Density theorem
In the mathematical theory of Kleinian groups, the density conjecture of Lipman Bers, Dennis Sullivan, and William Thurston, later proved independently by Namazi & Souto (2012) and Ohshika (2011), states that every finitely generated Kleinian group is an algebraic limit of geometrically finite Kleinian groups.
References
Primary source
Progress summary
The conjecture is now a theorem: every finitely generated Kleinian group can be approximated by geometrically finite ones, with no current dispute recorded.
The Bers–Sullivan–Thurston conjecture asserts that every finitely generated Kleinian group is an algebraic limit of geometrically finite Kleinian groups. Bers proposed the surface-group version in 1970; the general conjecture was proved independently by Namazi–Souto and Ohshika.
Known results
- Cusp-free, incompressible-end manifolds: Namazi–Souto, 2002.
- Cusp-free singly degenerate surface groups: Bromberg, 2007.
- Freely indecomposable groups without parabolics: Brock–Bromberg, 2004.
- Surface groups: quasifuchsian groups are dense in .
Full resolution, 2011–2012; confirmation in 2024
Ohshika (2011) and Namazi–Souto (2012) established the full density theorem, using tameness, uniformisation, and the Ending Lamination Theorem. A 2024 paper explicitly treats the conjecture as resolved and cites an even stronger geometric-limit formulation. No counterexample, proof gap, retraction, or contrary verification was found.
Current status (as of August 2026): The density conjecture is resolved as a theorem by the independent work of Ohshika and Namazi–Souto, with no recorded outstanding objection.
Solutions 0
No solutions have been posted yet.