Simplicial volume via foliated simplices and duality

Let $M$ be a triangulated oriented closed connected manifold with universal cover $\widetilde{M}\to M$ and fundamental group $\Gamma=\pi_1(M)$ and consider an essentially free measure preserving action $\Gamma\curvearrowright (X,\mu)$ on a standard Borel probability space. We study the space $\Gamma\backslash(\widetilde{M}\times X)$ equipped with the measured foliation defined by Sauer and the theory of singular foliated simplices in this setting. We define its real singular foliated homology and compare it to classical singular homology. In particular, we construct a foliated fundamental class and prove that its norm coincides with the simplicial volume of $M$, formalizing ideas of Gromov. Passing to the dual chain complex, we define the singular foliated bounded cohomology. When $M$ is aspherical we establish an isometric isomorphism with the measurable bounded cohomology of the action groupoid $\Gamma\curvearrowright X$. As a consequence of a foliated duality principle, we odeduce vanishing criteria for the simplicial volume of $M$ in terms of the vanishing of the measurable bounded cohomology of the action groupoid and/or of its transverse groupoids.