A major open problem in descriptive set theory asks whether amenability of a countable Borel equivalence relation implies a stronger finite approximation property known as hyperfiniteness. Although open in general, there have been many recent advances in important cases, producing techniques which have proven to be fruitful in all sorts of other situations. One of the landmark results is due to Gao and Jackson, who showed that every orbit equivalence relation of a countable abelian group is hyperfinite. An orthogonal result due to Dougherty, Jackson and Kechris shows hyperfiniteness for every countable Borel equivalence generated by a single function, i.e. an action of the additive monoid of natural numbers. We prove a common generalization of these results, namely that the orbit equivalence relation of a bounded-to-one action of a commutative monoid is hyperfinite, where an action is said to be bounded-to-one if each function has a uniform finite bound on the size of its fibers. The argument goes through Borel asymptotic dimension, a tool introduced by Conley, Jackson, Marks, Seward and Tucker-Drob to prove hyperfiniteness. This is joint work with Felix Weilacher and Jing Yu.
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In 2004, I conjectured that the quasi-isometry problem for finitely generated groups was strictly harder than the isomorphism problem; but unfortunately I was only able to prove that the quasi-isometry problem was not smooth. In this talk, making essential use of techniques developed by Harvey Friedman and Robin Tucker-Drob, I will show that the quasi-isometry problem does not Borel reduce to an essentially free countable Borel equivalence relation.