Actions of a countable group on the Cantor set or a standard measure space naturally give rise to larger groups of transformations of the space which act piecewise via elements of the original acting group. These “full groups” come with a Polish topology which may be discrete (in the topological setting) or non-locally-compact (in the measure-theoretic setting). They are closely related to both orbit equivalence theory and operator algebras and have revealed themselves over the last fifteen years to be a rich source of novel phenomena at the abstract group-theoretic level.

This introductory seminar will treat both topological and measurable full groups, with an eye towards properties such as simplicity, finite generation, amenability, and nonamenability and their relation to the underlying dynamics.