Simple and large equivalence relations (Q2832822)

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: NEWLINE\[NEWLINE\mathcal R_\Gamma=\{(x,y): \exists\gamma\in\Gamma, y=\gamma\cdot x\}.NEWLINE\]NEWLINE The author studies subequivalence relations of such equivalence relations, following the pioneering work of \textit{J. Feldman} et al. [Ergodic Theory Dyn. Syst. 9, No. 2, 239--269 (1989; Zbl 0654.22003)].NEWLINENEWLINEThe 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 NEWLINE\[NEWLINE\Gamma\cdot x=\bigsqcup_{i\in I}\Lambda\cdot (\gamma_i\cdot x).NEWLINE\]NEWLINE 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 NEWLINE\[NEWLINE[x]_{\mathcal R}=\bigsqcup_{i\in I}[\varphi_i(x)]_{\mathcal S}.NEWLINE\]NEWLINE The maps \((\varphi_i)_{i\in I}\) are called \textit{choice functions} for \(\mathcal S\).NEWLINENEWLINEReturning 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 \textit{endomorphism} of \(\mathcal S\), i.e., for all \(i\in I\) we have \((x,y)\in\mathcal S\) implies \((\gamma_i x,\gamma_i y)\in\mathcal S\).NEWLINENEWLINEThis 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 \textit{normal} if there are choice functions \((\varphi_i)_{i\in I}\) for \(\mathcal S\) which are endomorphisms of \(\mathcal S\).NEWLINENEWLINEOne can then construct the \textit{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.NEWLINENEWLINETheorem. 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.NEWLINENEWLINEWhen \(\mathcal S\) 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.NEWLINENEWLINETheorem. There is a measure-preserving ergodic equivalence relation without proper normal ergodic equivalence subrelations and without proper finite index equivalence subrelations.NEWLINENEWLINEIn the opposite direction, the author provides examples of ``large'' equivalence relations.NEWLINENEWLINETheorem. 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.NEWLINENEWLINEAs 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.