Countable Borel treeable equivalence relations are classifiable by $\ell_1$

Abstract

Gao and Jackson showed that any countable Borel equivalence relation (CBER) induced by a countable abelian Polish group is hyperfinite. This prompted Hjorth to ask if this is in fact true for all CBERs classifiable by (uncountable) abelian Polish groups. We describe reductions involving free Banach spaces to show that every treeable CBER is classifiable by an abelian Polish group. As there exist treeable CBERs that are not hyperfinite, this answers Hjorth's question in the negative. On the other hand, we show that any CBER classifiable by a countable product of locally compact abelian Polish groups (such as $\mathbb{R}^\omega$) is indeed hyperfinite. We use a small fragment of the Hjorth analysis of Polish group actions, which is Hjorth's generalization of the Scott analysis of countable structures to Polish group actions.