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 Φ be a locally free action of L on M having class Cr with r ≥ 2. Let R be the radical of L and let χ1, . . ., χn be the characters of the adjoint action of {itR}. Finally, let Δ be the modular function of R. Under the assumption that none of the identities Δ×|χi| = |χj|α hold for any α ∈ [0, 1], one shows that Φ is the restriction to L of a locally free and transitive Cr action of a larger Lie group. A second result is the existence of a unique Φ-invariant probability measure on {itM}; 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.