Geometry & Topology

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

Nicolas Tholozan

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Abstract

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.

Article information

Source
Geom. Topol., Volume 21, Number 1 (2017), 193-214.

Dates
Received: 23 September 2014
Revised: 29 January 2016
Accepted: 29 February 2016
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1510859132

Digital Object Identifier
doi:10.2140/gt.2017.21.193

Mathematical Reviews number (MathSciNet)
MR3608712

Zentralblatt MATH identifier
1361.30067

Subjects
Primary: 57M50: Geometric structures on low-dimensional manifolds 58E20: Harmonic maps [See also 53C43], etc.
Secondary: 53C50: Lorentz manifolds, manifolds with indefinite metrics 32G15: Moduli of Riemann surfaces, Teichmüller theory [See also 14H15, 30Fxx]

Keywords
anti-de Sitter representations of surface groups Teichmüller harmonic maps deformation space

Citation

Tholozan, Nicolas. Dominating surface group representations and deforming closed anti-de Sitter $3$–manifolds. Geom. Topol. 21 (2017), no. 1, 193--214. doi:10.2140/gt.2017.21.193. https://projecteuclid.org/euclid.gt/1510859132


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