Iterated monodromy group

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
• \digamma :\pi_1 (X, t)\rightarrow \mathrm{Sym},f^{-1}(t) 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\ge 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

Iterated monodromy groups of rational functions

Let :

• f be a complex rational function
• P_f be the union of forward orbits of its critical points (the post-critical set).

If P_f is finite (or has a finite set of accumulation points), then the iterated monodromy group of f is the iterated monodromy group of the covering f:\hat C\setminus f^{-1}(P_f)\rightarrow \hat C\setminus P_f, where \hat C is the Riemann sphere.

Iterated monodromy groups of rational functions usually have exotic properties from the point of view of classical group theory. Most of them are infinitely presented, many have intermediate growth.

IMG of polynomials

The Basilica group is the iterated monodromy group of the polynomial z^2 - 1

References

• Volodymyr Nekrashevych, Self-Similar Groups, Mathematical Surveys and Monographs Vol. 117, Amer. Math. Soc., Providence, RI, 2005; .
• Kevin M. Pilgrim, Combinations of Complex Dynamical Systems, Springer-Verlag, Berlin, 2003; .