Locally Homogeneous Aspherical Sasaki Manifolds.

Abstract

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.