Geometric Structures and Triangulations

Our initial motivation was to unify the theory of spherical CR-structures and the theory of hyperbolic structures on 3-manifolds, by viewing them in the common framework of so-called flag structures. In terms of holonomy representations, which take values in \(\mathrm{SL}(2,\mathbb{C})\) and \(SU(2,1)\) for hyperbolic and spherical CR-structures respectively, we view these two groups as subgroups of \(SL(3,\mathbb{C})\).

We intend to study representations of 3-manifold groups into \(SL(3,\mathbb{C})\) that come from geometric structures on 3-manifolds. More specifically, we study representations that come from a triangulation of the 3-manifold. With a suitable interpretation, one can think of both hyperbolic and spherical CR structures as being described by associating to each vertex of the tetrahedra in the triangulation a flag, i.e. a point in the complex projective plane together with a projective line that contains that point. We shall refer to such a structure as a flag-structure on the 3- manifold. These are parametrized by the invariants of quadruples of flags. The gluing of tetrahedra in the triangulation imposes certain relations between the invariants, which can be expressed as a system of algebraic equations.

One goal is to develop efficient methods to solve these compatibility equations, and to develop a computer program that finds all solutions and computes their invariants. Eventually, we intend to study the existence and rigidity properties of the solutions, and to give a geometric interpretation of the results.

