## Logic Seminar

A deep result of Gao-Jackson is that orbit equivalence relations induced by Borel actions of countable discrete abelian groups on Polish spaces are hyperfinite. Hjorth asked if indeed any orbit equivalence relation induced by a Borel action of an abelian Polish group on a Polish space, which is also essentially countable, must be essentially hyperfinite. We show that any countable Borel equivalence relation (CBER) which is treeable must be classifiable by an abelian Polish group. As the free part of the Bernoulli shift action of $F_2$ is a treeable CBER, and not hyperfinite, this answers Hjorth’s question in the negative.

On the other hand, for certain abelian Polish groups such as $\mathbb{R}^\omega$, Hjorth’s question has a positive answer. Indeed, we show that any orbit equivalence relation induced by a Borel action of a countable product of locally compact abelian Polish groups which is also potentially $\mathbf{\Pi}^0_3$ must be Borel-reducible to $E_0^\omega$. By a dichotomy result of Hjorth-Kechris, this implies that essentially countable such orbit equivalence relations are hyperfinite. This uses a result of Cotton that locally compact abelian Polish groups yield essentially hyperfinite orbit equivalence relations, as well as the Hjorth analysis of Polish group actions.