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
Let be a covering of a path-connected and locally path-connected topological space X by its subset , let be the fundamental group of X and let be the monodromy action for f. Now let be the monodromy action of the iteration of f, .
The Iterated monodromy group of f is the following quotient group:
The iterated monodromy group acts by automorphism on the rooted tree of preimages
where a vertex
is connected by an edge with
be a complex rational function
be the union of forward orbits
of its critical points
(the post-critical set
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
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.
- 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.