Groups and Topological Dynamics

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This project explores newly discovered connections between two branches of Mathematics: the theory of dynamical systems and group theory. The former studies behavior of complex and chaotic systems, while the latter usually studies groups of symmetries of (for example geometric) structures. It has become clear that interesting groups of symmetries are naturally associated with many chaotic dynamical systems. This connection makes it possible to answer some longstanding open questions in group theory, as well as in the theory of dynamical systems. For example, it was shown that dynamical systems of low complexity can be used to construct simple groups of sub-exponential growth, the existence of which was an open question for 30 years. The principal investigator plans to use the methods of the theory of chaotic dynamical systems to solve problems in group theory and to use group theory to better understand dynamical systems. A direction which will be explored in this project is the study of relations between different measures of complexity of dynamical systems and groups: rates of growth, entropy, pattern complexity, "finite type" conditions, etc.This project will study groups naturally associated with dynamical systems: topological full groups of dynamical systems and etale groupoids, iterated monodromy groups of self-coverings of topological spaces, holonomy groups of laminations, etc. They are often exotic from the point of view of "classical" group theory and are good sources of interesting examples, especially from the point of view of asymptotic properties of groups (for example amenability and growth). New topological and dynamical methods have proven to be effective tools in the study of such groups. For instance, almost all currently known examples of non-elementary amenable groups are constructed and studied using topological dynamical systems. This research program aims to shed new light on several group theoretic questions using topological dynamics and etale groupoids, and to work on some long-standing open questions (for example, on possible slow intermediate word growth, amenability of groups, new examples of torsion groups, constructing finitely presented groups with exotic properties, studying growth of iterated monodromy groups of quadratic polynomials, etc.). The PI also anticipate that group theory will be effective in studying problems in dynamical systems.