Self-similar groups and their geometry

This is an overview of results concerning applications of self-similar groups generated by automata to fractal geometry and dynamical systems. Few proofs are given, interested reader can find the rest of the proofs in the monograph [Nek05]. We associate to every contracting self-similar action a topological space JG called limit space together with a surjective continuous map s : JG −→ JG. On the other hand if we have an expanding self-covering f : M1 −→ M of a topological space by its open subset, then we construct the iterated monodromy group (denoted IMG(f)) of f , which is a contracting self-similar group. These two constructions (dynamical system (JG, s) from a self-similar group and self-similar group IMG(f) from a dynamical system) are inverse to each other. The action of f on its Julia set is topologically conjugate to the action of s on the limit space JIMG(f) (see Theorem 6.1). We get in this way on one hand interesting examples of groups from dynamical systems (like the “basilica group” IMG ( z − 1 ) , which is a first example of an amenable group not belonging to the class of the sub-exponentially amenable groups). On the other hand, iterated monodromy groups are algebraic tools giving full information about combinatorics of self-coverings. The paper has the following structure. Section “Self-similar actions and automata” provides the basic notions from automata theory and theory of groups acting on rooted trees. It also gives some classical examples of self-similar groups. Section “Permutational bimodules” develops algebraic tools which are used in the study of self-similar groups. We define the notion of a permutational bimodule, which gives a convenient algebraic interpretation of automata. A closely related notion is virtual endomorphism, which can be used to construct explicit self-similar actions. We describe at the end of the section self-similar actions of free abelian groups and show how they are related to numeration systems on Z. Section 4 defines iterated monodromy groups. We show how to compute them (their standard actions) as groups generated by automata. Section 5 studies contracting self-similar actions and defines their limit spaces JG. We also prove some basic properties of the limit spaces, limit G-spaces and tiles. The last section shows connections of the obtained results with other topics of Mathematics. Subsection 6.2 shows that Julia sets of post-critically finite rational functions are limit spaces of their iterated monodromy groups. Next two subsections show a connection between topology of the limit spaces and a notion of bounded automata from [Sid00] and construct an iterative algorithm finding approximations of the limit space of actions by bounded automata. In Subsection 6.5 automata generating iterated monodromy groups of complex polynomials are described. We will see in particular, that iterated monodromy groups of complex polynomials are generated by bounded automata, so that the algorithm of the previous subsection can be used to draw topological approximations of the Julia sets of polynomials. We study in the last subsection the limit spaces of free Abelian groups and fit the theory of self-affine “digit” tilings in the framework of self-similar groups and their limit spaces.