The paper is about non-singular equivalence relations on a standard probability space \((X,\mu)\), which by the Feldman-Moore theorem, always arise from some non-singular action of some countable group \(\Gamma\) on \((X,\mu)\) as the equivalence relation \(\mathcal R_\Gamma\) whose classes are the \(\Gamma\)-orbits: \[ \mathcal R_\Gamma=\{(x,y): \exists\gamma\in\Gamma, y=\gamma\cdot x\}. \] The author studies subequivalence relations of such equivalence relations, following the pioneering work of J. Feldman et al. [Ergodic Theory Dyn. Syst. 9, No. 2, 239–269 (1989; Zbl 0654.22003)].
The basic examples of subequivalence relations come from subgroups: if \(\Lambda\) is a subgroup of \(\Gamma\) and \(\Gamma\) acts on \((X,\mu)\) by non-singular transformations, then \(\mathcal R_\Lambda\) is a subequivalence relation of \(\mathcal R_\Gamma\). Suppose that the \(\Gamma\)-action is moreover free (no non-trivial element fixes a point), then we can enumerate nicely the \(\mathcal R_\Lambda\)-classes as follows: fix \((\gamma_i)_{i\in I}\) such that \(\Gamma=\bigsqcup_{i\in I}\Lambda\gamma_i\), then for each \(x\in X\) the \(\mathcal R_\Gamma\)-class decomposes as \[ \Gamma\cdot x=\bigsqcup_{i\in I}\Lambda\cdot (\gamma_i\cdot x). \] One of the first results about subequivalence relations is that this is actually a general fact: by the above-mentioned work, if \(\mathcal R\) is ergodic non-singular and \(\mathcal S\) is a Borel subequivalence relation of \(\mathcal R\) then there is some countable index set \(I\) and for each \(i\in I\) a Borel map \(\varphi_i: X\to X\) such that for all \(x\in X\) we have \[ [x]_{\mathcal R}=\bigsqcup_{i\in I}[\varphi_i(x)]_{\mathcal S}. \] The maps \((\varphi_i)_{i\in I}\) are called choice functions for \(\mathcal S\).
Returning to the example where \(\Gamma\) acts freely, \(\Lambda\leq\Gamma\) and \(\Gamma=\bigsqcup_{i\in I}\Lambda\gamma_i\), observe that when \(\Lambda\) is normal the \(\gamma_i\)’s have the further property that they are endomorphism of \(\mathcal S\), i.e., for all \(i\in I\) we have \((x,y)\in\mathcal S\) implies \((\gamma_i x,\gamma_i y)\in\mathcal S\).
This motivates the general definition by Feldman, Sutherland and Zimmer of a normal subequivalence relation: a subrelation \(\mathcal S\) of an ergodic equivalence relation \(\mathcal R\) is normal if there are choice functions \((\varphi_i)_{i\in I}\) for \(\mathcal S\) which are endomorphisms of \(\mathcal S\).
One can then construct the quotient groupoid \(\mathcal R/\mathcal S\) whose unit space is the space of ergodic components of \(\mathcal S\). There is a natural groupoid morphism \(\mathcal R\to \mathcal R/\mathcal S\) whose kernel is \(\mathcal S\). Conversely, we have the following result of the author, which removes some technical conditions from a similar result of Feldman, Sutherland and Zimmer.
Theorem. Suppose that \(\mathcal R\) is a non-singular ergodic equivalence relation. Then a subequivalence \(\mathcal S\) of \(\mathcal R\) is normal if and only if it arises as the kernel of a morphism \(\mathcal R\to\mathcal G\) where \(\mathcal G\) is a discrete Borel groupoid.
When \(\mathcal S\) is ergodic, the quotient groupoid \(\mathcal R/\mathcal S\) is a countable group and if \(\mathcal R=\mathcal R_\Gamma\) comes from a free \(\Gamma\)-action, the groupoid morphism \(\mathcal R\to\mathcal R/\mathcal S\) induces a cocycle \(X\times \Gamma\to \mathcal R/\mathcal S\). Using Popa’s cocycle super-rigidity theorem, the author provides examples of “simple” equivalence relations.
Theorem. There is a measure-preserving ergodic equivalence relation without proper normal ergodic equivalence subrelations and without proper finite index equivalence subrelations.
In the opposite direction, the author provides examples of “large” equivalence relations.
Theorem. Let \(\mathcal R\) be an ergodic treeable measure-preserving equivalence relation which is not hyperfinite and admits a primitive ergodic proper subequivalence relation. Then \(\mathcal R\) surjects onto every countable group.
As noted by the author, a result announced by Tucker-Drob guarantees the existence of a primitive ergodic proper subequivalence under the other hypotheses of the theorem, so that every ergodic non-hyperfinite treeable measure-preserving equivalence relations surjects onto every countable group.