Sharpness of proper and cocompact actions on reductive homogeneous spaces

TL;DR

This paper proves that discrete subgroups acting properly discontinuously and cocompactly on certain reductive homogeneous spaces are quasi-isometrically embedded and sharp, with applications to spectral analysis and classification of such actions.

Contribution

It establishes the sharpness of discrete subgroups acting on reductive homogeneous spaces and characterizes proper actions in real corank one using Anosov representations.

Findings

Discrete subgroups are quasi-isometrically embedded and sharp.

Properly discontinuous and cocompact actions are characterized in real corank one.

Certain homogeneous spaces do not admit compact quotients.

Abstract

We prove that if $G$ is a noncompact connected real reductive linear Lie group, then any discrete subgroup of $G$ acting properly discontinuously and cocompactly on some homogeneous space $G/H$ of $G$ is quasi-isometrically embedded and sharp for $G/H$, 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.