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Related publications

  • [DOI] E. Falbel, A. Guilloux, P. Koseleff, F. Rouillier, and M. Thistlethwaite, “Character Varieties For SL(3,C): The Figure Eight Knot,” Experimental Mathematics, vol. 25, iss. 2, p. 17, 2016.
    [Bibtex]
    @article{falbel:hal-01362208,
    TITLE = {{Character Varieties For SL(3,C): The Figure Eight Knot}},
    AUTHOR = {Falbel, Elisha and Guilloux, Antonin and Koseleff, Pierre-Vincent and Rouillier, Fabrice and Thistlethwaite, Morwen},
    URL = {https://hal.inria.fr/hal-01362208},
    JOURNAL = {{Experimental Mathematics}},
    PUBLISHER = {{Taylor \& Francis}},
    VOLUME = {25},
    NUMBER = {2},
    PAGES = {17},
    YEAR = {2016},
    DOI = {10.1080/10586458.2015.1068249},
    KEYWORDS = { figure eight knot ; character varieties ; decorated representations ; },
    HAL_ID = {hal-01362208},
    HAL_VERSION = {v1}
    }
  • [DOI] E. Falbel and J. M. Veloso, “Diffeomorphisms preserving R-circles in three dimensional CR manifolds,” Mathematical News / Mathematische Nachrichten, vol. 289, iss. 8-9, pp. 962-973, 2016.
    [Bibtex]
    @article{falbel:hal-01374792,
    TITLE = {{Diffeomorphisms preserving R-circles in three dimensional CR manifolds}},
    AUTHOR = {Falbel, Elisha and Veloso, Jose M.},
    URL = {https://hal.archives-ouvertes.fr/hal-01374792},
    JOURNAL = {{Mathematical News / Mathematische Nachrichten}},
    PUBLISHER = {{Wiley-VCH Verlag}},
    VOLUME = {289},
    NUMBER = {8-9},
    PAGES = {962-973},
    YEAR = {2016},
    DOI = {10.1002/mana.201400301},
    HAL_ID = {hal-01374792},
    HAL_VERSION = {v1},
    }
  • E. Falbel and Q. Wang, “Duality and invariants of representations of fundamental groups of 3-manifolds into PGL(3,C),” Bulletin of the London Mathematical Society, 2016.
    [Bibtex]
    @article{falbel:hal-01374795,
    TITLE = {{Duality and invariants of representations of fundamental groups of 3-manifolds into PGL(3,C)}},
    AUTHOR = {Falbel, Elisha and Wang, Qingxue},
    URL = {https://hal.archives-ouvertes.fr/hal-01374795},
    JOURNAL = {{Bulletin of the London Mathematical Society}},
    PUBLISHER = {{London Mathematical Society}},
    YEAR = {2016},
    HAL_ID = {hal-01374795},
    HAL_VERSION = {v1},
    }
  • [DOI] E. Falbel, P. Koseleff, and F. Rouillier, “Representations of fundamental groups of 3-manifolds into PGL(3,C): Exact computations in low complexity,” Geometriae Dedicata, vol. 177, iss. 1, p. 52, 2015.
    [Bibtex]
    @article{falbel:hal-00908843,
    TITLE = {{Representations of fundamental groups of 3-manifolds into PGL(3,C): Exact computations in low complexity}},
    AUTHOR = {Falbel, Elisha and Koseleff, Pierre-Vincent and Rouillier, Fabrice},
    URL = {https://hal.inria.fr/hal-00908843},
    JOURNAL = {{Geometriae Dedicata}},
    PUBLISHER = {{Springer Verlag}},
    VOLUME = {177},
    NUMBER = {1},
    PAGES = {52},
    YEAR = {2015},
    MONTH = Aug,
    DOI = {10.1007/s10711-014-9987-x},
    HAL_ID = {hal-00908843},
    HAL_VERSION = {v1},
    }
  • [PDF] [DOI] M. Deraux and E. Falbel, “Complex hyperbolic geometry of the figure eight knot,” Geometry and Topology, vol. 19, p. 237–293, 2015.
    [Bibtex]
    @article{deraux:hal-00805427,
    TITLE = {{Complex hyperbolic geometry of the figure eight knot}},
    AUTHOR = {Deraux, Martin and Falbel, Elisha},
    URL = {https://hal.archives-ouvertes.fr/hal-00805427},
    JOURNAL = {{Geometry and Topology}},
    HAL_LOCAL_REFERENCE = {IF\_PREPUB},
    PUBLISHER = {{Mathematical Sciences Publishers}},
    VOLUME = {19},
    PAGES = {237--293},
    YEAR = {2015},
    MONTH = Feb,
    DOI = {10.2140/gt.2015.19.237},
    PDF = {https://hal.archives-ouvertes.fr/hal-00805427/file/figure8.pdf},
    HAL_ID = {hal-00805427},
    HAL_VERSION = {v2},
    }
  • [PDF] E. Falbel and R. Santos Thebaldi, “A Flag structure on a cusped hyperbolic 3-manifold with unipotent holonomy.,” Pacific Journal of Mathematics, vol. 278, iss. 1, pp. 51-78, 2015.
    [Bibtex]
    @article{falbel:hal-00958255,
    TITLE = {{A Flag structure on a cusped hyperbolic 3-manifold with unipotent holonomy.}},
    AUTHOR = {Falbel, Elisha and Santos Thebaldi, Rafael},
    URL = {https://hal.archives-ouvertes.fr/hal-00958255},
    JOURNAL = {{Pacific Journal of Mathematics}},
    PUBLISHER = {{Mathematical Sciences Publishers}},
    VOLUME = {278},
    NUMBER = {1},
    PAGES = {51-78},
