Orbit Inequivalent Measure-Preserving Actions of Non-Amenable Groups

Nils Jacobsen will speak on the following.

Dye's Theorem tells us that all ergodic measure-preserving actions of Z on a standard prob- ability space are orbit equivalent. Ornstein and Weiss extended this statement to include all amenable groups and Hjorth proved that amenability can be characterizes by its orbit structure. In this talk three different cases of measure-preserving actions of non-amenable countable discrete groups on a standard probability space will be presented. These cases are

1. Countably infnite discrete groups with Property T (Hjorth)
2. Free groups of rank two or higher (Törnquist)
3. Countable discrete groups containing a copy of the free group of rank two (Ioana)

In each case we will be able to conclude that the orbit structure is rather rich and complex compared to amenable cases, as each group admits uncountably many orbit inequivalent ergodic measure-preserving actions.