Locally Homogeneous Aspherical Sasaki Manifolds

Let $G/H$ be a contractible homogeneous Sasaki manifold. A compact locally homogeneous aspherical Sasaki manifold $\Gamma\big\backslash G/H$ is by definition a quotient of $G/H$ by a discrete uniform subgroup $\Gamma\leq G$. We show that a compact locally homogeneous aspherical Sasaki manifold is always quasi-regular, that is, $\Gamma\big\backslash G/H$ is an $S^{1}$-Seifert bundle over a locally homogeneous aspherical K\"ahler orbifold. We discuss the structure of the isometry group $\mathrm{Isom}(G/H)$ for a Sasaki metric of $G/H$ in relation with the pseudo-Hermitian group $\mathrm{Psh} (G/H)$ for the Sasaki structure of $G/H$. We show that a Sasaki Lie group $G$, when $\Gamma\big\backslash G$ is a compact locally homogeneous aspherical Sasaki manifold, is either the universal covering group of $SL(2,R)$ or a modification of a Heisenberg nilpotent Lie group with its natural Sasaki structure. In addition, we classify all aspherical Sasaki homogeneous spaces for semisimple Lie groups.