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.

Definition

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

\mathrm{IMG}f := \frac{\pi_1 (X, t)}{\bigcap_{n\in\mathbb{N}}\mathrm{Ker}\,\digamma^n}

where :

$f:X_1\rightarrow X$ is a covering of a path-connected and locally path-connected topological space X by its subset $X_1$ ,
$\pi_1 (X, t)$ is the fundamental group of X and

$\digamma :\pi_1 (X, t)\rightarrow \mathrm{Sym}\,f^{-1}(t)$ is the monodromy action for f.

$\digamma^n:\pi_1 (X, t)\rightarrow \mathrm{Sym}\,f^{-n}(t)$ is the monodromy action of the $n^\mathrm{th}$ iteration of f, $\forall n\in\mathbb{N}_0$ .

Action

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

T_f := \bigsqcup_{n\ge 0}f^{-n}(t),

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

Examples