Also appears in collection : Centre Émile Borel - Programmes presentations

Fractals play a key role in several topics in group theory. Many group actions have an intrinsically fractal nature. A prominent such scenario arises for a group acting on a tree when its restriction to a proper subtree is isomorphic to the original action. This is how Grigorchuk’s famous examples arise. Some groups have a boundary at infinity that is a compact fractal. This is the case for Gromov’s word-hyperbolic groups whose properties are reflected in the shape of this fractal boundary. Some groups are naturally related to dynamics of substitutions, leading naturally to fractalness of limit objects. This is the case for automorphism groups of free groups acting on the space of words, or automorphism groups of surface groups acting on the space of curves. We articulate our viewpoint of fractalness in group theory around these three topics: fractal groups acting on rooted trees, hyperbolic groups, and automorphism groups.