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The BIC of a singular foliation defined by an abelian group of isometries

We study the cohomology properties of the singular foliation $\F$ determined by an action $Φ\colon G \times M\to M$ where the abelian Lie group $G$ preserves a riemannian metric on the compact manifold $M$. More precisely, we prove that the basic intersection cohomology $\lau{\IH}{*}{\per{p}}{\mf}$ is finite dimensional and verifies the Poincaré Duality. This duality includes two well-known situations: -- Poincaré Duality for basic cohomology (the action $Φ$ is almost free). -- Poincaré Duality for intersection cohomology (the group $G$ is compact and connected).