#### Volume 21, issue 1 (2017)

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

### Nicolas Tholozan

Geometry & Topology 21 (2017) 193–214
##### Abstract

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 $\mathsc{T}\left(S\right)$ 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 ${\Psi }_{\rho }:\mathsc{T}\left(S\right)\to Dom\left(\rho \right)$ 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.

##### Keywords
anti-de Sitter, representations of surface groups, Teichmüller, harmonic maps, deformation space
##### Mathematical Subject Classification 2010
Primary: 57M50, 58E20
Secondary: 53C50, 32G15