Rigidity of group actions on homogeneous spaces, III

TL;DR

This paper investigates the rigidity of group actions on homogeneous spaces, showing that measurable morphisms and factors are actually algebraically constrained, and classifies all measurable maps commuting with the lattice action under ergodicity.

Contribution

It provides new algebraic classifications of measurable group actions and commuting maps on homogeneous spaces, extending rigidity results in the context of S-algebraic groups.

Findings

Measurable morphisms are algebraically constrained.

Classified all measurable maps commuting with the lattice action under ergodicity.

Established rigidity results for group actions on homogeneous spaces.

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 apriori 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 they are algebraically defined.