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Almost strict domination and anti-de Sitter 3-manifolds

Nathaniel Sagman

Abstract

We define a condition called almost strict domination for pairs of representations $ρ_1:π_1(S_{g,n})\to \textrm{PSL}(2,\mathbb{R})$, $ρ_2:π_1(S_{g,n})\to G$, where $G$ is the isometry group of a Hadamard manifold $(X,ν)$, and prove it holds if and only if one can find a $(ρ_1,ρ_2)$-equivariant spacelike maximal surface in a certain pseudo-Riemannian manifold, unique up to fixing some parameters. The proof amounts to setting up and solving an interesting variational problem that involves infinite energy harmonic maps. Adapting a construction of Tholozan, we construct all such representations and parametrize the deformation space. When $(X,ν)=(\mathbb{H},σ)$, an almost strictly dominating pair is equivalent to the data of an anti-de Sitter 3-manifold with specific properties. The results on maximal surfaces provide a parametrization of the deformation space of such $3$-manifolds as a union of components in a $\textrm{PSL}(2,\mathbb{R})\times \textrm{PSL}(2,\mathbb{R})$ relative representation variety.

Almost strict domination and anti-de Sitter 3-manifolds

Abstract

We define a condition called almost strict domination for pairs of representations , , where is the isometry group of a Hadamard manifold , and prove it holds if and only if one can find a -equivariant spacelike maximal surface in a certain pseudo-Riemannian manifold, unique up to fixing some parameters. The proof amounts to setting up and solving an interesting variational problem that involves infinite energy harmonic maps. Adapting a construction of Tholozan, we construct all such representations and parametrize the deformation space. When , an almost strictly dominating pair is equivalent to the data of an anti-de Sitter 3-manifold with specific properties. The results on maximal surfaces provide a parametrization of the deformation space of such -manifolds as a union of components in a relative representation variety.

Paper Structure

This paper contains 37 sections, 33 theorems, 162 equations.

Table of Contents

  1. Introduction
  2. On AdS $3$-manifolds
  3. Recent work
  4. Main theorems: maximal surfaces
  5. Main theorems: parametrizations
  6. Main theorems: AdS $3$-manifolds
  7. Future Work
  8. Outline
  9. Acknowledgements
  10. Preliminaries
  11. Teichmüller spaces
  12. Harmonic maps
  13. Infinite energy harmonic maps
  14. Spacelike maximal surfaces
  15. The derivative of the energy functional
  16. ...and 22 more sections

Key Result

Theorem 1.3

A finitely generated discrete group $\Gamma_{\rho_1,\rho_2}\subset \textrm{PSL}(2,\mathbb{R})\times \textrm{PSL}(2,\mathbb{R})$ of the form (form) acts properly discontinuously and without torsion on $\textrm{AdS}^3$ if and only if $\rho_1$ is Fuchsian and strictly dominates $\rho_2$, up to intercha

Theorems & Definitions (79)

  • Remark 1.1
  • Definition 1.2
  • Theorem 1.3: Guéritaud-Kassel, Theorem 1.8 in GK
  • Theorem 1.4: Deroin-Tholozan, Theorem A in DT
  • Theorem 1.5: Tholozan, Theorem 1 in T
  • Definition 1.6
  • Proposition 1.7
  • Definition 1.8
  • Proposition 1.9
  • Remark 1.10
  • ...and 69 more