A DIFFERENTIABLE CLASSIFICATION OF CERTAIN LOCALLY FREE ACTIONS OF LIE GROUPS

Let L be a Lie group and let M be a compact manifold with dimension dim(L) + 1. Let Phi be a locally free action of L on M having class C-r with r >= 2. Let R be the radical of L and let chi(1), . . ., chi (n) be the characters of the adjoint action of R. Finally, let Delta be the modular function of R. Under the assumption that none of the identities Delta x vertical bar chi(i)vertical bar = vertical bar chi(j)vertical bar(alpha) hold for any alpha is an element of[0, 1], one shows that Phi is the restriction to L of a locally free and transitive C-r action of a larger Lie group. A second result is the existence of a unique Phi-invariant probability measure on M; that measure is induced by a Cr-1 nonsingular volume form. What makes that theorem all the more interesting is that certain of the Lie groups under consideration are not amenable.