Geometric Analysis and Topology Seminar

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On Growth of Iterated Monodromy Groups: A Geometric Point of View

Speaker: Misha Hlushchanka, Jacobs University Bremen

Location: Warren Weaver Hall 517

Date: Friday, May 6, 2016, 11 a.m.

Synopsis:

Iterated monodromy group (IMG) is a self-similar group associated to every branched covering f of the 2-sphere (in particularly to every rational map). It was observed that even very simple maps generate groups with complicated structure and exotic properties which are hard to find among groups defined by more classical methods. For instance, IMG(z^2+i) is a group of intermediate growth and IMG(z^2-1) is an amenable group of exponential growth. Unfortunately, we still face a lack of general theory which would unify and explain these nice examples. In the talk I will first make a detour to the theory of growth of groups and overview the current state of studies of algebraic properties of IMGs. Then I will concentrate on a specific example of a rational map, whose Julia set is given by the whole Riemann sphere, and prove that its IMG has exponential growth. The proof is based on the Geometry of the tilings associated to the map. This is a joint project with Mario Bonk and Daniel Meyer.