- Iterated monodromy group
In
geometric group theory anddynamical systems the iterated monodromy group of acovering map is a group describing themonodromy action of thefundamental group on all iterations of the covering. It encodes the combinatorics andsymbolic dynamics of the covering and is an example of aself-similar group .Definition
Let f:X_1 ightarrow X be a covering of a path-connected and locally path-connected topological space X by its subset X_1, let pi_1 (X, t) be the
fundamental group of X and let mathrm{md}_f :pi_1 (X, t) ightarrow mathrm{Sym},f^{-1}(t) be the monodromy action for f. Now let mathrm{md}_{f^n}:pi_1 (X, t) ightarrow mathrm{Sym},f^{-n}(t) be the monodromy action of the n^mathrm{th} iteration of f, forall ninmathbb{N}_0.The Iterated monodromy group of f is the following
quotient group ::mathrm{IMG}f := frac{pi_1 (X, t)}{igcap_{ninmathbb{Nmathrm{Ker},mathrm{md}_{f^n.The iterated monodromy group acts by automorphism on the rooted tree of pre
T_f := igsqcup_{nge 0}f^{-n}(t),where a vertex zin f^{-n}(t) is connected by an edge with f(z)in f^{-(n-1)}(t).Examples
Let f be a complex
rational function and let P_f be the union of forward orbits of its critical points (thepost-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^{-1}(P_f) ightarrow hat Csetminus P_f, where hat C is theRiemann 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 .ee also
*
Growth rate (group theory)
*amenable group
*complex dynamics
*Julia set References
* Volodymyr Nekrashevych, [http://www.ams.org/bookstore-getitem/item=surv-117 "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.External links
* [http://www.arxiv.org/find/grp_physics,grp_nlin,grp_math/1/abs:+EXACT+iterated%5fmonodromy%5fgroup/0/1/0/all/0/1?skip=0&query_id=02226432472b51fb arXiv.org - Iterated Monodromy Group] - preprints about the Iterated Monodromy Group.
* [http://mad.epfl.ch/%7Elaurent/pub/rabbit/index.html Laurent Bartholdi's page] - Movies illustrating the Dehn twists about aJulia set .
* [http://mathworld.wolfram.com/MonodromyGroup.html mathworld.wolfram.com] - The Monodromy Group page.
Wikimedia Foundation. 2010.