MEASURABLE RIGIDITY OF ACTIONSON INFINITE MEASURE HOMOGENEOUS SPACES, II

摘要

The starting point of our discussion is the following beautiful result of Yehuda Shalom and Tim Steger: Theorem 1.1 (Shalom and Steger, [21]). Measurable isomorphisms between linear actions on R~2 of abstractly isomorphic lattices in SL2 (R) are algebraic.More precisely, if τ:Γ_1→Γ_2 is an isomorphism between two lattices in SL_2 (R) and T : R~2 →R~2 is a measure class preserving map with T(γx)=γ~TT(x) for a.e. x ∈ R~2 and all γ ∈ Γ_1￡? then there exists A ∈ GL_2 (R) so that γ~τ = AγA~(-1) for all γ ∈ Γ_1 and T(x) = Ax a.e. on R~2.The linear SL2(R)-action on R2—{0} is G = SL2 (R)-action on the homogeneous space G/H where H is the horocyclic subgroup H = {(01 i) : s E R} .