Iterated monodromy group

In geometric group theory and dynamical systems the iterated monodromy group of a covering map is a group describing the monodromy action of the fundamental group on all iterations of the covering. A single covering map between spaces is therefore used to create a tower of coverings, by placing the covering over itself repeatedly. In terms of the Galois theory of covering spaces, this construction on spaces is expected to correspond to a construction on groups. The iterated monodromy group provides this construction, and it is applied to encode the combinatorics and symbolic dynamics of the covering, and provide examples of self-similar groups.

The iterated monodromy group of f is the following quotient group:

where :

The iterated monodromy group acts by automorphism on the rooted tree of preimages

where a vertex ${\displaystyle z\in f^{-n}(t)}$ is connected by an edge with ${\displaystyle f(z)\in f^{-(n-1)}(t)}$.

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