Descriptive Set Theory and Measured Group Theory

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This project aims to address several problems in the areas of descriptive combinatorics and orbit equivalence. These topics lie at the interface of descriptive set theory, measured group theory, graph theory, ergodic theory, probability theory, and operator algebras. In recent years, mathematicians in these fields have come to realize that problems concerning algebraic, dynamical, and descriptive structural complexity of countable group and equivalence relations can be fruitfully studied via descriptive combinatorial and graph theoretic means. The principal investigator and collaborators have employed this combinatorial perspective to create new tools used to answer several open problems in these fields. In addition to a combinatorial perspective, research in these fields is also facilitated by a global perspective, from which problems in ergodic theory, for example, may be seen as topological-dynamical and descriptive problems concerning actions of the group of automorphisms of a standard probability space. This research project aims to generate more new general tools and to promote fruitful interactions among these fields. Measured group theory seeks to understand countably infinite groups through ergodic theoretic properties of their measurable actions on standard probability spaces, and particularly through structural properties of the orbit equivalence relations generated by these actions. The motivating phenomenon is that algebraic properties of an acting group are often expressed through measurable properties of the associated equivalence relation. An extreme form of this phenomenon is seen in orbit equivalence and cocycle rigidity and superrigidity theorems, which state that in certain settings an equivalence relation completely remembers the group from which it was generated. At the other extreme are what might be called orbit equivalence anti-rigidity theorems, stating that certain groups and actions cannot be distinguished by looking at the equivalence relations they generate. This research project touches upon phenomena at both of these extremes. The questions are motivated by rigidity/anti-rigidity results established by the principal investigator and collaborators, which have led to new questions addressed in this project. The project will address problems concerning algebraic, dynamical, and descriptive structural complexity of countable groups and equivalence relations, specifically in the areas of orbit equivalence, treeability, cocycle superrigidity, weak equivalence rigidity, and measurable combinatorial properties and parameters of graphs and group actions. The project pursues a descriptive combinatorial and graph theoretic approach to these problems.