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).
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), , 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
January 26, 2023
density, theorem, kleinian, groups, mathematical, theory, kleinian, groups, density, conjecture, lipman, bers, dennis, sullivan, william, thurston, later, proved, independently, namazi, souto, 2012, ohshika, 2011, states, that, every, finitely, generated, klei. In the mathematical theory of Kleinian groups the density conjecture of Lipman Bers Dennis Sullivan and William Thurston later proved independently by Namazi amp 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 amp 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 amp Souto 2012 and Ohshika 2011 References EditBers Lipman 1970 On boundaries of Teichmuller 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 Universita di Trieste 37 1 1 38 ISSN 0049 4704 MR 2227047 archived from the original on 2011 07 22 Retrieved from https en wikipedia org w index php title Density theorem for Kleinian groups amp oldid 1132962301, wikipedia, wiki, book, books, library,
