Iterated monodromy group

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 defining ingredient::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.