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.
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Sergey L. Gefter, Outer Automorphism Group of the Ergodic Equivalence Relation Generated by Translations of Dense Subgroup of Compact Group on its Homogeneous Space. Publ. Res. Inst. Math. Sci. 32 (1996), no. 3, pp. 517–538DOI 10.2977/PRIMS/1195162855