The Volume of complete anti-de Sitter 3-manifolds

Up to a finite cover, closed anti-de Sitter $3$-manifolds are quotients of $\mathrm{SO}_0(2,1)$ by a discrete subgroup of $\mathrm{SO}_0(2,1) \times \mathrm{SO}_0(2,1)$ of the form \[j\times \rho(\Gamma)~,\] where $\Gamma$ is the fundamental group of a closed oriented surface, $j$ a Fuchsian representation and $\rho$ another representation which is "strictly dominated" by $j$. Here we prove that the volume of such a quotient is proportional to the sum of the Euler classes of $j$ and $\rho$. As a consequence, we obtain that this volume is constant under deformation of the anti-de Sitter structure. Our results extend to (not necessarily compact) quotients of $\mathrm{SO}_0(n,1)$ by a discrete subgroup of $\mathrm{SO}_0(n,1) \times \mathrm{SO}_0(n,1)$.