automata) appeared initially as examples of groups that are easy to define but that enjoy exotic properties like nontrivial torsion, intermediate growth, etc.
The book studies the self-similarity phenomenon in group theory and shows its intimate relation with dynamical systems and more classical self-similar structures, such as fractals, Julia sets, and self-affine tilings. The relation is established through the notions of the iterated monodromy group and the limit space, which are the central topics of the book.
A wide variety of examples and different applications of self-similar groups to dynamical systems and vice versa are discussed. It is shown in particular how Julia sets can be reconstructed from the respective iterated monodromy groups and that groups with exotic properties appear now not just as isolated examples but as naturally defined iterated monodromy groups of rational functions.
The book is intended to be accessible to a wide mathematical readership, including graduate students interested in group theory and dynamical systems.
Graduate students and research mathematicians interested in group theory and dynamical systems.
Table of Contents
- 1. Basic definitions and examples
- 2. Algebraic theory
- 3. Limit spaces
- 4. Orbispaces
- 5. Iterated monodromy groups
- 6. Examples and applications