[Paper Review] The classification of Kleinian surface groups, II: The Ending Lamination Conjecture

Jeffrey Brock, Richard D. Canary+1 more|arXiv (Cornell University)|Dec 1, 2004

Geometric and Algebraic Topology33 references20 citations

TL;DR

This paper proves the Ending Lamination Conjecture for Kleinian surface groups by establishing a uniformly bilipschitz model for the quotient of hyperbolic 3-space by such groups, showing that the hyperbolic 3-manifold is uniquely determined by its end invariants—geodesic laminations or conformal structures on subsurfaces. The key result is the Bilipschitz Model Theorem, which constructs a model manifold dependent only on end invariants, leading to the Ending Lamination Theorem for surface groups and its generalization to manifolds with incompressible ends.

ABSTRACT

Thurston's Ending Lamination Conjecture states that a hyperbolic 3-manifold N with finitely generated fundamental group is uniquely determined by its topological type and its end invariants. In this paper we prove this conjecture for Kleinian surface groups; the general case when N has incompressible ends relative to its cusps follows readily. The main ingredient is the establishment of a uniformly bilipschitz model for a Kleinian surface group. The first half of the proof appeared in math.GT/0302208, and a subsequent paper will establish the Ending Lamination Conjecture in general.

Motivation & Objective

• To prove the Ending Lamination Conjecture for Kleinian surface groups, which posits that a hyperbolic 3-manifold is uniquely determined by its topological type and end invariants.
• To construct a uniformly bilipschitz model manifold $M_\nu$ that depends only on the end invariants $\nu = (\nu_+, \nu_-)$, not on the specific group representation.
• To extend the classification of hyperbolic 3-manifolds with finitely generated fundamental group to the case of incompressible ends by leveraging the surface group result.
• To correct and clarify a technical error in a prior result (Theorem 7.1 in [54]) concerning quasiconvexity of short-curve projections in annuli, ensuring the validity of the broader proof framework.

Proposed method

• Construct a bilipschitz model manifold $M_\nu$ from the end invariants $\nu_+$ and $\nu_-$, using a scaffold and partial order structure on subsurfaces to organize geometric data.
• Define 'addresses' and 'regions' to control the geometry of subsurfaces and their interactions, particularly near short geodesics.
• Use uniform bilipschitz embeddings of Lipschitz surfaces to relate the model manifold to the actual hyperbolic 3-manifold $N_\rho$, ensuring controlled distortion.
• Apply the Short Curve Theorem and Margulis tube theory to bound the Teichmüller parameters of torus neighborhoods around short curves, linking them to the end invariants.
• Establish a uniform bound on the Teichmüller distance between the intrinsic parameter $\omega$ of a curve and $2\pi i / \lambda$, where $\lambda$ is the holonomy parameter, via the Poincaré metric on the upper half-plane.
• Use Sullivan’s rigidity theorem to conclude that the model manifold is isometric to the original hyperbolic manifold, proving the uniqueness of $\rho$ up to conjugacy.

Experimental results

Research questions

• RQ1 Can the end invariants—geodesic laminations or conformal structures—uniquely determine a hyperbolic 3-manifold arising from a Kleinian surface group?
• RQ2 Is there a uniformly bilipschitz model manifold $M_\nu$ that depends only on the end invariants $\nu = (\nu_+, \nu_-)$ and not on the specific group representation $\rho$?
• RQ3 Does the Bilipschitz Model Theorem imply that the hyperbolic 3-manifold $N_\rho$ is uniquely determined by its end invariants, up to conjugacy in $\mathrm{PSL}_2(\mathbb{C})$?
• RQ4 Can the result for surface groups be extended to general hyperbolic 3-manifolds with incompressible ends relative to cusps?
• RQ5 What is the correct statement of the projection quasiconvexity result in [54] when the subsurface $Y$ is an annulus, and how does this affect the overall proof?

Key findings

• The Bilipschitz Model Theorem establishes a $K$-bilipschitz map from the model manifold $M_\nu$ to the hyperbolic 3-manifold $N_\rho$, where $K$ is uniform across all surface groups.
• For curves with $|\omega| > k_2$, the Teichmüller parameter $\omega$ of the tube neighborhood is uniformly close to $2\pi i / \lambda$, with distance bounded in the Poincaré metric on $\mathbb{H}^2$, ensuring control over holonomy parameters.
• The corrected version of Theorem 7.1 in [54] states that for non-annular subsurfaces $Y$ with $\xi(Y) \neq 2,3$, the projection of the short-curve set is $B$-quasiconvex in $\mathcal{A}(Y)$, and $d_Y(v, \Pi_{\rho,L}(v)) \leq D_2$ for all vertices $v$ in the hierarchy.
• The proof shows that the end invariants $(\nu_+, \nu_-)$ uniquely determine the Kleinian surface group $\rho$ up to conjugacy in $\mathrm{PSL}_2(\mathbb{C})$, confirming the Ending Lamination Theorem for surface groups.
• The result extends to hyperbolic 3-manifolds with incompressible ends: such manifolds are uniquely determined by the marked homeomorphism type of their relative compact core and the end invariants on the relative boundary.
• The error in [54]—where quasiconvexity was incorrectly claimed for annuli—is corrected: the quasiconvexity fails for annuli, but the second part of the theorem (the uniform bound on projection distance) remains valid and sufficient for the proof.