Treeable equivalence relations (Q2909620)

An equivalence relation \(E\) on a standard Borel space \(X\) is countable if all equivalence classes are countable. It is treeable if there is a Borel acyclic graph whose connected components are the \(E\)-classes. If \(X\) carries a standard Borel probability measure \(\mu\), \(E\) is measure-preserving if for each Borel bijection \(f\) between Borel sets \(A\) and \(B\) with \(f(x) E x\) for all \(x \in A\), we have \(\mu (A) = \mu (B)\). \(E\) is ergodic if any Borel \(E\)-invariant set is either null or conull. Given two equivalence relations \(E\) and \(F\) on standard Borel spaces \(X\) and \(Y\), \(E\) is Borel-reducible to \(F\) if there is a Borel function \(\theta : X \to Y\) such that \(x_1 E x_2\) iff \(\theta (x_1) F \theta (x_2)\) holds for all \(x_1, x_2 \in X\).NEWLINENEWLINEThe author shows that for \(n \geq 2\), there are standard Borel probability spaces \((X_s, \mu_s)_{s \in {\mathbb R}}\), each equipped with a free measure-preserving ergodic action \(a_s\) of the free group \({\mathbb F}_n\) on \(n\) generators, such that for any \(s \neq t\), the orbit equivalence relation \(E_{a_s}\) is not Borel-reducible to \(E_{a_t}\). In particular, there are continuum many treeable countable Borel equivalence relations which are incomparable with respect to Borel reducibility. This should be contrasted to the well-known result that there are only countably many smooth countable Borel equivalence relations and one which is hyperfinite and non-smooth. The existence of continuum many incomparable countable Borel equivalence relations was established earlier by \textit{S. Adams} and \textit{A. S. Kechris} [J. Am. Math. Soc. 13, No. 4, 903--943 (2000; Zbl 0952.03057)] but their examples are not treeable.