Iterated Monodromy Groups

Texas A&M Research Foundation, College Station TX

Investigators

Abstract

Iterated monodromy groups {First abstract.} The project explores new connections between geometric group theory and dynamical systems via the notion of the iterated monodromy group of a self-covering. It relates a very active branch of geometric group theory (automata groups and groups acting on rooted trees) with symbolic and holomorphic dynamics. Before introduction of iterated monodromy groups, groups generated by finite automata were mainly isolated exotic (counter) examples. It is clear now that such groups appear naturally in connection with iterations of self-coverings of topological spaces (in particular iterations of post-critically finite rational functions). This opens new perspectives of application of holomorphic dynamics to group theory. On the other hand, computational effectiveness of automata groups helps to study deeper the combinatorics and symbolic dynamics of self-coverings. For example, if the self-covering is expanding, then its Julia set is uniquely determined by the iterated monodromy group. The project suggests further investigation of the connections between dynamical systems and automata groups. We hope, in particular, to get new results in the theory of groups acting on rooted trees, groups generated by automata, theory of growth of groups, find new applications of geometric group theory to combinatorics of iterations of rational functions and polynomials. {Second abstract.} The project brings together two branches of Mathematics, which where not so close before: geometric group theory and holomorphic dynamics. Geometric groups theory is a part of algebra which studies large-scale properties of groups of symmetries. Holomorphic dynamics studies iterations of rational and polynomial functions and is a source of complex and beautiful fractal structures such as Mandelbrot and Julia sets. Iterated monodromy groups, introduced by the investigator, are naturally associated with rational iterations and are groups encoding in a compact symbolic form all the complex topological behavior of the iterations and geometry of their Julia sets. The construction of the iterated monodromy group is very natural, but groups obtained in this way are very exotic from the point of view of classical group theory. Difference between classical groups and iterated monodromy 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 of iterated monodromy groups remain to be rather mysterious and a part of the project is to understand them better. We are also hoping that this new connection between group theory and dynamical systems will be fruitful for both parts of mathematics, that we will be able to use group theory to obtain new results in holomorphic dynamics and to apply holomorphic dynamics to study new classes of groups.