Volume and non-existence of compact Clifford-Klein forms
Résumé
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 that, in all known examples of compact reductive Clifford-Klein forms that admit deformations, the volume is constant under these deformations. We also derive a new obstruction to the existence of compact Clifford-Klein forms for certain homogeneous spaces. In particular, we obtain that $\mathrm{SO}(2p+1,q+1)/\mathrm{SO}(2p+1,q)$ and $\mathrm{SL}(2m+k,\mathbb{R})/\mathrm{SL}(2m, \mathbb{R})$ do not admit compact quotients.