Logic Seminar

It has been a long-lasting problem, posed by Weiss (1984), whether any Borel action of a countable amenable group on a standard Borel space gives rise to a hyperfinite Borel equivalence relation. This can be equivalently reformuated as whether countable amenable group action in a Borel way gives rise to the orbit equivalence relation being hyperfinite, liminf of a sequence of finite Borel equivalence relations, Borel embeddable into E₀ or Borel reducible to E₀. It is possible to consider these questions in a continuous setting, with the action being continuous, and a space being zero dimensional second countable Hausdorff, to see whether the induced orbit equivalence relation is continuously hyperfinite, liminf of G-clopen finite equivalence relations, continuously embeddable to E₀ or continuously reducible to E₀. We discuss that the one implies the other, but not necessarily the other way around. Also, we discuss the continuous analogue of the Borel asymptotic dimension, discuss its continuous embeddability and show that the shift action of G to 2^G should have infinite continuous asymptotic dimension when G has an element of infinite order. Finally, we discuss the continuous embedding of G=Zⁿ+T into E₀, where T is a torsion abelian group.