A differentiable classification of certain locally free actions of Lie groups |
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Authors: | Michel Belliart |
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Institution: | 1.Laboratoire Paul Painlevé U.M.R. C.N.R.S. 8524 U.F.R. de Mathématiques,Université Lille I,Villeneuve d’Ascq Cedex,France |
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Abstract: | 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 C r with r ≥ 2. Let R be the radical of L and let χ1, . . ., χ n 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×|χ i | = |χ j |α hold for any α ∈ 0, 1], one shows that Φ 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 Φ-invariant probability measure on {itM}; that measure is induced by a C r?1 nonsingular volume form. What makes that theorem all the more interesting is that certain of the Lie groups under consideration are not amenable. |
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