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 independently by (Namazi Souto) 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 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) 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) 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
- 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., 299, Cambridge University Press, pp. 75–93, doi:10.1017/CBO9780511542817.004
- Brock, Jeffrey F.; Bromberg, Kenneth W. (2004), "On the density of geometrically finite Kleinian groups", Acta Mathematica 192 (1): 33–93, doi:10.1007/BF02441085, ISSN 0001-5962
- Bromberg, K. (2007), "Projective structures with degenerate holonomy and the Bers density conjecture", Annals of Mathematics, Second Series 166 (1): 77–93, doi:10.4007/annals.2007.166.77, ISSN 0003-486X
- 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, https://projecteuclid.org/journals/acta-mathematica/volume-209/issue-2/Non-realizability-and-ending-laminations--Proof-of-the-density/10.1007/s11511-012-0088-0.full
- Ohshika, Ken'ichi (2011), "Realising end invariants by limits of minimally parabolic, geometrically finite groups", Geometry and Topology 15 (2): 827–890, doi:10.2140/gt.2011.15.827, ISSN 1364-0380, http://www.msp.warwick.ac.uk/gt/2011/15-02/p023.xhtml, 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, archived from the original on 2011-07-22, https://web.archive.org/web/20110722063916/http://www.dmi.units.it/~rimut/volumi/37/
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