Caltech/UCLA Logic Seminar

A well known and long-standing open problem in the theory of Borel equivalence relations asks if the orbit equivalence relation generated by a Borel action of a countable amenable group is hyperfinite. Previous progress on this problem has been confined to groups possessing coarse euclidean geometry and polynomial volume growth (ultimately leading to a positive answer for groups that are either virtually nilpotent or locally nilpotent). In this talk I will discuss the coarse geometric notion of asymptotic dimension and its recently discovered applications to this problem. Relying upon the framework of asymptotic dimension, it is possible to both significantly simplify the proofs of prior results and uncover the first examples of solvable groups of exponential volume growth all of whose Borel actions generate hyperfinite equivalence relations. This is joint work with Clinton Conley, Steve Jackson, Andrew Marks, and Robin Tucker-Drob.