We associated with every covering F of a topological space by an open subspace the iterated monodromy group IMG(F) of F together with a natural action of IMG(F) on a regular rooted tree. I will show how to compute this action and the group in terms of automata theory. The algebraic properties of these group are still not well known. They are rather exotic from the point of view of classical group theory. In particular, some of them have intermediate growth.
I will show how the Julia set of F can be reconstructed from IMG(F), which shows a very close connections between the dynamics of F and the properties of its iterated monodromy group. In particular, this applies to the case when F is a rational function on the complex sphere or an expanding endomorphism of a Riemannian manifold.Volodymyr Nekrashevych