Abelian group actions and hypersmooth equivalence relations

Abstract

We show that a Borel action of a standard Borel group which is isomorphic to a sum of a countable abelian group with a countable sum of real lines and circles induces an orbit equivalence relation which is hypersmooth, i.e., Borel reducible to eventual agreement on sequences of reals, and it follows from this result along with the structure theory for locally compact abelian groups that Borel actions of Polish LCA groups induce orbit equivalence relations which are essentially hyperfinite, extending a result of Gao and Jackson and answering a question of Ding and Gao.