In the mathematical theory of Kleinian groups, the density conjecture of Lipman Bers, Dennis Sullivan, and William Thurston, later proved independently by Namazi & Souto (2012) and Ohshika (2011), states that every finitely generated Kleinian group is an algebraic limit of geometrically finite Kleinian groups.
History
editBers (1970) suggested the Bers density conjecture, that singly degenerate Kleinian surface groups are on the boundary of a Bers slice. This was proved by Bromberg (2007) for Kleinian surface groups with no parabolic elements. A more general version of Bers's conjecture due to Sullivan and Thurston in the late 1970s and early 1980s states that every finitely generated Kleinian group is an algebraic limit of geometrically finite Kleinian groups. Brock & Bromberg (2004) proved this for freely indecomposable Kleinian groups without parabolic elements. The density conjecture was finally proved using the tameness theorem and the ending lamination theorem by Namazi & Souto (2012) and Ohshika (2011).
References
edit- Bers, Lipman (1970), "On boundaries of Teichmüller spaces and on Kleinian groups. I", Annals of Mathematics, Second Series, 91: 570–600, doi:10.2307/1970638, ISSN 0003-486X, JSTOR 1970638, MR 0297992
- Brock, Jeffrey F.; Bromberg, Kenneth W. (2003), "Cone-manifolds and the density conjecture", Kleinian groups and hyperbolic 3-manifolds (Warwick, 2001), London Math. Soc. Lecture Note Ser., vol. 299, Cambridge University Press, pp. 75–93, arXiv:math/0210484, doi:10.1017/CBO9780511542817.004, MR 2044545
- Brock, Jeffrey F.; Bromberg, Kenneth W. (2004), "On the density of geometrically finite Kleinian groups", Acta Mathematica, 192 (1): 33–93, arXiv:math/0212189, doi:10.1007/BF02441085, ISSN 0001-5962, MR 2079598
- Bromberg, K. (2007), "Projective structures with degenerate holonomy and the Bers density conjecture", Annals of Mathematics, Second Series, 166 (1): 77–93, arXiv:math/0211402, doi:10.4007/annals.2007.166.77, ISSN 0003-486X, MR 2342691
- Namazi, Hossein; Souto, Juan (2012), "Non-realizability and ending laminations: Proof of the density conjecture", Acta Mathematica, 209 (2): 323–395, doi:10.1007/s11511-012-0088-0, ISSN 0001-5962
- Ohshika, Ken'ichi (2011), "Realising end invariants by limits of minimally parabolic, geometrically finite groups", Geometry and Topology, 15 (2): 827–890, arXiv:math/0504546, doi:10.2140/gt.2011.15.827, ISSN 1364-0380, archived from the original on 2014-05-25, retrieved 2011-08-01
- Series, Caroline (2005), "A crash course on Kleinian groups", Rendiconti dell'Istituto di Matematica dell'Università di Trieste, 37 (1): 1–38, ISSN 0049-4704, MR 2227047, archived from the original on 2011-07-22