The talk will be about the Polish group of all measure preserving transformations. The behavior of a measure preserving transformation, even a generic one, is highly non-uniform. 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 Polish group of all Lebesgue measurable functions from $[0,1]$ to the circle.

In fact, Glasner and Weiss asked if this is the case.

I will describe the background touched on above. I will indicate 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 {\bf not} topologically isomorphic to $L^0$. The proof rests on an analysis of unitary representations of the non-locally compact group $L^0$.