Equivalence Relations, Symbolic Dynamics, and Descriptive Set Theory

本项目拟将新近发展的一些技术引入可数Borel等价关系的研究中。这些技术是近年来发展起来的，涉及到等价关系上的新型标记结构。例如，这些新结构证明了可数阿贝尔群作用的所有轨道等价关系都是超有限的。一个重要的目标是将结果扩展到更大的群类，并描述超有限的程度。第二种新方法涉及任意可数群上的标记结构，也称为“蓝图”。例如，这些蓝图证明了每一个可数群的伯努利位移作用都有一个自由子流。它们也被用来给出一般作用的其他结果，例如拓扑共轭关系的复杂性的结果。本项目的另一个目标是探索群上可能存在的蓝图与这些群的作用所引起的等价关系上的标记结构之间的联系。预期沿着这些方向的进展将提高我们对特殊类型群的作用和一般可数群的Borel作用性质的理解。可数Borel等价关系是在许多数学环境中出现的基本数学对象。除了他们的内在兴趣，他们的理论与其他重要的数学领域，如动力学，遍历理论和几何群论相互作用。因此，这个领域的工作涉及到逻辑、动力学、组合学和其他领域的技术。考虑古希腊人研究的等价关系，即可通约性：如果两个正实数的比值是有理数，则它们是等价的。关于这个简单关系的一个非常基本的问题直到最近才为人所知，即它是否可以有效地描述为有限关系的递增联合。这个问题的几个自然的概括仍然是开放的。该项目旨在进一步开发一些新技术，以进一步研究这些基本问题。预计这项研究也将与其他数学领域建立新的联系。

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.