Rigidity of group actions on homogeneous spaces, III

Abstract

Consider homogeneous $G/H$ and $G/F$, for an $S$-algebraic group $G$. A lattice $\Gamma$ acts on the left strictly conservatively. The following rigidity results are obtained: morphisms, factors, and joinings defined a priori only in the measurable category are in fact algebraically constrained. Arguing in an elementary fashion, we manage to classify all the measurable $\Phi$ commuting with the $\Gamma$-action: assuming ergodicity, we find that they are algebraically defined.