Outer Automorphism Group of the Ergodic Equivalence Relation Generated by Translations of Dense Subgroup of Compact Group on its Homogeneous Space

We study the outer automorphism group Out_Rr_ of the ergodic equivalence relation_Rr_ generated by the action of a lattice Γ in a semisimple Lie group on the homogeneos space of a compact group K . It is shown that Out_Rr_ is locally compact. If K is a connected simple Lie group, we prove the compactness of 0ut_Rr_ using the D. Witte's rigidity theorem. Moreover, an example of an equivalence relation without outer automorphisms is constructed.