# Iterated monodromy group

From Topospaces

## 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 is a path-connected locally path-connected topological space, , , and is a covering map. We define the **iterated monodromy group** of , denoted , as follows:

where:

- is the fundamental group of at basepoint .
- represents the homomorphism from to viewed in the context of the covering map . Note that in the case , we get the usual monodromy action for 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* sets.