AMENABLE SUBGROUPS OF SEMI-SIMPLE GROUPS AND PROXIMAL FLOWS

We present a classification of maximal amenable subgroups of a semi-simple group G. The result is that modulo a technical connectivity condition, there are precisely 2′ conjugacy classes of such subgroups of G and we shall describe them explicitly. Here l is the split rank of the group G. These groups are the isotropy groups of the action of G on the Satake-Furstenberg compactification of the associated symmetric space and our results give necessary and sufficient conditions for a subgroup to have a fixed point in this compactification. We also study the action of G on the set of all measures on its maximal boundary. One consequence of this is a proof that the algebraic hull of an amenable subgroup of a linear group is amenable. © 1979 The Weizmann Science Press of Israel.