Iterated monodromy group

From Wikipedia, the free encyclopedia

Let $f:\tilde{X}\rightarrow X$ be an open covering of a path-connected and locally path-connected topological space $X$ , let $π 1 (X)$ be the fundamental group of $X$ and let $\mathrm{md}_f :\pi_1 (X)\rightarrow \mathrm{Sym}\,f^{-1}(X)$ be the monodromy action for $f$ . Now let $\mathrm{md}_{f^n}:\pi_1 (X)\rightarrow \mathrm{Sym}\,f^{-n}(X)$ be the monodromy action of the $n th$ iteration of $f$ , $\forall n\in\mathbb{N}_0$ .

The Iterated monodromy group of $f$ is the following quotient group:

$\mathrm{IMG}f := \frac{\pi_1 (X)}{\bigcap_{n\in\mathbb{N}}\mathrm{Ker}\,\mathrm{md}_{f^n}}$ .

[edit] See also

[edit] References

Volodymyr Nekrashevych, Self-Similar Groups, Mathematical Surveys and Monographs Vol. 117, Amer. Math. Soc., Providence, RI, 2005; ISBN 0-412-34550-1.
Kevin M. Pilgrim, Combinations of Complex Dynamical Systems, Springer-Verlag, Berlin, 2003; ISBN 3-540-20173-4.