A differentiable classification of certain locally free actions of Lie groups

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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 Cr with r ≥ 2. Let R be the radical of L and let χ1,...,χ n be the characters of the adjoint action of R. Finally, let Δ be the modular function of R. Under the assumption that none of the identities Δ×{χ i { = {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 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.

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© Hebrew University of Jerusalem 2018

Authors and Affiliations

  1. 1.Laboratoire Paul Painlevé U.M.R. C.N.R.S. 8524, U.F.R. de MathématiquesUniversité Lille IVilleneuve d’Ascq CedexFrance

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