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

Abstract

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 + (H 2 ) strictly dominates a representation ρ : π1(S) → 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 DeroinTholozan, 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 ≤ −1, the image of Ψρ lies (almost always) in the domain Dom(ρ) of Fuchsian representations stricly 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 + (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.

AdS 3-manifolds and Higgs bundles

“…Moreover, the conformal structure of this minimal surface is equivalent to the special holomorphic structure on S ′ with vanishing Pfaffian found in [17].…”

“…The latter notion was introduced in [15] and it only depends on the two representations ρ 1 , ρ 2 . The following proposition comes from [17] and [5]. Proof.…”

“…Given representations ρ 1 , ρ 2 such that ρ 1 strictly dominates ρ 2 , the set of holomorphic structures on S such that g 1 > g 2 is an open subset of the Teichmüller space that plays an important role in this work. This open subset has a special point, the unique structure found by Tholozan in [17] such that αβ = γδ (in our terminology, vanishing Pfaffian). In section 6 we will show how to give a geometric interpretation to this structure as the conformal structure of an ρ 1 ⊗ ρ 2 -equivariant minimal surface in the Klein quadric.…”

“…The converse is the only interesting implication. It is proved in [17] that if ρ 1 strictly dominates ρ 2 , there exists a unique holomorphic structure Σ such that the corresponding Higgs bundles have αβ = γδ. In this case, it is proved in [5] that the difference of the pull-back metrics is positive definite.…”

Margulis spacetimes via the arc complex

“…It would be interesting to obtain a parameterization of Adm + (ρ) by some simple combinatorial object in this situation as well. See the recent paper [45] for an approach via harmonic maps.…”