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Iterated monodromy groups

Published online by Cambridge University Press:  05 July 2011

By
Volodymyr Nekrashevych
Edited by
C. M. Campbell ,
M. R. Quick ,
E. F. Robertson ,
C. M. Roney-Dougal ,
G. C. Smith  and
G. Traustason
Volodymyr Nekrashevych
Affiliation:
Texas A&M University, USA
C. M. Campbell
Affiliation:
University of St Andrews, Scotland
M. R. Quick
Affiliation:
University of St Andrews, Scotland
E. F. Robertson
Affiliation:
University of St Andrews, Scotland
C. M. Roney-Dougal
Affiliation:
University of St Andrews, Scotland
G. C. Smith
Affiliation:
University of Bath
G. Traustason
Affiliation:
University of Bath
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Summary

Abstract

The paper is a survey of topics related to the theory of iterated monodromy groups and its applications. We also present a collection of examples illustrating different aspects of the theory.

Introduction

Iterated monodromy groups are algebraic invariants of topological dynamical systems (e.g., rational functions acting on the Riemann sphere). They encode in a computationally efficient way combinatorial information about the dynamical systems. In hyperbolic (expanding) case the iterated monodromy group contains all essential information about the dynamical system. For instance, the Julia set of the system can be reconstructed from the iterated monodromy group).

Besides their applications to dynamical systems (see, for instance [BN06] and [Nek08b]) iterated monodromy groups are interesting from the point of view of group theory, as they often possess exotic properties. In some sense their complicated structure is parallel to the complicated structure of the associated fractal Julia sets. In some cases the relation with the dynamical systems can be used to understand algebraic properties of the iterated monodromy groups.

Even though the main application of the iterated monodromy groups is dynamics, their origins are in algebra (however, some previous works in holomorphic dynamics contained constructions directly related to the iterated monodromy groups, see [HOV95, LM97, Pil00]). They were defined in 2001 in connection with the following construction due to R. Pink. Let F(x) be a rational function over ℂ.

Type
Chapter
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Groups St Andrews 2009 in Bath , pp. 41 - 93
Publisher: Cambridge University Press
Print publication year: 2011

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