  A long standing open problem in the theory of hyperfinite equivalence
relations asks if the orbit equivalence relation generated by a Borel action of
a countable amenable group is hyperfinite. In this paper we show that this
question has a positive answer when the acting group is locally nilpotent. This
extends previous results obtained by Gao-Jackson for abelian groups and by
Jackson-Kechris-Louveau for finitely generated nilpotent-by-finite groups. Our
proof is based on a mixture of coarse geometric properties of locally nilpotent
groups together with an adaptation of the Gao-Jackson machinery.
