The Volume of complete anti-de Sitter 3-manifolds

Abstract

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 \[j\times ρ(Γ)~,\] 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 ${\mathrm{SO}}_0(n,1)\times {\mathrm{SO}}_0(n,1)$.

References in corpus (2)

• Proper actions on corank-one reductive homogeneous spaces
• Some open questions on anti-de Sitter geometry

Cited by in corpus (7)

• Higgs bundles and geometric structures on manifolds
• Volume and non-existence of compact Clifford-Klein forms
• Strict contractions and exotic SO_0(d,1) quotients
• ADS 3-manifolds and Higgs bundles
• Chern-Simons theory and cohomological invariants of representation varieties
• Almost strict domination and anti-de Sitter 3-manifolds
• Geodesic completeness of some Lorentzian simple Lie groups