Sharpness of proper and cocompact actions on reductive homogeneous spaces

Abstract

We prove that if $G$ is any noncompact connected real reductive linear Lie group, and $\Gamma$ any discrete subgroup of $G$ acting properly discontinuously and cocompactly on some homogeneous space $G/H$ of $G$, then $\Gamma$ is quasi-isometrically embedded in $G$ and the action of $\Gamma$ on $G/H$ is sharp, i.e. satisfies a strong, quantitative form of proper discontinuity. For noncompact reductive $H$, this was known as the Sharpness Conjecture, with applications to spectral analysis on pseudo-Riemannian locally symmetric spaces developed in arXiv:1209.4075. For $G/H$ rational of real corank one, we use sharpness to fully characterize properly discontinuous and cocompact actions on $G/H$ in terms of Anosov representations. This enables us to show that in real corank one, acting properly discontinuously and cocompactly on $G/H$ is an open property, and also to prove that a number of homogeneous spaces do not admit compact quotients, such as $\mathrm{SL}(n+1,\mathbb{K})/\mathrm{SL}(n,\mathbb{K})$ for $n>1$ and $\mathbb{K}=\mathbb{R}$, $\mathbb{C}$, or the quaternions.