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Theses

Espace de Teichmüller du fibré des repères d’une 3-variété hyperbolique réelle

Abstract : The aim of this thesis is to continue and generalize, using the global point of view offered by the fields, the local study made by Ghys concerning the deformations of complex structures of homogeneous spaces of SL2pCq.Ghys' local study of deformations of complex structures of SL2pCq homogeneous spaces.In this paper, the author shows that the deformation of the holonomy of the pSL2pCq ˆ SL2pCq, SL2pCq-structureof a quotient SL2pCq{Γ (where Γ is a discrete subgroup of SL2pCq, co-compact and torsion-free) allowsto construct a family of complex structures on this quotient. More precisely, he shows that theanalytic of the representation variety RpΓq of Γ in SL2pCq, pointed to the trivial morphism, determines theKuranishi space of SL2pCq{Γ. We show that by this same construction, the tautological family overRpΓq remains complete at every point corresponding to an admissible representation, i.e., one that corresponds tothe holonomy of a complete structure. Moreover, Kassel's work on the admissibility of these representationsallow us to state that the set of admissible representations RpΓq a constitutes an open of RpΓq. Notethat SL2pCq acts by conjugation on RpΓq a preserving equivalence classes of complex structures and,that, in general, the non-triviality of SL2pCq-orbits defeats the versa lity criterion of this family. Finally,the computation of the group of C 8-isotopic automorphisms at the identity, which corresponds to the isotropy group of acomplex structure in Teichmüller space, allows us to state that the quotient field RpΓq a { SL2pCq is an open subfield of the Teichmüller fieldopen subfield of the Teichmüller field of SL2pCq{Γ. Finally, let us note that the fibred of the markersof a closed hyperbolic compact variety V of dimension 3 is naturally identified with the quotient of PSL2pCq bya PSL2pCq-faithful and discrete representation of π1pV q. By a result of Thurston, this representation is alwaysalways to SL2pCq and so one can see the quotient SL2pCq{πČ1pV q as a dual covering of the fibred ofbenchmarks of V . This justifies the abundance of such spaces as well as the name given to this thesis
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Submitted on : Thursday, June 9, 2022 - 4:18:10 PM
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Théo Jamin. Espace de Teichmüller du fibré des repères d’une 3-variété hyperbolique réelle. Mathématiques générales [math.GM]. Université d'Angers, 2021. Français. ⟨NNT : 2021ANGE0041⟩. ⟨tel-03692271⟩

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