Self-similar groups. (English) Zbl 1087.20032

Mathematical Surveys and Monographs 117. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3831-8/hbk). xi, 231 p. (2005).
A self-similar group action is defined as follows. Let \(X\) be a finite alphabet and let \(X^*\) be the set of all finite words on \(X\). A faithful action of a group \(G\) on \(X^*\) is said to be ‘self-similar’ if for every \(g\in G\) and \(x\in X\) there exists \(h\in G\) and \(y\in X\) such that \(g(xw)=yh(w)\) for all words \(w\in X^*\). Thus, self-similar actions agree with the self-similarity of the set \(X^*\) given by the shift map \(xw\mapsto w\). The self-similar groups (groups generated by automata) first appeared in the early eighties as examples of groups with properties which are hard to find among groups built with traditional methods (the Grigorchuk group, for example). One of the aims of this investigation is to show that self-similar groups are not just isolated examples, but they have close connections with dynamics and fractal geometry.
In the first chapter the basic definitions and examples are given. In the second chapter the self-similarity of groups is studied from the algebraic point of view. It is shown that self-similarity can be interpreted as a ‘permutational bimodule’, that is, a set with two (left and right) commuting actions of a group. In the third chapter ‘limit spaces’ and ‘limit dynamical systems’ of contracting self-similar actions are constructed and studied. In the fourth chapter some technical definitions are collected about orbispaces, which are structures represented locally as quotients of topological spaces by finite homomorphism groups. In the fifth chapter ‘iterated monodromy groups’ are defined and studied. If \(p\colon{\mathcal M}_1\to{\mathcal M}\) is a covering of a topological space (or an orbispace) \(\mathcal M\) by an open subspace (an open sub-orbispace) \({\mathcal M}_1\), then the fundamental group \(\pi_1({\mathcal M},t)\) acts naturally by the monodromy action of the set of preimages \(p^{-1}(t)\) of the basepoint under the \(n\)-th iteration of \(p\). Let us denote by \(K_n\) the kernel of the action. Then the iterated monodromy group of \(p\) (denoted by \(\text{IMG}(p)\)) is the quotient \(\pi_1({\mathcal M},t)/\bigcap_{n\geq 0}K_n\). In the sixth chapter different examples of iterated monodromy groups and their applications are shown.


20F65 Geometric group theory
20E08 Groups acting on trees
20-02 Research exposition (monographs, survey articles) pertaining to group theory
37F20 Combinatorics and topology in relation with holomorphic dynamical systems
37F15 Expanding holomorphic maps; hyperbolicity; structural stability of holomorphic dynamical systems
37B10 Symbolic dynamics