The talk will be focused on the study of topological properties of the following "large" groups: the group Aut(X,m) of all non-singular automorphisms of a standard measure space (X,m), the group Aut(X) of a standard Borel space X, and the group H(Y) of all homeomorphisms of a Cantor set Y.
There are two well known topologies on Aut(X,m), the uniform and weak topologies, which were defined and studied by P.Halmos in 1940's. We define their analogs in the context of Borel and Cantor dynamics. We show that many results, proved in ergodic theory, have their counterparts for the groups Aut(X) and H(Y). In particular, we discuss the Rokhlin lemma, periodic approximation, density, genericity, and closures of various classes of transformations.