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Sur les triangulations des structures CR-sphériques

Abstract : Thurston gives a construction of the hyperbolic structure on the complement of the figure-eight knot by gluing together two regular tetrahedra. Falbel extends this method in spherical CR geometry. In doing so, he endows the complement of the figure-eight knot with a branched spherical CR structure. This approach uses the resolution of polynomial equations whose unknowns are invariants characterizing the tetrahedra. By solving these equations, we construct representations of fundamental groups in PU(2,1) for non-compact manifolds. In the real case, the rigidity of hyperbolic structures is ensured by Mostow's theorem,  while some representations of compact spherical CR manifolds admit deformations. The calculation of the rank of the system of the equations previously mentioned allows us to conclude for the rigidity of a triangulated spherical CR structure as soon as it exists . For the representations we have constructed, the rank of the system is always maximal. In the general case, we give some lower bound for the rank. In an independent part, we study the trace field for subgroups of SU(n,1). We prove that for a Zariski dense group G in SU(2,1) that contains a parabolic transformation, up to conjugation, its trace field is exactly the field generated by the coefficients of its matrices.
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Contributor : Juliette Genzmer <>
Submitted on : Tuesday, July 13, 2010 - 6:58:21 PM
Last modification on : Wednesday, December 9, 2020 - 3:10:16 PM
Long-term archiving on: : Thursday, October 14, 2010 - 3:45:58 PM



  • HAL Id : tel-00502363, version 1


Juliette Genzmer. Sur les triangulations des structures CR-sphériques. Mathématiques [math]. Université Pierre et Marie Curie - Paris VI, 2010. Français. ⟨tel-00502363⟩



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