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 $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi></math>$ be a closed oriented surface of negative Euler characteristic and $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>M</mi></math>$ a complete contractible Riemannian manifold. A Fuchsian representation $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>j</mi> <mo class="MathClass-punc">:</mo> <msub><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo class="MathClass-open">(</mo><mrow><mi>S</mi></mrow><mo class="MathClass-close">)</mo></mrow> <mo class="MathClass-rel">→</mo><msup><mrow><mo class="qopname"> Isom</mo></mrow><mrow><mo class="MathClass-bin">+</mo></mrow></msup><mrow><mo class="MathClass-open">(</mo><mrow><msup><mrow><mi>ℍ</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mo class="MathClass-close">)</mo></mrow></math>$ strictly dominates a representation $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ρ</mi><mo class="MathClass-punc">:</mo> <msub><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo class="MathClass-open">(</mo><mrow><mi>S</mi></mrow><mo class="MathClass-close">)</mo></mrow> <mo class="MathClass-rel">→</mo><mo class="qopname"> Isom</mo><mrow><mo class="MathClass-open">(</mo><mrow><mi>M</mi></mrow><mo class="MathClass-close">)</mo></mrow></math>$ if there exists a $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mo class="MathClass-open">(</mo><mrow><mi>j</mi><mo class="MathClass-punc">,</mo><mi>ρ</mi></mrow><mo class="MathClass-close">)</mo></mrow></math>$ –equivariant map from $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mrow><mi>ℍ</mi></mrow><mrow><mn>2</mn></mrow></msup></math>$ to $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>M</mi></math>$ that is $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>λ</mi></math>$ –Lipschitz for some $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>λ</mi> <mo class="MathClass-rel">&lt;</mo> <mn>1</mn></math>$ . In a previous paper by Deroin and Tholozan, the authors construct a map $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>Ψ</mi></mrow><mrow><mi>ρ</mi></mrow></msub></math>$ from the Teichmüller space $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="bold-script">T</mi> <mrow><mo class="MathClass-open">(</mo><mrow><mi>S</mi></mrow><mo class="MathClass-close">)</mo></mrow></math>$ of the surface $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi></math>$ to itself and prove that, when $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>M</mi></math>$ has sectional curvature at most $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"> <mo class="MathClass-bin">−</mo> <mn>1</mn></math>$ , the image of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>Ψ</mi></mrow><mrow><mi>ρ</mi></mrow></msub></math>$ lies (almost always) in the domain $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo class="qopname"> Dom</mo><mrow><mo class="MathClass-open">(</mo><mrow><mi>ρ</mi></mrow><mo class="MathClass-close">)</mo></mrow></math>$ of Fuchsian representations strictly dominating $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ρ</mi></math>$ . Here we prove that $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>Ψ</mi></mrow><mrow><mi>ρ</mi></mrow></msub><mo class="MathClass-punc">:</mo> <mi mathvariant="bold-script">T</mi> <mrow><mo class="MathClass-open">(</mo><mrow><mi>S</mi></mrow><mo class="MathClass-close">)</mo></mrow> <mo class="MathClass-rel">→</mo><mo class="qopname"> Dom</mo><mrow><mo class="MathClass-open">(</mo><mrow><mi>ρ</mi></mrow><mo class="MathClass-close">)</mo></mrow></math>$ is a homeomorphism. As a consequence, we are able to describe the topology of the space of pairs of representations $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mo class="MathClass-open">(</mo><mrow><mi>j</mi><mo class="MathClass-punc">,</mo><mi>ρ</mi></mrow><mo class="MathClass-close">)</mo></mrow></math>$ from $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo class="MathClass-open">(</mo><mrow><mi>S</mi></mrow><mo class="MathClass-close">)</mo></mrow></math>$ to $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mrow><mo class="qopname"> Isom</mo></mrow><mrow><mo class="MathClass-bin">+</mo></mrow></msup><mrow><mo class="MathClass-open">(</mo><mrow><msup><mrow><mi>ℍ</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mo class="MathClass-close">)</mo></mrow></math>$ with $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>j</mi></math>$ Fuchsian strictly dominating $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ρ</mi></math>$ . In particular, we obtain that its connected components are classified by the Euler class of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ρ</mi></math>$ . 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 $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>3</mn></math>$ –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)