Iterated monodromy group

Definition

Motivation

The notion of iterated monodromy group is defined in the context of a covering map from a subspace of a space to the whole space (the covering map obviously differs from the inclusion of the subspace). The key word here is iterated -- the fact that the cover itself can also be identified as a subset of the space allows us to consider iterating the function used for the covering map.

Full definition

Suppose $X$ is a path-connected locally path-connected topological space, $t\in X$, $X_{1}\subseteq X$, and $f:X_{1}\to X$ is a covering map. We define the iterated monodromy group of $f$, denoted $\operatorname {IMG} f$, as follows:

$\operatorname {IMG} f:={\frac {\pi _{1}(X,t)}{\displaystyle \bigcap _{n\in \mathbb {N} }\operatorname {Ker} F^{n}}}$

where:

• $\pi _{1}(X,t)$ is the fundamental group of $X$ at basepoint $t$.
• $F^{n}$ represents the homomorphism from $\pi _{1}(X,t)$ to $\operatorname {Sym} (f^{-n}(t))$ viewed in the context of the covering map $f^{-n}(X)\to X$. Note that in the case $n=1$, we get the usual monodromy action for $F$ and the quotient by the kernel is the usual monodromy group.

In other words, the iterated monodromy group stores the fundamental group modulo those loops that act trivially on all $f^{-n}(t)$ sets.