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Sharpness of proper and cocompact actions on reductive homogeneous spaces

Fanny Kassel, Nicolas Tholozan

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.

Sharpness of proper and cocompact actions on reductive homogeneous spaces

Abstract

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

Paper Structure

This paper contains 43 sections, 35 theorems, 111 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Compact quotients of reductive homogeneous spaces
  3. Sharpness
  4. Application 1: spectral analysis on pseudo-Riemannian locally symmetric spaces
  5. Application 2: openness for proper and cocompact actions
  6. Sharpness and Anosov representations
  7. The case where $\mu(H)$ is a union of walls
  8. Geometric aspects in one example
  9. The general corank-one case
  10. Application 3: nonexistence of compact quotients
  11. Application 4: compact quotients of group manifolds of type $A_2$
  12. Organization of the paper
  13. Acknowledgements
  14. Reminders: Cartan and polar decompositions
  15. Cartan decomposition
  16. ...and 28 more sections

Key Result

Theorem 1.5

Let $G$ be a connected real linear reductive Lie group and $H$ a closed subgroup of $G$. Then the Sharpness Conjecture holds for $G/H$. More precisely, any discrete subgroup $\Gamma$ of $G$ acting properly discontinuously and cocompactly on $G/H$ is finitely generated and sharply embedded in $G$ wit

Theorems & Definitions (81)

  • Definition 1.4: kk16
  • Theorem 1.5
  • Definition 1.6
  • Conjecture 1.7
  • Remark 1.8
  • Theorem 1.9
  • Theorem 1.10
  • Theorem 1.11
  • Corollary 1.12
  • Proposition 1.13
  • ...and 71 more