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Chirurgies de Dehn sur des variétés CR-sphériques et variétés de caractères pour les formes réelles de SL(n,C)

Abstract : In this thesis, we study the construction and deformation of spherical-CR structures on three dimensional manifolds. In order to do it, we give a detailed description of the complex hyperbolic plane, its group of isometries and some geometric objects attached to this space such as bisectors and extors. We show a surgery theorem which allows to construct spherical-CR on Dehn surgeries of a cusped spherical-CR manifold : this theorem can be applied for the Deraux-Falbel structure on the figure eight knot complement and for Schwartz's and Parker-Will structures on the Whitehead link complement. We also define the character varieties for a real form of SL(n,C) for finitely generated groups as some subsets of the SL(n,C)-character variety invariant under an anti-holomorphic involution. We study in detail the example of the group Z/3Z*Z/3Z. These character varieties give deformation spaces for the holonomy representations of spherical-CR structures. With these deformation spaces and tools related to the visual spheres of a point in CP^2, we construct an explicit deformation of the Ford domain constructed by Parker and Will, which gives a spherical-CR uniformisation of the Whitehead link complement. This deformation provides infinitely many spherical-CR uniformisations of a particular Dehn surgery of the manifold, and spherical-CR unifomisations for infinitely many Dehn surgeries of the Whitehead link complement.
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Submitted on : Wednesday, March 21, 2018 - 11:45:07 AM
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Miguel Acosta. Chirurgies de Dehn sur des variétés CR-sphériques et variétés de caractères pour les formes réelles de SL(n,C). Géométrie métrique [math.MG]. Université Pierre et Marie Curie - Paris VI, 2017. Français. ⟨NNT : 2017PA066368⟩. ⟨tel-01739680⟩



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