Closed groups generated by generic measure preserving transformations

The behavior of a measure preserving transformation, even a generic one, is highly nonuniform. In contrast to this observation, a different picture of a very uniform behavior of the closed group generated by a generic measure preserving transformation T has emerged. This picture included substantial evidence that pointed to these groups (for a generic T) being all topologically isomorphic to a single group, namely, L^0 -- the topological group of all Lebesgue measurable functions from [0,1] to the circle. In fact, Glasner and Weiss asked if this is the case.

We will describe the background touched on above (including all the relevant definitions) and outline a proof of the following theorem that answers the Glasner--Weiss question in the negative: for a generic measure preserving transformation T, the closed group generated by T is not topologically isomorphic to L^0. The proof rests on an analysis of unitary representations of L^0.