
Mathematics 2002
From fractal groups to fractal setsAbstract: This paper is a survey, with few proofs, of ideas and notions related to selfsimilarity of groups, semigroups and their actions. It attempts to relate these concepts to more familiar ones, such as fractals, selfsimilar sets, and renormalizable dynamical systems. In particular, it presents a plausible definition of what a "fractal group" should be, and gives many examples of such groups. A particularly interesting class of examples, derived from monodromy groups of iterated branch coverings, or equivalently from Galois groups of iterated polynomials, is presented. This class contains interesting groups from an algebraic point of view (justnonsolvable groups, groups of intermediate growth, branch groups,...), and at the same time the geometry of the group is apparent in that a limit of the group identifies naturally with the Julia set of the covering map. In its survey, the paper discusses finitestate transducers, growth of groups and languages, limit spaces of groups, hyperbolic spaces and groups, dynamical systems, Hecketype operators, C^*algebras, random matrices, ergodic theorems and entropy of noncommuting transformations. Selfsimilar groups appear then as a natural weaving thread through these seemingly different topics.
