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
${\displaystyle \digamma$ 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\geq 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