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# Iterated monodromy group

In geometric group theory and dynamical systems the iterated monodromy group of a covering map is a group describing the monodromy action of the fundamental group on all iterations of the covering. It encodes the combinatorics and symbolic dynamics of the covering and is an example of a self-similar group.

## Definition

Let $f:X_1rightarrow X$ be a covering of a path-connected and locally path-connected topological space X by its subset $X_1$, let $pi_1 \left(X, t\right)$ be the fundamental group of X and let $mathrm\left\{md\right\}_f :pi_1 \left(X, t\right)rightarrow mathrm\left\{Sym\right\},f^\left\{-1\right\}\left(t\right)$ be the monodromy action for f. Now let $mathrm\left\{md\right\}_\left\{f^n\right\}:pi_1 \left(X, t\right)rightarrow mathrm\left\{Sym\right\},f^\left\{-n\right\}\left(t\right)$ be the monodromy action of the $n^mathrm\left\{th\right\}$ iteration of f, $forall ninmathbb\left\{N\right\}_0$.

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

$mathrm\left\{IMG\right\}f := frac\left\{pi_1 \left(X, t\right)\right\}\left\{bigcap_\left\{ninmathbb\left\{N\right\}\right\}mathrm\left\{Ker\right\},mathrm\left\{md\right\}_\left\{f^n\right\}\right\}$.

The iterated monodromy group acts by automorphism on the rooted tree of preimages

$T_f := bigsqcup_\left\{nge 0\right\}f^\left\{-n\right\}\left(t\right),$
where a vertex $zin f^\left\{-n\right\}\left(t\right)$ is connected by an edge with $f\left(z\right)in f^\left\{-\left(n-1\right)\right\}\left(t\right)$.

## Examples

Let f be a complex rational function and let $P_f$ 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 $f:hat Csetminus f^\left\{-1\right\}\left(P_f\right)rightarrow hat Csetminus P_f$, 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.

## 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.