Essential laminations in seifert-fibered spaces

Brittenham, Mark

doi:10.1016/0040-9383(93)90038-w

Cited by 43 publications

(52 citation statements)

References 13 publications

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“…This result, as well as those of [1] and [2], illustrates a technique for attacking problems whose solutions for incompressible surfaces relies on induction (on the number of points of intersection with a 1-dimensional object). The technique used here is, in fact, simply a different way of thinking about this induction process; instead, it is something we might call 'eventual stability under a sequence of isotopies'.…”

Section: Corollary M Contains An Essential Lamination Iff It Containmentioning

confidence: 94%

“…We shall describe an algorithm for performing a (typically infinite time) isotopy of £, controlled along the intersection of C with the 1-skeleton τ^ of r. After identifying a collection of points which are left fixed by all of the isotopies, we then study what happens when we take the 'limit' of these isotopies. At this point the proof diverges markedly from [1]; seeing that pieces of the lamination stabilize around these fixed points is easier, but in the situation encountered here, the union of these stable pieces is not a lamination -it is a disjoint union of 1-to-l immersed surfaces, but it is not, in general, a closed set.…”

Section: Corollary M Contains An Essential Lamination Iff It Containmentioning

confidence: 94%

“…The proof of the theorem is in broad outline very similar to the arguments found in [1]. We shall describe an algorithm for performing a (typically infinite time) isotopy of £, controlled along the intersection of C with the 1-skeleton τ^ of r. After identifying a collection of points which are left fixed by all of the isotopies, we then study what happens when we take the 'limit' of these isotopies.…”

Section: Corollary M Contains An Essential Lamination Iff It Containmentioning

confidence: 99%

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Essential laminations and Haken normal form

Brittenham¹

1995

Pacific J. Math.

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Section: Corollary M Contains An Essential Lamination Iff It Containmentioning

confidence: 94%

Section: Corollary M Contains An Essential Lamination Iff It Containmentioning

confidence: 94%

Section: Corollary M Contains An Essential Lamination Iff It Containmentioning

confidence: 99%

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Essential laminations and Haken normal form

Brittenham¹

1995

Pacific J. Math.

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“…An application of [38, Theorem 1.8 (2)] shows that if ∆(r, r 0 ) > 1, then X (r) is not an I-bundle over a surface F with ∂ v X (r) the I-bundle over ∂F . But then by [4], M (r) cannot be atoroidal Seifert fibred and hence ∆(r, r 0 ) ≤ 1 as claimed.…”

Section: Fillings Of Manifolds Of Large First Betti Numbermentioning

confidence: 99%

“…Also, if S is an essential closed surface in M K and C(S) is finite, each of its slopes is a singular slope of S. In particular this is true for µ K . (4). Let r i be a very big Seifert surgery slope on ∂M K .…”

Section: Proof Of Part (1) Whenmentioning

confidence: 99%

Dehn Fillings of Large Hyperbolic 3-Manifolds

Boyer

Gordon²,

Zhang

2001

J. Differential Geom.

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Let M be a compact, connected, orientable, hyperbolic 3-manifold whose boundary is a torus and which contains an essential closed surface S. It is conjectured that 5 is an upper bound for the distance between two slopes on ∂M whose associated fillings are not hyperbolic manifolds. In this paper we verify the conjecture when the first Betti number of M is at least 2 by showing that given a pseudo-Anosov mapping class f of a surface and an essential simple closed curve γ in the surface, then 5 is an upper bound for the diameter of the set of integers n for which the composition of f with the n th power of a Dehn twist along γ is not pseudo-Anosov. This sharpens an inequality of Albert Fathi. For large manifolds M of first Betti number 1 we obtain partial results. SetA singular slope for S is a slope r 0 ∈ C(S) such that any other slope in C(S) is at most distance 1 from r 0 . We prove that the distance between two exceptional filling slopes is at most 5 if either (i) there is a closed essential surface S in M with C(S) finite, or (ii) there are singular slopes r 1 = r 2 for closed essential surfaces S 1 , S 2 in M .

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Research Announcement: Partially Hyperbolic Diffeomorphisms Homotopic to the Identity on 3-Manifolds

et al. 2020

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We study partially hyperbolic diffeomorphisms homotopic to the identity in 3-manifolds. Under a general minimality condition, we show a dichotomy for the dynamics of the (branching) foliations in the universal cover. This allows us to give a full classification in certain settings: partially hyperbolic diffeomorphisms homotopic to the identity on Seifert fibered manifolds (proving a conjecture of Pujals [BW05] in this setting), and dynamically coherent partially hyperbolic diffeomorphisms on hyperbolic 3-manifolds (proving a classification conjecture of Hertz-Hertz-Ures [CRRU15] in this setting). In both cases, up to iterates we prove that the diffeomorphism is leaf conjugate to the time one map of an Anosov flow. Several other results of independent interest are obtained along the way.

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Essential laminations in seifert-fibered spaces

Cited by 43 publications

References 13 publications

Essential laminations and Haken normal form

Essential laminations and Haken normal form

Dehn Fillings of Large Hyperbolic 3-Manifolds

Research Announcement: Partially Hyperbolic Diffeomorphisms Homotopic to the Identity on 3-Manifolds

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