Abstract. Riesz transforms appear naturally in harmonic analysis, even for harmonic analysis on Riemannian manifolds. Indeed, the Ricci curvature is important when it comes to boundedness of the transforms as operators on L_p. In this talk we study noncommutative Riesz transforms in the context of discrete groups and Fourier multipliers. Motivated by the analytical properties of these transforms, we can define some notion of cotangent space. We will show that the space of derivatives is topologically complemented in the cotangent space. As a by product we obtain differentiable structures for finite discrete groups.