Gromov's asymptotic dimension has beenrecently extended to the Borel setting and used to prove new results
on the problem of determining when Borel equivalencerelations are hyperfinite. I will review Gromov's definition and
the new one and explain how these are used to provide newclasses of examples of amenable groups that have the property
that their Borel actions define hyperfinite equivalencerelations. The talk is based on a recent preprint
by C. Conley, S. Jackson, A. Marks, B. Seward and R.Tucker-Drob with the title above.