Geometry & Topology
- Geom. Topol.
- Volume 21, Number 1 (2017), 193-214.
Dominating surface group representations and deforming closed anti-de Sitter $3$–manifolds
Let be a closed oriented surface of negative Euler characteristic and a complete contractible Riemannian manifold. A Fuchsian representation strictly dominates a representation if there exists a –equivariant map from to that is –Lipschitz for some . In a previous paper by Deroin and Tholozan, the authors construct a map from the Teichmüller space of the surface to itself and prove that, when has sectional curvature at most , the image of lies (almost always) in the domain of Fuchsian representations strictly dominating . Here we prove that is a homeomorphism. As a consequence, we are able to describe the topology of the space of pairs of representations from to with Fuchsian strictly dominating . In particular, we obtain that its connected components are classified by the Euler class of . 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 –manifolds.
Geom. Topol., Volume 21, Number 1 (2017), 193-214.
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
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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]
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