Iterated Monodromy Groups

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Iterated monodromy groups{First abstract.} The project explores new connectionsbetween geometric group theory and dynamical systems via thenotion of the iterated monodromy group of a self-covering. Itrelates a very active branch of geometric group theory (automatagroups and groups acting on rooted trees) with symbolic andholomorphic dynamics. Before introduction of iterated monodromygroups, groups generated by finite automata were mainly isolatedexotic (counter) examples. It is clear now that such groups appearnaturally in connection with iterations of self-coverings oftopological spaces (in particular iterations of post-criticallyfinite rational functions). This opens new perspectives ofapplication of holomorphic dynamics to group theory. On the otherhand, computational effectiveness of automata groups helps tostudy deeper the combinatorics and symbolic dynamics ofself-coverings. For example, if the self-covering is expanding,then its Julia set is uniquely determined by the iteratedmonodromy group. The project suggests further investigation of theconnections between dynamical systems and automata groups. Wehope, in particular, to get new results in the theory of groupsacting on rooted trees, groups generated by automata, theory ofgrowth of groups, find new applications of geometric group theoryto combinatorics of iterations of rational functions andpolynomials.{Second abstract.} The project brings together two branchesof Mathematics, which where not so close before: geometric grouptheory and holomorphic dynamics. Geometric groups theory is a partof algebra which studies large-scale properties of groups ofsymmetries. Holomorphic dynamics studies iterations of rationaland polynomial functions and is a source of complex and beautifulfractal structures such as Mandelbrot and Julia sets. Iteratedmonodromy groups, introduced by the investigator, are naturallyassociated with rational iterations and are groups encoding in acompact symbolic form all the complex topological behavior of theiterations and geometry of their Julia sets. The construction ofthe iterated monodromy group is very natural, but groups obtainedin this way are very exotic from the point of view of classicalgroup theory. Difference between classical groups and iteratedmonodromy groups resembles in some way the difference between the``smooth'' geometric shapes of the classical geometry and fractal``wild'' shapes of holomorphic dynamics. Algebraic properties ofiterated monodromy groups remain to be rather mysterious and apart of the project is to understand them better. We are alsohoping that this new connection between group theory and dynamicalsystems will be fruitful for both parts of mathematics, that wewill be able to use group theory to obtain new results inholomorphic dynamics and to apply holomorphic dynamics to studynew classes of groups.