Self-similarity and branching in group theory. (English) Zbl 1185.20044
Campbell, C.M. (ed.) et al., Groups St. Andrews 2005. Vol. I. Selected papers of the conference, St. Andrews, UK, July 30–August 6, 2005. Cambridge: Cambridge University Press (ISBN 978-0-521-69469-8/pbk). London Mathematical Society Lecture Note Series 339, 36-95 (2007).
From the introduction: This survey article is based on the course of four talks that were given by the first author at St. Andrews Group Theory Conference 2005 (although we do indicate here some new examples and links). We hope that it will serve as an accessible and quick introduction into the subject.
The article is organized as follows. After a quick overview of several self-similar objects and basic notions related to actions on rooted trees in Section 1 and Section 2, we define the notion of a self-similar group in Section 3 and explain how such groups are related to finite automata. Among the examples we consider are the Basilica group, the 3-generated 2-group of intermediate growth known as “the first group of intermediate growth” and the Hanoi Towers groups \(H^k\), which model the popular (in life and in mathematics) Hanoi Towers Problem on \(k\) pegs, \(k\geq 3\).
Section 4 contains a quick introduction to the theory of iterated monodromy groups developed by V. Nekrashevych [Self-similar groups. Math. Surv. Monogr. 117. Providence: AMS (2005; Zbl 1087.20032)]. This theory is a wonderful example of application of group theory in dynamical systems and, in particular, in holomorphic dynamics. We mention here that the well known Hubbard Twisted Rabbit Problem in holomorphic dynamics was recently solved by L. Bartholdi and V. Nekrashevych [Acta Math. 197, No. 1, 1-51 (2006; Zbl 1176.37020)] by using self-similar groups arising as iterated monodromy groups.
Section 5 deals with branch groups. We give two versions of the definition (algebraic and geometric) and mention some of basic properties of these groups. We show that the Hanoi Group \(\mathcal H^{(3)}\) is branch and hence the other Hanoi groups are at least weakly branch.
Section 6 and Section 7 deal with important asymptotic characteristics of groups such as growth and amenability. Basically, all currently known results on groups of intermediate growth and on amenable but not elementary amenable groups are based on self-similar and/or branch groups. Among various topics related to amenability we discus (following the article [T. Ceccherini-Silberstein, R. I. Grigorchuk, and P. de la Harpe, Proc. Steklov Inst. Math. 224, 57-97 (1999); translation from Tr. Mat. Inst. Steklova 224, 68-111 (1999; Zbl 0968.43002)]) the question on the range of Tarski numbers and amenability of groups generated by bounded automata and their generalizations, introduced by Said Sidki.
In the last sections we give an account of the use of Schreier graphs in the circle of questions described above, related to self-similarity, amenability and geometry of Julia sets and other fractal type sets, substitutional systems and the spectral problem. We finish with an example of a computation of the spectrum in a problem related to Sierpiński gasket.
Some of the sections end with a short list of open problems.
The subject of self-similarity and branching in group theory is quite young and the number of different directions, open questions, and applications is growing rather quickly. We hope that this article will serve as an invitation to this beautiful, exciting, and extremely promising subject.
For the entire collection see [Zbl 1105.20301].
The article is organized as follows. After a quick overview of several self-similar objects and basic notions related to actions on rooted trees in Section 1 and Section 2, we define the notion of a self-similar group in Section 3 and explain how such groups are related to finite automata. Among the examples we consider are the Basilica group, the 3-generated 2-group of intermediate growth known as “the first group of intermediate growth” and the Hanoi Towers groups \(H^k\), which model the popular (in life and in mathematics) Hanoi Towers Problem on \(k\) pegs, \(k\geq 3\).
Section 4 contains a quick introduction to the theory of iterated monodromy groups developed by V. Nekrashevych [Self-similar groups. Math. Surv. Monogr. 117. Providence: AMS (2005; Zbl 1087.20032)]. This theory is a wonderful example of application of group theory in dynamical systems and, in particular, in holomorphic dynamics. We mention here that the well known Hubbard Twisted Rabbit Problem in holomorphic dynamics was recently solved by L. Bartholdi and V. Nekrashevych [Acta Math. 197, No. 1, 1-51 (2006; Zbl 1176.37020)] by using self-similar groups arising as iterated monodromy groups.
Section 5 deals with branch groups. We give two versions of the definition (algebraic and geometric) and mention some of basic properties of these groups. We show that the Hanoi Group \(\mathcal H^{(3)}\) is branch and hence the other Hanoi groups are at least weakly branch.
Section 6 and Section 7 deal with important asymptotic characteristics of groups such as growth and amenability. Basically, all currently known results on groups of intermediate growth and on amenable but not elementary amenable groups are based on self-similar and/or branch groups. Among various topics related to amenability we discus (following the article [T. Ceccherini-Silberstein, R. I. Grigorchuk, and P. de la Harpe, Proc. Steklov Inst. Math. 224, 57-97 (1999); translation from Tr. Mat. Inst. Steklova 224, 68-111 (1999; Zbl 0968.43002)]) the question on the range of Tarski numbers and amenability of groups generated by bounded automata and their generalizations, introduced by Said Sidki.
In the last sections we give an account of the use of Schreier graphs in the circle of questions described above, related to self-similarity, amenability and geometry of Julia sets and other fractal type sets, substitutional systems and the spectral problem. We finish with an example of a computation of the spectrum in a problem related to Sierpiński gasket.
Some of the sections end with a short list of open problems.
The subject of self-similarity and branching in group theory is quite young and the number of different directions, open questions, and applications is growing rather quickly. We hope that this article will serve as an invitation to this beautiful, exciting, and extremely promising subject.
For the entire collection see [Zbl 1105.20301].
MSC:
| 20F65 | Geometric group theory |
| 20E08 | Groups acting on trees |
| 20F10 | Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) |
| 20F05 | Generators, relations, and presentations of groups |
| 43A07 | Means on groups, semigroups, etc.; amenable groups |
| 37A15 | General groups of measure-preserving transformations and dynamical systems |
| 37F10 | Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets |
| 57M07 | Topological methods in group theory |