    YEAR = {2015},
    KEYWORDS = {SL(3 ; cusped 3-manifold ; Geometric structure ; Flag structure ; R)},
    PDF = {https://hal.archives-ouvertes.fr/hal-00958255/file/tetraprojective.pdf},
    HAL_ID = {hal-00958255},
    HAL_VERSION = {v1},
    }
  • [PDF] A. Guilloux, “Représentations et Structures Géométriques,” Accreditation to supervise research PhD Thesis, 2015.
    [Bibtex]
    @phdthesis{guilloux:tel-01370296,
    TITLE = {{Repr{\'e}sentations et Structures G{\'e}om{\'e}triques}},
    AUTHOR = {Guilloux, Antonin},
    URL = {https://hal.archives-ouvertes.fr/tel-01370296},
    SCHOOL = {{UPMC - Universit{\'e} Paris 6 Pierre et Marie Curie}},
    YEAR = {2015},
    MONTH = Nov,
    KEYWORDS = {Character variety ; 3-manifolds ; Geometric structures ; structures g{\'e}om{\'e}triques ; repr{\'e}sentation ; Vari{\'e}t{\'e} de caract{\`e}res},
    TYPE = {Accreditation to supervise research},
    PDF = {https://hal.archives-ouvertes.fr/tel-01370296/file/hdr.pdf},
    HAL_ID = {tel-01370296},
    HAL_VERSION = {v1},
    }
  • E. Falbel and Q. Wang, “A combinatorial invariant for spherical cr structures.,” Asian j. math., vol. 17, iss. 2, p. 391–422, 2012.
    [Bibtex]
    @article{FW,
    Author = {E. Falbel and Q. Wang },
    Date-Added = {2015-01-13 13:49:42 +0100},
    Date-Modified = {2015-01-13 13:50:47 +0100},
    Journal = {Asian J. Math.},
    Number = {2},
    Pages = {391--422},
    Title = {A combinatorial invariant for spherical CR structures.},
    Volume = {17},
    Year = {2012}
    }
  • E. Falbel and J. M. Veloso, “A lorentz form associated to contact sub-conformal and cr manifolds.,” Kodai math. j., vol. 37, iss. 2, p. 405–426, 2014.
    [Bibtex]
    @article{FV,
    Author = {E. Falbel and J. M. Veloso},
    Date-Added = {2015-01-13 13:48:03 +0100},
    Date-Modified = {2015-01-13 13:49:25 +0100},
    Journal = {Kodai Math. J.},
    Number = {2},
    Pages = {405--426},
    Title = {A Lorentz form associated to contact sub-conformal and CR manifolds.},
    Volume = {37},
    Year = {2014}}
  • E. Falbel and J. Wang, “Branched spherical cr structures on the complement of the figure-eight knot,” Michigan math. j., vol. 63, iss. 3, p. 635–667, 2014.
    [Bibtex]
    @article{FWbranch,
    Author = {Falbel, E. and J. Wang},
    Date-Added = {2015-01-13 13:41:23 +0100},
    Date-Modified = {2015-01-13 13:47:45 +0100},
    Journal = {Michigan Math. J.},
    Number = {3},
    Pages = { 635--667},
    Publisher = { },
    Title = {Branched spherical CR structures on the complement of the figure-eight knot},
    Volume = {63},
    Year = {2014}}
  • N. Bergeron, E. Falbel, and A. Guilloux, “Tetrahedra of flags, volume and homology of SL(3),” Geometry and topology, vol. 18, 2014.
    [Bibtex]
    @article{BFG14,
    Author = {Bergeron, N. and Falbel, E. and Guilloux, A.},
    Journal = {Geometry and Topology},
    Keywords = {Mathematics - Geometric Topology, Mathematics - K-Theory and Homology},
    Pages = 1911-1971,
    Title = {{Tetrahedra of flags, volume and homology of SL(3)}},
    Volume = {18},
    Year = 2014}
  • A. Guilloux, “Deformation of hyperbolic manifolds in and discreteness of the peripheral representations,” Proceedings of the american mathematical society, 2014.
    [Bibtex]
    @article{Gui14,
    Author = {Guilloux, A.},
    Journal = {Proceedings of the American Mathematical Society},
    Title = {{Deformation of hyperbolic manifolds in and discreteness of the peripheral representations}},
    Year = 2014}
  • [DOI] N. Bergeron, A. Guilloux, E. Falbel, P. Koseleff, and F. Rouillier, “Local rigidity for sl (3,c) representations of 3-manifolds groups,” Experimental Mathematics, vol. 22, iss. 4, p. 10, 2013.
    [Bibtex]
    @article{BGFKR13,
    Author = {Bergeron, Nicolas and Guilloux, Antonin and Falbel, Elisha and Koseleff, Pierre-Vincent and Rouillier, Fabrice},
    Doi = {10.1080/10586458.2013.832441},
    Hal_Id = {hal-00803837},
    Hal_Version = {v1},
    Journal = {{Experimental Mathematics}},
    Number = {4},
    Pages = {10},
    Publisher = {{AK Peters}},
    Title = {Local rigidity for SL (3,C) representations of 3-manifolds groups},
    Url = {https://hal.inria.fr/hal-00803837},
    Volume = {22},
    Year = {2013},
    Bdsk-Url-1 = {https://hal.inria.fr/hal-00803837},
    Bdsk-Url-2 = {http://dx.doi.org/10.1080/10586458.2013.832441}}

 

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