The Volume of Complete Anti-de Sitter 3-Manifolds

Nicolas Tholozan¹

¹Dép. de Mathématiques et Applications, Ecole Normale Supérieure, PSL Research University, 45 rue d'Ulm, 75005 Paris, France

Journal of Lie Theory, Tome 28 (2018) no. 3, pp. 619-642

Résumé

\def\SO{\mathop{\rm SO}\nolimits} Up to a finite cover, closed anti-de Sitter $3$-manifolds are quotients of $\SO_0(2,1)$ by a discrete subgroup of $\SO_0(2,1) \times \SO_0(2,1)$ 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 $j$.\par 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 $\SO_0(n,1)$ by a ``geometrically finite'' subgroup of $\SO_0(n,1) \times \SO_0(n,1)$.

Détail

Classification : 53C50, 22E40
Mots-clés : Anti-de Sitter, (G,X)-structures, Clifford-Klein forms, volume of 3-manifolds