Journal of Lie Theory

no. 3
The Volume of Complete Anti-de Sitter 3-Manifolds
Journal of Lie Theory, Volume 28 (2018) no. 3, pp. 619-642
\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 $\SO_0(n,1) \times \SO_0(n,1)$.
DOI: 10.5802/jolt.1017
Classification: 53C50, 22E40
Keywords: Anti-de Sitter, (G,X)-structures, Clifford-Klein forms, volume of 3-manifolds 
@article{JOLT_2018_28_3_a1,
     author = {N. Tholozan},
     title = {The {Volume} of {Complete} {Anti-de} {Sitter} {3-Manifolds}},
     journal = {Journal of Lie Theory},
     pages = {619--642},
     year = {2018},
     volume = {28},
     number = {3},
     doi = {10.5802/jolt.1017},
     zbl = {1403.53059},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.1017/}
}
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N. Tholozan. The Volume of Complete Anti-de Sitter 3-Manifolds. Journal of Lie Theory, Volume 28 (2018) no. 3, pp. 619-642. doi: 10.5802/jolt.1017

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