Course info

Prerequisites.
Elementary group theory (groups, homomorphisms, group actions, etc.; see, e.g., Part I of Dummit and Foote’s “Abstract algebra”, or sections 1-5 of lecture notes for Algebra I by Peter
Stevenhagen (in Dutch), https://websites.math.leidenuniv.nl/algebra/algebra1.pdf). Elementary point-set topology (see, e.g., Chapters I-III of Gamelin’s “Introduction to topology”,
or “Topology” by Munkres). The basics of homotopy theory, covering spaces, and fundamental groups will be briefly reviewed during the course, though prior knowledge is strongly
recommended.

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 theory of dynamical systems. An extra emphasis will be put on the algorithmic aspects.

The first part of the course will focus on combinatorial and geometric group theory. In the second part of the course, we will discuss self-similar groups and how they 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) Groups acting on trees (limit spaces, Grigorchuk’s groups, Basilica group, growth and amenability) Dynamics of rational maps (Julia sets, symbolic dynamics, iterated monodromy groups).

Rules about Homework/Exam.

During the course, the students will complete four sets of hand-in assignments (homework). One lowest or missing grade for homework will be dropped when computing the final grade. In addition, each student will make a presentation on one topic from a list suggested by the lecturers. At the end of the semester, there will be a written exam. The final grade for the course is determined by the average homework grade (30%), the grade for the presentation (15%), and the grade for a written (or retake) exam (55%). To pass the course, the grade for the written (or retake) exam must be 5.0 or higher, regardless of your grade for the homework assignments and the presentation.

Lecture Notes/Literature
Mode of instruction. Weekly lectures and tutorials: 2x45 min lectures and 45 min tutorial per week. Assessed homework, assessed presentations, written exam.

Literature.

Roger C. Lyndon and Paul E. Schupp, Combinatorial Group Theory. Springer-Verlag, Berlin, Heidelberg,
New York, 1977.
Clara Löh. Geometric group theory. Springer International Publishing AG, 2017.

• Volodymyr Nekrashevych, Iterated monodromy groups, London Mathematical Society Lecture Note Series, vol. 1, pp. 41–93, 2011.

Volodymyr Nekrashevych. Self-similar groups. No. 117. American Mathematical Society, 2005.
The first two books are available via your university's subscription to SpringerLink. The electronic versions of the other references are also available online.
In addition to these texts, relevant research papers will be used in the preparation of the student presentations. The list of papers will be communicated to students once the list of presentation topics is fixed.

Teaching Assistant. Dean Wardell , PhD student, Leiden University.