Dominating surface group representations and deforming closed anti-de Sitter 3–manifolds

Tholozan, Nicolas¹

¹ University of Luxembourg, Campus Kirchberg, Mathematics Research Unit, BLG, 6, rue Richard Coudenhove-Kalergi, L-1359, Luxembourg

Geometry & topology, Tome 21 (2017) no. 1, pp. 193-214.

Résumé

Let $S$ be a closed oriented surface of negative Euler characteristic and $M$ a complete contractible Riemannian manifold. A Fuchsian representation $j:{\pi }_1\left(S\right)\to {Isom}^{+}\left({ℍ}^2\right)$ strictly dominates a representation $\rho :{\pi }_1\left(S\right)\to Isom\left(M\right)$ if there exists a $\left(j,\rho \right)$–equivariant map from ${ℍ}^2$ to $M$ that is $\lambda$–Lipschitz for some $\lambda <1$. In a previous paper by Deroin and Tholozan, the authors construct a map ${\Psi }_{\rho }$ from the Teichmüller space $\mathcal{T}\left(S\right)$ of the surface $S$ to itself and prove that, when $M$ has sectional curvature at most $-1$, the image of ${\Psi }_{\rho }$ lies (almost always) in the domain $Dom\left(\rho \right)$ of Fuchsian representations strictly dominating $\rho$. Here we prove that ${\Psi }_{\rho }:\mathcal{T}\left(S\right)\to Dom\left(\rho \right)$ is a homeomorphism. As a consequence, we are able to describe the topology of the space of pairs of representations $\left(j,\rho \right)$ from ${\pi }_1\left(S\right)$ to ${Isom}^{+}\left({ℍ}^2\right)$ with $j$ Fuchsian strictly dominating $\rho$. In particular, we obtain that its connected components are classified by the Euler class of $\rho$. The link with anti-de Sitter geometry comes from a theorem of Kassel, stating that those pairs parametrize deformation spaces of anti-de Sitter structures on closed $3$–manifolds.

Détail

DOI : 10.2140/gt.2017.21.193

Classification : 57M50, 58E20, 53C50, 32G15
Keywords: anti-de Sitter, representations of surface groups, Teichmüller, harmonic maps, deformation space

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