In the mathematical theory of

Kleinian group In mathematics, a Kleinian group is a discrete subgroup of the group of orientation-preserving isometries of hyperbolic 3-space . The latter, identifiable with , is the quotient group of the 2 by 2 complex matrices of determinant 1 by their ...

s, the density conjecture of

Lipman Bers Lipman Bers ( Latvian: ''Lipmans Berss''; May 22, 1914 – October 29, 1993) was a Latvian-American mathematician, born in Riga, who created the theory of pseudoanalytic functions and worked on Riemann surfaces and Kleinian groups. He was also ...

Dennis Sullivan Dennis Parnell Sullivan (born February 12, 1941) is an American mathematician known for his work in algebraic topology, geometric topology, and dynamical systems. He holds the Albert Einstein Chair at the City University of New York Graduate C ...

, 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. Thursto ...

, later proved independently by and , states that every finitely generated

is an algebraic limit of

geometrically finite Kleinian group In geometry, a group of isometries of hyperbolic space is called geometrically finite if it has a well-behaved fundamental domain. A hyperbolic manifold is called geometrically finite if it can be described in terms of geometrically finite gro ...

History

suggested the Bers density conjecture, that singly degenerate Kleinian surface groups are on the boundary of a

Bers slice In the mathematical theory of Kleinian groups, Bers slices and Maskit slices, named after Lipman Bers and Bernard Maskit, are certain slices through the moduli space of Kleinian groups. Bers slices For a quasi-Fuchsian group, the limit set is ...

. 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 In mathematics, the tameness theorem states that every complete hyperbolic 3-manifold with finitely generated fundamental group is topologically tame, in other words homeomorphic to the interior of a compact 3-manifold. The tameness theorem wa ...

and the

ending lamination theorem In hyperbolic geometry, the ending lamination theorem, originally conjectured by , states that hyperbolic 3-manifolds with finitely generated fundamental groups are determined by their topology together with certain "end invariants", which are g ...

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