  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.
