Density theorem for Kleinian groups

Last updated January 30, 2023

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

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 (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).

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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 (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
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

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