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

Tholozan, Nicolas

doi:10.2140/gt.2017.21.193

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

Nicolas Tholozan

Geometry & Topology 21 (2017) 193–214

DOI: 10.2140/gt.2017.21.193

Abstract

Let $S$ be a closed oriented surface of negative Euler characteristic and $M$ a complete contractible Riemannian manifold. A Fuchsian representation $j : π_{1} (S) \to {Isom}^{+} (ℍ^{2})$ strictly dominates a representation $ρ : π_{1} (S) \to Isom (M)$ if there exists a $(j, ρ)$ –equivariant map from $ℍ^{2}$ to $M$ that is $λ$ –Lipschitz for some $λ < 1$ . In a previous paper by Deroin and Tholozan, the authors construct a map $Ψ_{ρ}$ 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 $Ψ_{ρ}$ lies (almost always) in the domain $Dom (ρ)$ of Fuchsian representations strictly dominating $ρ$ . Here we prove that $Ψ_{ρ} : T (S) \to Dom (ρ)$ is a homeomorphism. As a consequence, we are able to describe the topology of the space of pairs of representations $(j, ρ)$ from $π_{1} (S)$ to ${Isom}^{+} (ℍ^{2})$ 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.

Keywords

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

Mathematical Subject Classification 2010

Primary: 57M50, 58E20

Secondary: 53C50, 32G15

References

Publication

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

Authors

Nicolas Tholozan
University of Luxembourg Campus Kirchberg Mathematics Research Unit, BLG 6, rue Richard Coudenhove-Kalergi L-1359 Luxembourg
http://math.uni.lu/~tholozan/

Volume 21, issue 1 (2017)