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

Nicolas Tholozan

Geometry & Topology 21 (2017) 193–214

Let S be a closed oriented surface of negative Euler characteristic and M a complete contractible Riemannian manifold. A Fuchsian representation j : π1(S) Isom+(2) strictly dominates a representation ρ: π1(S) Isom(M) if there exists a (j,ρ)–equivariant map from 2 to M that is λ–Lipschitz for some λ < 1. In a previous paper by Deroin and Tholozan, the authors construct a map Ψρ from the Teichmüller space T (S) of the surface S to itself and prove that, when M has sectional curvature at most 1, the image of Ψρ lies (almost always) in the domain Dom(ρ) of Fuchsian representations strictly dominating ρ. Here we prove that Ψρ: T (S) Dom(ρ) is a homeomorphism. As a consequence, we are able to describe the topology of the space of pairs of representations (j,ρ) from π1(S) to Isom+(2) with j Fuchsian strictly dominating ρ. In particular, we obtain that its connected components are classified by the Euler class of ρ. 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.

anti-de Sitter, representations of surface groups, Teichmüller, harmonic maps, deformation space
Mathematical Subject Classification 2010
Primary: 57M50, 58E20
Secondary: 53C50, 32G15
Received: 23 September 2014
Revised: 29 January 2016
Accepted: 29 February 2016
Published: 10 February 2017
Proposed: Jean-Pierre Otal
Seconded: Ian Agol, Bruce Kleiner
Nicolas Tholozan
University of Luxembourg
Campus Kirchberg
Mathematics Research Unit, BLG
6, rue Richard Coudenhove-Kalergi