Density theorem for Kleinian groups
In the mathematical theory of Kleinian groups, the density conjecture of Lipman Bers, Dennis Sullivan, and William Thurston, later proved by Namazi & Souto (2010) and Ohshika (2011), states that every finitely generated Kleinian group is an algebraic limit of geometrically finite Kleinian groups.
History[]
Bers (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 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 (2010) and Ohshika (2011).
References[]
- 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 (2010), Non-realizability, ending laminations and the density conjecture, archived from the original on 2009-07-15
- 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
- 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
- Kleinian groups