Density theorem for Kleinian groups

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In the mathematical theory of Kleinian groups, the density conjecture of Lipman Bers, Dennis Sullivan, and William Thurston, later proved by Namazi & Suoto (2010) and Ohshika (2011), states that every finitely generated Kleinian group is an algebraic limit of geometrically finite Kleinian groups.

History[edit]

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 groups with no parabolic elements. A more general version of Bers's conjecture due to Sullivan and Thurston 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 & Suoto (2010) and Ohshika (2011).

References[edit]