Kleinian groups and the complex of curves

Abstract

We examine the internal geometry of a Kleinian surface group and its relations to the asymptotic geometry of its ends, using the combinatorial structure of the complex of curves on the surface. Our main results give necessary conditions for the Kleinian group to have `bounded geometry' (lower bounds on injectivity radius) in terms of a sequence of coefficients (subsurface projections) computed using the ending invariants of the group and the complex of curves. These results are directly analogous to those obtained in the case of punctured-torus surface groups. In that setting the ending invariants are points in the closed unit disk and the coefficients are closely related to classical continued-fraction coefficients. The estimates obtained play an essential role in the solution of Thurston's ending lamination conjecture in that case.

32 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTVol4/paper3.abs.html

References in corpus (3)

• Geometry of the Complex of Curves I: Hyperbolicity
• Hyperbolic Structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle
• Hyperbolic Structures on 3-manifolds, III: Deformations of 3-manifolds with incompressible boundary

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• Shadows of mapping class groups: capturing convex cocompactness
• Subgroups of the mapping class group from the geometrical viewpoint
• Convergence and divergence of Kleinian surface groups
• On the density of geometrically finite Kleinian groups
• Uniform models and short curves for random 3-manifolds
• The Heegaard genus of bundles over S^1
• Ranks of mapping tori via the curve complex
• Cone-manifolds and the density conjecture
• Geometry of the complex of curves II: Hierarchical structure
• The Thurston boundary of Teichmuller space and complex of curves
• Thurston's Bounded image theorem