Self-Similar Groups (Mathematical Surveys and Monographs)

Self-similar groups (groups generated by automata) initially appeared as examples of groups that are easy to define but have exotic properties like nontrivial torsion, intermediate growth, etc. This book studies the self-similarity phenomenon in group theory and shows its intimate relationship with dynamical systems and more classical self-similar structures, such as fractals, Julia sets, and self-affine tilings. This connection is established through the central topics of the book, which are the notions of the iterated monodromy group and limit space. A wide variety of examples and different applications of self-similar groups to dynamical systems and vice versa are discussed. In particular, it is shown that Julia sets can be reconstructed from the respective iterated monodromy groups and that groups with exotic properties can appear not just as isolated examples, but as naturally defined iterated monodromy groups of rational functions. The book offers important, new mathematics that will open new avenues of research in group theory and dynamical systems. It is intended to be accessible to a wide readership of professional mathematicians 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.--book Jacket. Ch. 1. Basic Definitions And Examples -- Ch. 2. Algebraic Theory -- Ch. 3. Limit Spaces -- Ch. 4. Orbispaces -- Ch. 5. Iterated Monodromy Groups -- Ch. 6. Examples And Applications. Volodymyr Nekrashevych. Includes Bibliographical References (p. 223-228) And Index. 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.