Volume and non-existence of compact Clifford-Klein forms

Abstract

This article studies the volume of compact quotients of reductive homogeneous spaces. Let $G/H$ be a reductive homogeneous space and $\Gamma$ a discrete subgroup of $G$ acting properly discontinuously and cocompactly on $G/H$. We prove that the volume of $\Gamma \backslash G/H$ is the integral, over a certain homology class of $\Gamma$, of a $G$-invariant form on $G/K$ (where $K$ is a maximal compact subgroup of $G$). As a corollary, we obtain a large class of homogeneous spaces the compact quotients of which have rational volume. For instance, compact quotients of pseudo-Riemannian spaces of constant curvature $-1$ and odd dimension have rational volume. This contrasts with the Riemannian case. We also derive a new obstruction to the existence of compact Clifford--Klein forms for certain homogeneous spaces. In particular, we obtain that $\mathrm{SO}(p,q+1)/\mathrm{SO}(p,q)$ does not admit compact quotients when $p$ is odd, and that $\mathrm{SL}(n,\mathbb{R})/\mathrm{SL}(m,\mathbb{R})$ does not admit compact quotients when $m$ is even.

New version with a lot of improvements. Instead of proving the local rigidity of the volume, we now prove its rationality (in many cases), and we give more examples of homogeneous spaces without compact quotients

References in corpus (4)

• Proper actions on corank-one reductive homogeneous spaces
• The Volume of complete anti-de Sitter 3-manifolds
• A cohomological obstruction to the existence of compact Clifford-Klein forms
• Some open questions on anti-de Sitter geometry

Cited by in corpus (5)

• A cohomological obstruction to the existence of compact Clifford-Klein forms
• Cartan projections of some non-reductive subgroups and proper actions on homogeneous spaces
• On locally homogeneous pseudo-Riemannian compact einstein manifolds
• Chern-Simons theory and cohomological invariants of representation varieties
• Semisimple symmetric spaces that do not model any compact manifold