Tameness of hyperbolic 3-manifolds

Abstract

We show that hyperbolic 3-manifolds with finitely generated fundamental group are tame, that is the ends are products. We actually work in slightly greater generality with pinched negatively curved manifolds with hyperbolic cusps. This answers a conjecture of Marden and implies the Ahlfors measure conjecture. Applications are given to other questions about Kleinian groups and 3-manifolds.

22 pages, 1 figure

References in corpus (1)

• Volume change under drilling

Cited by in corpus (75)

• Geometry and dynamics in Gromov hyperbolic metric spaces: With an emphasis on non-proper settings
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• Separation of Relatively Quasiconvex Subgroups
• Effective bilipschitz bounds on drilling and filling
• The classification of Kleinian surface groups, II: The Ending Lamination Conjecture
• The rank of the fundamental group of certain hyperbolic 3-manifolds fibering over the circle
• Geometric inflexibility and 3-manifolds that fiber over the circle
• Marden's Tameness Conjecture: history and applications
• Twisted cohomology for hyperbolic three manifolds
• Local topology in deformation spaces of hyperbolic 3-manifolds
• On dynamics of Out(F_n) on PSL(2,C) characters
• Unimodular measures on the space of all Riemannian manifolds
• Geometry, Heegaard splittings and rank of the fundamental group of hyperbolic 3-manifolds
• Skinning maps
• Geometric limits of knot complements
• A finiteness theorem for hyperbolic 3-manifolds
• Geometry and Rank of Fibered Hyperbolic 3-Manifolds
• End reductions, fundamental groups, and covering spaces of irreducible open 3-manifolds
• Topological tameness of Margulis spacetimes
• Fuchsian Groups, Circularly Ordered Groups, and Dense Invariant Laminations on the Circle
• Some open 3-manifolds and 3-orbifolds without locally finite canonical decompositions
• Special subgroups of Bianchi groups
• Shadows of mapping class groups: capturing convex cocompactness
• The critical exponent, the Hausdorff dimension of the limit set and the convex core entropy of a Kleinian group
• Subgroups of the mapping class group from the geometrical viewpoint
• Minimal Surfaces in Hyperbolic 3-manifolds
• NonLERFness of arithmetic hyperbolic manifold groups and mixed 3-manifold groups
• The PL-methods for hyperbolic 3-manifolds to prove tameness
• Commensurability of hyperbolic manifolds with geodesic boundary
• Kleinian groups in higher dimensions
• The Double Limit Theorem and its Legacy
• Conical limit points and the Cannon-Thurston map
• Residual properties of 3-manifold groups I: Fibered and hyperbolic 3-manifolds
• The Geometry and Arithmetic of Kleinian Groups
• Homological dimension and critical exponent of Kleinian groups
• On smooth moduli space of Riemann surfaces
• Small optimal Margulis numbers force upper volume bounds
• A Cantor set with hyperbolic complement
• Property (QT) for 3-manifold groups
• Quasiconvexity in $3$-manifold groups
• Windows, cores and skinning maps
• Arithmetic Kleinian groups generated by elements of finite order
• Measurable Rigidity for Kleinian groups
• Quasi-isometric rigidity of three manifold groups
• Effective drilling and filling of tame hyperbolic 3-manifolds
• Characterization of Fuchsian groups as laminar groups & the structure theorem of hyperbolic 2-orbifolds
• Symmetric Decompositions of Free Kleinian Groups and Hyperbolic Displacements
• Thick hyperbolic 3-manifolds with bounded rank
• All finitely generated Kleinian groups of small Hausdorff dimension are classical Schottky groups
• Margulis numbers for Haken manifolds
• Hyperbolization of infinite-type 3-manifolds
• Hausdorff dimension and conformal measures of Feigenbaum Julia sets
• Equivalent trace sets for arithmetic Fuchsian groups
• Ranks of mapping tori via the curve complex
• On density of horospheres in dynamical laminations
• On the limit set of a geometrically infinite Kleinian group
• Singular surfaces, mod 2 homology, and hyperbolic volume, I
• Separable subgroups of mapping class groups
• Twisting Alexander Invariants with Periodic Representations
• A spherical decomposition for Riemannian open 3-manifolds
• Geometrically infinite surfaces in 3-manifolds with hyperbolic fundamental group
• Kleinian groups with ubiquitous surface subgroups
• Constructing geometrically infinite groups on boundaries of deformation spaces
• Determining hyperbolic 3-manifolds by their surfaces
• Unknotting tunnels in hyperbolic 3-manifolds
• Bounded cohomology and negatively curved manifolds
• Lengths of edges in carrier graphs
• Algebraic and geometric convergence of discrete representations into PSL(2,C)
• Hausdorff dimension of non-conical limit sets
• Effective contraction of skinning maps
• Margulis numbers and number fields
• 3-manifolds of rank 3 have filling links
• Isospectral finiteness on convex cocompact hyperbolic 3-manifolds
• Singular measures on the limit set of a Kleinian group
• Kleinian groups of small Hausdorff dimension are classical Schottky groups I