Tameness of hyperbolic 3-manifolds

By Ian Agol. 2004

241 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/84a21503dec221f762465a894ea94ec527fe6d2b

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In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Cannon-Thurston Maps for Surface Groups (2006, methodology): “o Fan [Fan99a] [Fan99b]). Remark 2: We implicitly use here the idea of drilling disk-busting curves introduced by Canary in [Can93] and used again by Agol in his resolution of the tameness conjecture [Ago04]. Remark 4.15. Recall that extended split components were deﬁned in Mf by adjoining Margulis ”
Hausdorff dimension and conformal measures of Feigenbaum Julia sets (2004, background): “Also, it has been recently announced that the limit set of any finitely generated Kleinian group has zero area, which settles a long-standing Ahlfors Conjecture, see [ Ag , CG].”
Shadows Of Mapping Class Groups: Capturing Convex Cocompactness (2005, background): “ct,we think of Γpossessing a cusp. Crudely then, coboundedness in T(S) is analogous to coboundedness with respect to the covering H3 → MΓ. Thanksto the resolutionof Marden’sTamenessConjecturebyI. Agol[2] and(independently) D. Calegari and D. Gabai [13], and to R. Canary’s Covering Theorem [16], a ﬁn”
Two-generator free Kleinian groups and hyperbolic displacements (2009, background): “Proof By the conclusions of Propositions 6.9 and 3.9 in [8] and tameness ([1], [5]) every Γ –invariant D –conformal density M is a constant multiple of the area density A , i.e., D = 2.”
Realising end invariants by limits of minimally parabolic, geometrically finite groups (2005, background): “y ﬁnitely generated Kleinian group is topologically tame so that ending laminations can be deﬁned. This is exactly what Marden’s conjecture says, which was recently solved by Agol and Calegari-Gabai ([Ag], [CG]). Once Marden’s conjecture is solved, to show that every (ﬁnitely generated and torsion-f”
The classification of Kleinian surface groups I : Models and bounds : preprint (2003, background): “When this article was originally written the question of tameness was still open, but it has since been resolved by Agol [2] and CalegariGabai [21].”
Twisted cohomology for hyperbolic three manifolds (2010, background): “hyperbolic 3-manifold M is said to be topologically ﬁnite if it is the interior of a compact manifold M. This is equivalent to say that π1(M) is ﬁnitely generated, by the proof of Marden’s conjecture [1, 9]. ∗Both authors partially supported by the Spanish Micinn through grant MTM2009- 0759 and by t”
Non-realizability and ending laminations: Proof of the density conjecture (2012, methodology): “(Agol, Calegari–Gabai) Suppose that N is a hyperbolic 3- manifold with finitely generated fundamental group and let ε be positive and smaller than the Margulis constant.”
KLEINIAN GROUPS IN HIGHER DIMENSIONS (2007, background): “A geometric proof (valid even in the context of manifolds of negative curvature) follows from the solution of Tameness Conjecture, see [2].”