The complexity of classi?cation problems in ergodic theory

Abstract

theorems in various areas of mathematics. In the last three lectures, we will show how these ideas can be applied in proving a strong non-classication theorem for orbit equivalence. Given a countable group , two free, measure-preserving, ergodic actions of on standard probability spaces are called orbit equivalent if, roughly speaking, they have the same orbit spaces. More precisely this means that there is an isomorphism of the underlying measure spaces that takes the orbits of one action to the orbits of the other. A remarkable result of Dye and Ornstein-Weiss asserts that any two such actions of amenable groups are orbit equivalent. Our goal will be to outline a proof of a dichotomy theorem which states that for any non-amenable group, we have the opposite situation: The structure of its actions up to orbit equivalence is so complex that it is impossible, in a vey strong sense, to classify them (Epstein-Ioana-Kechris-Tsankov). Beyond the method of turbulence, an interesting aspect of this proof is the use of many diverse of tools from ergodic theory. These include: unitary representations and their associated Gaussian actions; rigidity properties of the action of SL2(Z) on the torus and separability arguments (Popa, Gaboriau-Popa, Ioana), Epstein’s co-inducing construction for generating actions of a group from actions of another, quantitative aspects of inclusions of equivalence relations (Ioana-KechrisTsankov) and the use of percolation on Cayley graphs of groups and the theory of costs in proving a measure theoretic analog of the von 1