The project proposes to bring some recently developed techniques to the study of countable Borel equivalence relations. These techniques, developed over the last several years, involve new types of marker structures on the equivalence relations. For example, these new structures have led to a proof that all orbit equivalence relations of countable abelian group actions are hyperfinite. An important goal is to extend the results to larger classes of groups, and to delineate the extent of hyperfiniteness. A second new method concerns marker structures on arbitrary countable groups, also referred to as ``blueprints". The bluprints, for example, give a proof that the Bernoulli shift action of every countable group has a free subflow. They have also been used to give other results for general actions, such as results on the complexity of the topological conjugacy relation. Another goal of this project is to explore the connections between the possible blueprints that can exist on groups and the marker structures on the equivalence relations induced by actions of these groups. It is expected that progress along these lines will improve our understanding both of actions by special types of groups, and of the nature of Borel actions for general countable groups.
Countable Borel equivalence relations are fundamental mathematical objects which occur in many mathematical contexts. Aside from their intrinsic interest, their theory interacts with other important areas of mathematics such as dynamics, ergodic theory, and geometric group theory. Thus, work in this area involves techniques from logic as well as dynamics, combinatorics, and other areas. Consider an equivalence relation studied by the ancient Greeks, that of commensurability: two positive real numbers are equivalent if their ratio is rational. A very basic question about this simple relation was not known until recently, namely whether it can be described in an effective way as an increasing union of finite relations. Several natural generalizations of this question are still open. This project seeks to further develop some of the new techniques to further the study of these fundamental questions. It is expected that this study will also make new connections with other areas of mathematics.