Ergodic actions of semisimple groups and product relations
The author shows that if \(\Gamma\) is a lattice in a connected simple Lie group with finite center and if S is an essentially free ergodic \(\Gamma\)-space with finite invariant measure then the \(\Gamma\)-action is not orbit equivalent to a product of two ergodic actions in a nontrivial way; that is, if \(\Gamma_ 1\) and \(\Gamma_ 2\) are countable (discrete) groups acting ergodically on measure spaces \(S_ 1\) and \(S_ 2\) respectively such that the \(\Gamma\)-action on S is orbit equivalent to the product action of \(\Gamma_ 1\times \Gamma_ 2\) on \(S_ 1\times S_ 2\) then either \(S_ 1\) or \(S_ 2\) is finite (modulo null sets). In fact, the assertion is upheld under the weaker condition of 'stable orbit equivalence'. Similar results are also proved for irreducible actions of connected semisimple Lie groups with finite center and also for equivalence relations defined by foliations in which the leaves have negative curvature (under other conditions analogous to the above); in fact, the result for lattices is deduced by considering the induced action of the ambient Lie group.
scientific article; zbMATH DE number 3867619