Volume and non-existence of compact Clifford-Klein forms

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.