[Paper Review] On the density of geometrically finite Kleinian groups

Jeffrey Brock, Kenneth Bromberg|ArXiv.org|Dec 13, 2002

Geometric and Algebraic Topology38 references4 citations

TL;DR

This paper proves the Bers-Sullivan-Thurston density conjecture for complete hyperbolic 3-manifolds with finitely generated fundamental group, incompressible ends, and no cusps, showing they are algebraic limits of geometrically finite hyperbolic 3-manifolds. The proof relies on cone-deformation techniques and a drilling theorem that controls bi-Lipschitz distortion when short geodesics are removed, enabling the construction of marking-preserving diffeomorphisms to realize limits in the Bers boundary.

ABSTRACT

The density conjecture of Bers, Sullivan and Thurston predicts that each complete hyperbolic 3-manifold M with finitely generated fundamental group is an algebraic limit of geometrically finite hyperbolic 3-manifolds. We prove that the conjecture obtains for each complete hyperbolic 3-manifold with no cusps and incompressible ends.

Motivation & Objective

• To prove the Bers-Sullivan-Thurston density conjecture for complete hyperbolic 3-manifolds with finitely generated fundamental group, incompressible ends, and no cusps.
• To establish that such manifolds are algebraic limits of geometrically finite hyperbolic 3-manifolds.
• To extend the applicability of cone-deformation theory to control geometric transitions when short geodesics are removed from hyperbolic 3-manifolds.
• To demonstrate that geometrically finite and degenerate ends in algebraic limits can be realized via bi-Lipschitz diffeomorphisms compatible with marking and conformal structure.

Proposed method

• Utilizes 3-dimensional hyperbolic cone-manifolds with cone-angle deformations to model the removal of short geodesics.
• Applies the drilling theorem to construct $L$-bi-Lipschitz diffeomorphisms between a hyperbolic manifold minus a tubular neighborhood of a short geodesic and the complete hyperbolic structure on the punctured manifold.
• Employs geometric and algebraic limits of quasi-Fuchsian manifolds to analyze convergence in the deformation space.
• Uses the covering theorem and Waldhausen’s theorem to ensure topological and geometric consistency in the limit manifold.
• Applies Sullivan’s theorem to conclude that a marking-preserving bi-Lipschitz diffeomorphism between the limit and original manifold is homotopic to an isometry.
• Relies on results from Bonahon and Thurston on ending laminations to rule out cusps in degenerate ends, ensuring the limit manifold is cusp-free.

Experimental results

Research questions

• RQ1 Can every complete hyperbolic 3-manifold with finitely generated fundamental group, incompressible ends, and no cusps be realized as an algebraic limit of geometrically finite hyperbolic 3-manifolds?
• RQ2 How does the geometry of a hyperbolic 3-manifold change when a short geodesic is removed and replaced by a cusp?
• RQ3 To what extent can cone-deformation techniques control the bi-Lipschitz distortion in the deformation space of infinite-volume hyperbolic 3-manifolds?
• RQ4 Under what conditions does the geometric limit of a sequence of quasi-Fuchsian manifolds coincide with the original manifold up to isometry?
• RQ5 How can the ending lamination theory be used to rule out the presence of cusps in the algebraic limit of a sequence of hyperbolic 3-manifolds?

Key findings

• The density conjecture holds for all complete hyperbolic 3-manifolds with finitely generated fundamental group, incompressible ends, and no cusps.
• The drilling theorem establishes the existence of an $L$-bi-Lipschitz diffeomorphism between a manifold minus a short geodesic and the complete structure on the punctured manifold, with $L$ depending only on the length bound $\ell$.
• Geometrically finite ends in the algebraic limit are compactified by closed surfaces, and their conformal structures match those of the original manifold.
• Degenerate ends in the limit are shown to have no cusps by using the filling property of ending laminations and the non-vanishing intersection number with simple closed curves.
• The existence of a marking-preserving bi-Lipschitz diffeomorphism between the algebraic limit and the original manifold implies that the limit is isometric to the original manifold.
• Strong convergence of the sequence of manifolds to the limit manifold follows from the cusp-free condition and results of Anderson and Canary, confirming the algebraic limit is geometrically realized.