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 {{harvtxt|Namazi|Souto|2012}} and {{harvtxt|Ohshika|2011}}, states that every finitely generated Kleinian group is an algebraic limit of geometrically finite Kleinian groups.
History
{{harvtxt|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 {{harvtxt|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. {{harvtxt|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 {{harvtxt|Namazi|Souto|2012}} and {{harvtxt|Ohshika|2011}}.
References
- {{Citation | last1=Bers | first1=Lipman |authorlink=Lipman Bers| title=On boundaries of Teichmüller spaces and on Kleinian groups. I | jstor=1970638 | mr=0297992 | year=1970 | journal=Annals of Mathematics |series=Second Series | issn=0003-486X | volume=91 | pages=570–600 | doi=10.2307/1970638}}
- {{Citation | last1=Brock | first1=Jeffrey F. | last2=Bromberg | first2=Kenneth W. | title=Kleinian groups and hyperbolic 3-manifolds (Warwick, 2001) | publisher=Cambridge University Press | series=London Math. Soc. Lecture Note Ser. | doi=10.1017/CBO9780511542817.004 | mr=2044545 | year=2003 | volume=299 | chapter=Cone-manifolds and the density conjecture | pages=75–93| arxiv=math/0210484 }}
- {{Citation | last1=Brock | first1=Jeffrey F. | last2=Bromberg | first2=Kenneth W. | title=On the density of geometrically finite Kleinian groups | doi=10.1007/BF02441085 | mr=2079598 | year=2004 | journal=Acta Mathematica | issn=0001-5962 | volume=192 | issue=1 | pages=33–93| arxiv=math/0212189 }}
- {{Citation | last1=Bromberg | first1=K. | title=Projective structures with degenerate holonomy and the Bers density conjecture | doi=10.4007/annals.2007.166.77 | mr=2342691 | year=2007 | journal=Annals of Mathematics |series=Second Series | issn=0003-486X | volume=166 | issue=1 | pages=77–93| arxiv=math/0211402 }}
- {{Citation |last1=Namazi |first1=Hossein |last2=Souto |first2=Juan |title=Non-realizability and ending laminations: Proof of the density conjecture |year=2012 |url=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 |journal=Acta Mathematica |volume=209 |issue=2 |pages=323–395 |doi=10.1007/s11511-012-0088-0 |issn=0001-5962|doi-access=free }}
- {{citation | last1=Ohshika | first1=Ken'ichi | title=Realising end invariants by limits of minimally parabolic, geometrically finite groups | url=http://www.msp.warwick.ac.uk/gt/2011/15-02/p023.xhtml | year=2011 | journal=Geometry and Topology | issn=1364-0380 | volume=15 | issue=2 | pages=827–890 | doi=10.2140/gt.2011.15.827 | arxiv=math/0504546 | access-date=2011-08-01 | archive-date=2014-05-25 | archive-url=https://web.archive.org/web/20140525195019/http://www.msp.warwick.ac.uk/gt/2011/15-02/p023.xhtml | url-status=dead }}
- {{Citation | last1=Series | first1=Caroline | title=A crash course on Kleinian groups | url=http://www.dmi.units.it/~rimut/volumi/37/ | mr=2227047 | year=2005 | journal=Rendiconti dell'Istituto di Matematica dell'Università di Trieste | issn=0049-4704 | volume=37 | issue=1 | pages=1–38 | url-status=dead | archiveurl=https://web.archive.org/web/20110722063916/http://www.dmi.units.it/~rimut/volumi/37/ | archivedate=2011-07-22 }}