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.

## Related publications

- 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} }`

- 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}, }`

- 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}, }`

- 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}, }`

- 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}, }`

- 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}`

- 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}}`