The overarching theme of this thesis is the study of orbit equivalence as an equivalence relation in the context of Borel reducibility. The thesis is divided in two parts: Chapter 1 is concerned with orbit equivalence among free ergodic actions of the free groups Fn. Building on work G. Hjorth, D. Gaboriau and S. Popa, the main result is that there are E0 many such actions, which not only shows that there are continuum many such actions, but also that orbit equivalence for these actions is not concretely classifiable.;Chapter 2 is concerned with the situation for countable groups with the relative property (T) over an infinite normal subgroup. Building on a Theorem of S. Popa, we show that there are TFA-many orbit inequivalent ergodic actions of such groups. In particular, orbit equivalence for such actions is not a Borel equivalence relation.
展开▼
