Full groups originated from the theory of measurable (and later Cantor)
dynamical systems and their von Neumann-algebra (C*-algebra) crossed-products.
For a given topological dynamical system (X,G), the full group [G] can be broadly defined
as the set of all homeomorphisms of X that act within the G-orbits.
Thus, the full groups can be viewed as a generalized symmetric group of the orbit equivalence
relation of (X,G).
In a series of papers by Giordano-Putnam-Skau, Matui, Medynets, Nekrashevych, and others,
it was shown that full groups (as abstract groups) encode complete information about the underlying
dynamical systems up to (topological) orbit equivalence. In recent years, the development of the theory
of full groups for Cantor minimal systems has been having considerable
impact on geometric group theory
driven primarily by the fact that by tweaking dynamical properties of
the underlying dynamical system
(X,G), we can produce a (countable) full group [G] with new and unusual properties,
which has been successfully used to solve some open problems in geometric group theory.
The goal of our talk is to give a gentle introduction into the theory of full groups,
discuss known results, and present open problems.

Meeting ID: 852 4277 3200
Passcode: 103121