Title: Existence of a global fundamental solution for Hörmander operators

Abstract: We prove that for a simply connected manifold $M$ - and a vast class of non-simply connected manifolds - the existence of a fundamental solution for a differential operator $\mathcal L=\sum_{\alpha\in\mathbb N^q} r_\alpha\cdot X^\alpha$ of finite degree over $M$, follows via a saturation method from the existence of a fundamental solution for the associated "lifted" operator over a group $G$. Given some smooth complete vector fields $X_1,\dots, X_q$ on $M$, suppose that they generate a finite dimensional Lie algebra $\mathfrak g$ satisfying the H\"ormander's condition. The simply connected Lie group $G$ is the unique such that $\mbox{Lie}(G)\cong\mathfrak g$. It has the property that a right $G$-action exists over $M$, faithful and transitive, inducing a natural projection $E\colon G\to M$. We generalize an approach developed by Biagi and Bonfiglioli. In particular we represent the group $G$ as a direct product $M\times G^z$ where the model fiber $G^z$ has a group structure. Our approach doesn't need any nilpotency hypothesis on the group $G$, therefore it broadens the spectrum of cases where techniques of the kind "lifting and approximation", as those introduce by the works of Rothschild, Stein, Goodman, can be applied.