From fractal groups to fractal sets

TL;DR

This survey explores the concept of fractal groups, relating self-similarity in algebraic structures to fractals and dynamical systems, and presents examples including monodromy and Galois groups with connections to Julia sets.

Contribution

It proposes a plausible definition of fractal groups and illustrates their properties through numerous examples, linking algebraic and geometric perspectives.

Findings

Examples of fractal groups derived from monodromy and Galois groups.

Connections between group limits and Julia sets of covering maps.

Discussion of diverse topics like growth, limit spaces, and ergodic theory in relation to self-similarity.

Abstract

This paper is a survey, with few proofs, of ideas and notions related to self-similarity of groups, semi-groups and their actions. It attempts to relate these concepts to more familiar ones, such as fractals, self-similar 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 (just-non-solvable 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 finite-state transducers, growth of groups and languages, limit spaces of groups, hyperbolic spaces and groups, dynamical systems, Hecke-type operators, C^*-algebras, random matrices, ergodic theorems and entropy of non-commuting transformations. Self-similar groups appear then as a natural weaving thread through these seemingly different topics.