Amenability, countable equivalence relations, and their full groups

This thesis consists of an introduction and four independent chapters.;In Chapter 1, we study homeomorphism groups of metrizable compactifications of the natural numbers. Those groups can be represented as almost zero-dimensional Polishable subgroups of the group Sinfinity. We show that all Polish groups are continuous homomorphic images of almost zero-dimensional Polishable subgroups of Sinfinity. We also find a sufficient condition for these groups to be one dimensional.;In Chapter 2, we study the connections between properties of the action of a countable group Gamma on a countable set X and the ergodic theoretic properties of the corresponding shift action of Gamma on MX, where M is a measure space. In particular, we show that the action of Gamma on X is amenable iff the corresponding shift has almost invariant sets. This is joint work with Alexander Kechris.;In Chapter 3, we prove that if the Koopman representation associated to a measure-preserving action of a countable group on a standard non-atomic probability space is non-amenable, then there does not exist a countable-to-one Borel homomorphism from its orbit equivalence relation to the orbit equivalence relation of any modular action (i.e., an action on the boundary of a countably splitting tree), generalizing previous results of Hjorth and Kechris. As an application, for certain groups, we connect antimodularity to mixing conditions. This is joint work with Inessa Epstein.;In Chapter 4, we study full groups of countable, measure-preserving equivalence relations. Our main results include that they are all homeomorphic to the separable Hilbert space and that every homomorphism from an ergodic full group to a separable group is continuous. We also find bounds for the minimal number of generators of a dense subgroup of full groups allowing us to distinguish full groups of equivalence relations generated by free, ergodic actions of the free groups Fn and F m if m and n are sufficiently far apart. We also show that an ergodic equivalence relation is generated by an action of a finitely generated group iff its full group has a finitely generated dense subgroup. This is joint work with John Kittrell.