
Journal of Lie Theory 28 (2018), No. 3, 619642 Copyright Heldermann Verlag 2018 The Volume of Complete Antide Sitter 3Manifolds Nicolas Tholozan Dép. de Mathématiques et Applications, Ecole Normale Supérieure, PSL Research University, 45 rue d'Ulm, 75005 Paris, France nicolas.tholozan@ens.fr [Abstractpdf] \def\SO{\mathop{\rm SO}\nolimits} Up to a finite cover, closed antide 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 antide Sitter structure. Our results extend to (not necessarily compact) quotients of $\SO_0(n,1)$ by a ``geometrically finite'' subgroup of $\SO_0(n,1) \times \SO_0(n,1)$. Keywords: Antide Sitter, (G,X)structures, CliffordKlein forms, volume of 3manifolds. MSC: 53C50, 22E40 [ Fulltextpdf (319 KB)] for subscribers only. 