Measurable rigidity of actions on infinite measure homogeneous spaces, II

Abstract

We consider the problems of measurable isomorphisms and joinings, measurable centralizers and quotients for certain classes of ergodic group actions on infinite measure spaces. Our main focus is on systems of algebraic origin: actions of lattices and other discrete subgroups Γ > G \Gamma >G on homogeneous spaces G / H G/H where H H is a sufficiently rich unimodular subgroup in a semi-simple group G G . We also consider actions of discrete groups of isometries Γ > I s o m ( X ) \Gamma >\mathrm {Isom}(X) of a pinched negative curvature space X X , acting on the space of horospheres H o r ( X ) \mathrm {Hor}(X) . For such systems we prove that the only measurable isomorphisms, joinings, quotients, etc., are the obvious algebraic (or geometric) ones. This work was inspired by the previous work of Shalom and Steger but uses completely different techniques which lead to more general results.