### Group Theory and Dynamical Systems - M1 - 8EC

Prerequisites.

• Basic knowledge of graph theory (see, e.g., Chapter I of Diestel’s “Graph Theory”, http://diestel-graph-theory.com/).
• Elementary group theory (groups, homomorphisms, group actions, etc.; see, e.g., Part I of Dummit and Foote’s “Abstract algebra”).
• Elementary point-set topology (see, e.g., Chapters I-III of Gamelin’s “Introduction to topology”). The basics of homotopy theory, covering spaces, and fundamental groups will be reviewed during the course, though prior knowledge is strongly recommended.
• Familiarity with basic complex analysis (e.g., Riemann Mapping Theorem) is recommended, but not required.

Aim of the course. We will explore various connections between two branches of mathematics: dynamical systems and group theory. The former studies the evolution of complex and chaotic systems, while the latter often focuses on groups of symmetries of certain (geometric or combinatorial) structures. We will discuss how interesting groups of symmetries may be 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. An extra emphasis will be put on the algorithmic aspects.

The first part of the course will focus on combinatorial group theory. Some aspects of mapping class groups and self-similar groups will be discussed in more detail. In the second part of the course, we will discuss how these groups naturally appear in the dynamics of rational maps.

Topics. Here is a tentative list of topics (The exact selection of topics is to be determined based on students' backgrounds and interests).

• Combinatorial group theory (Cayley graphs, Nielsen method, HNN extensions, graphs of groups)
• Mapping class groups (Dehn-Nielsen-Baer theorem, Birman exact sequence, Nielsen-Thurston classification, train tracks)
• Groups acting on trees (limit spaces, Grigorchuk group, Basilica group, growth, and amenability)
• Dynamics of rational maps (Julia sets, symbolic dynamics, iterated monodromy groups, Thurston’s characterization, twisting problems)

Lecturer
The course will be taught by Mikhail Hlushchanka, JUD at Utrecht University.