Ends of hyperbolic 3-manifolds

Canary, Richard D.

doi:10.1090/s0894-0347-1993-1166330-8

Cited by 132 publications

(144 citation statements)

References 40 publications

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“…The end E of N associated to E is simply degenerate if it is not convex cocompact and it does not contain a cusp. It is a consequence of the work of Thurston [52] and Canary [11] that associated to every such end there is an ending lamination λ E which is an element of EL(E) and belongs to the Masur domain. This in particular implies that if (γ n ) is a sequence of essential simple loops on E, which converges to λ E in the Gromov boundary of C(E), then the geodesic representatives of (γ n ) in N exit E, that is, given every compact subset K of E, the geodesic representative of γ n is in E \ K for n sufficiently large.…”

Section: Convex Cocompact and Simply Degenerate Endsmentioning

confidence: 99%

Bounded combinatorics and uniform models for hyperbolic 3-manifolds

Brock

Minsky²,

Namazi³

et al. 2016

J Topology

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Bounded-type 3-manifolds arise as combinatorially bounded gluings of irreducible 3-manifolds chosen from a finite list. We prove effective hyperbolization and effective rigidity for a broad class of 3-manifolds of bounded type and large gluing heights. Specifically, we show the existence and uniqueness of hyperbolic metrics on 3-manifolds of bounded type and large heights, and prove existence of a bilipschitz diffeomorphism to a combinatorial model described explicitly in terms of the list of irreducible manifolds, the topology of the identification, and the combinatorics of the gluing maps.

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Section: Convex Cocompact and Simply Degenerate Endsmentioning

confidence: 99%

Bounded combinatorics and uniform models for hyperbolic 3-manifolds

Brock

Minsky²,

Namazi³

et al. 2016

J Topology

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“…In particular, any Γinvariant Beltrami differential extends continuously via an averaging process to a harmonic strain field η on M with the pointwise norms of η and Dη uniformly bounded. If M is tame, then the limit set of Γ has measure zero or is all of C, by Canary's [18] result that tameness implies Ahlfors' measure conjecture. In the former case, any Beltrami differential supported on the limit set is trivial.…”

Section: Infinitesimal Inflexibilitymentioning

confidence: 94%

“…Two standard constructions of such maps are Thurston's pleated surfaces and the related simplicial hyperbolic surfaces, also introduced in [39] and used extensively by Bonahon [7] and Canary [18]. Though we shall employ both constructions, we need only their consequences rather than the constructions themselves.…”

Section: Lipschitz Mapsmentioning

confidence: 99%

Geometric inflexibility and 3-manifolds that fiber over the circle

Brock

Bromberg

2011

Journal of Topology

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We prove that hyperbolic 3-manifolds are geometrically inflexible: a unit quasiconformal deformation of a Kleinian group extends to an equivariant bi-Lipschitz diffeomorphism between quotients whose pointwise bi-Lipschitz constant decays exponentially in the distance from the boundary of the convex core for points in the thick part. Estimates at points in the thin part are controlled by similar estimates on the complex lengths of short curves. We use this inflexibility to give a new proof of the convergence of pseudo-Anosov double iteration on the quasi-Fuchsian space of a closed surface, and the resulting hyperbolization theorem for closed 3-manifolds that fiber over the circle with pseudo-Anosov monodromy.Theorem 1.1 (Geometric Inflexibility). Given a hyperbolic 3-manifold M, a K-bi-Lipschitz diffeomorphic hyperbolic 3-manifold M and an > 0, there is a diffeomorphism Φ: M → M whose bi-Lipschitz distortion in the -thick part of the convex core C(M ) decays exponentially with the distance from ∂C(M ) with the rate of decay depending only on , K and the topology of ∂M . See Theorem 5.6 for a more precise version.Proposition 3.1 (Hodgson-Kerkchoff). Let M be a compact manifold with piecewise smooth boundary and η be a harmonic strain field. ThenThe following inequality will allow us to control the boundary term in terms of pointwise bounds on the norms of η and Dη.Lemma 3.2. We have η 2 + Dη 2 2 * Dη ∧ η .Proof. The inequality follows from the fact that η − * Dη 2 0.The following lemma is the first step in showing that the formula from Proposition 3.1 holds on some noncompact manifolds if the strain field is bounded.Lemma 3.3. Let M be a complete hyperbolic 3-manifold that is exhausted by compact submanifolds M n with the area of ∂M n bounded above. If η is a harmonic strain field with the pointwise norms η and Dη bounded above, then the L 2 -norm of η and Dη is finite.Proof. By Proposition 3.1,Since both the area of ∂M n and the pointwise norms of η and Dη are bounded, Lemma 3.2 implies that the right-hand side is bounded. This implies that the L 2 -norm on M is finite.Let P n be a finite (1/n)-net on ∂M . DefineLemma 3.4. For all but an isolated set of t > 1/n, M n (t) is a manifold with piecewise smooth boundary.Proof. If the boundary of M n (t) is not a manifold with piecewise smooth boundary, then there is a geodesic of length 2t in M with endpoints in P n . The set of lengths of geodesics in M with endpoints in P n is a discrete subset of R, so M n (t) must be a manifold with piecewise smooth boundary for all but an isolated set of values for t.

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“…A list U of words in F n is said to be diskbusting if one cannot write F n = A * B in such a way that A, B = {1} and each word in U is conjugate into A or B (see [5,23,24]). We note that D(U ) is one-ended if and only if U is diskbusting [10].…”

Section: Tiling Conjecture and Its Implicationmentioning

confidence: 99%

Hyperbolic surface subgroups of one-ended doubles of free groups

Kim

Oum

2014

Journal of Topology

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Gromov asked whether every one-ended word-hyperbolic group contains a hyperbolic surface group. We prove that every one-ended double of a free group has a hyperbolic surface subgroup if (1) the free group has rank 2 or (2) every generator is used the same number of times in a minimal automorphic image of the amalgamating words. To prove this, we formulate a stronger statement on Whitehead graphs and prove its specialization by combinatorial induction for (1) and the characterization of perfect matching polytopes by Edmonds for (2).

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Ends of hyperbolic 3-manifolds

Abstract: Let N = H 3 / Γ N = {{\mathbf {H}}^3}/\Gamma be a hyperbolic 3 3 -manifold which is homeomorphic to the interior of a compact 3 3 -manifold. We prove that N N is geometrically tame. As a consequence, we prove that Γ \Gamma ’s limit set L Γ … Show more

Cited by 132 publications

References 40 publications

Bounded combinatorics and uniform models for hyperbolic 3-manifolds

Bounded combinatorics and uniform models for hyperbolic 3-manifolds

Geometric inflexibility and 3-manifolds that fiber over the circle

Hyperbolic surface subgroups of one-ended doubles of free groups

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