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In the mathematical theory of
Kleinian group In mathematics, a Kleinian group is a discrete subgroup of the group (mathematics), group of orientation-preserving Isometry, isometries of hyperbolic 3-space . The latter, identifiable with PSL(2,C), , is the quotient group of the 2 by 2 complex ...
s, the density conjecture of Lipman Bers, Dennis Sullivan, and
William Thurston William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds. Thurst ...
, later proved independently by and , states that every finitely generated
Kleinian group In mathematics, a Kleinian group is a discrete subgroup of the group (mathematics), group of orientation-preserving Isometry, isometries of hyperbolic 3-space . The latter, identifiable with PSL(2,C), , is the quotient group of the 2 by 2 complex ...
is an algebraic limit of geometrically finite Kleinian groups.


History

suggested the Bers density conjecture, that singly degenerate Kleinian surface groups are on the boundary of a Bers slice. This was proved by 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. 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 and .


References

* * * * * * *{{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 Kleinian groups Conjectures that have been proved Theorems in differential geometry