Non-treeability for product group actions (Q926431)

This paper continues (and in one direction naturally completes) a series of results on the complexity of the orbit equivalence relation for actions of products of groups, building on earlier work of \textit{S. Adams} [Isr. J. Math. 64, No.~3, 362--380 (1989; Zbl 0678.28010)], \textit{D. Gaboriau} [Invent. Math. 139, No.~1, 41--98 (2000; Zbl 0939.28012)], \textit{A. S. Kechris} [J. Lond. Math. Soc., II. Ser. 62, No.~2, 437--450 (2000; Zbl 1041.20038)], \textit{R. Pemantle} and \textit{Y. Peres} [Isr. J. Math. 118, 147--155 (2000; Zbl 0961.43002)]. The main result here is that if \(G\) and \(H\) are locally compact second countable Hausdorff groups with \(G\times H\) acting measurably by measure-preserving transformations of a standard Borel probability space so that the orbit equivalence relation induced by \(G\) is nonamenable while \(H\) does not act essentially transitively on any ergodic component of its action, then the orbit equivalence relation induced by the product action is not treeable (treeable is a generalization of hyperfiniteness). A consequence is that if \(G\times H\) is nonamenable and acts freely, then the induced orbit equivalence relation is not treeable.