The classification of Kleinian surface groups, II: The Ending Lamination Conjecture

Brock, Jeffrey; Canary, Richard D.; Minsky, Yair N.

doi:10.4007/annals.2012.176.1.1

Cited by 190 publications

(336 citation statements)

References 69 publications

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“…In fact, generically there will be many such -they arise whenever we have a large subsurface projection distance between the two end invariants of M . (See [Mi,BrCM,Bow2] etc. ) Suppose now that Σ is any compact surface.…”

Section: Statement Of Resultsmentioning

confidence: 99%

“…By tameness [Bon], Ψ(M, η) is homeomorphic to Σ × R. The proof of the Ending Lamination Conjecture [Mi, BrCM] has lead to a reasonably good understanding of such manifolds in terms of model spaces. An important feature of their geometry are "bands" -subsets homeomorphic to a subsurface of Σ times an interval, where the boundary curves of the subsurface are represented by short geodesics in M (see for example, [Mi,BrCM,Mj1,Mj2,Bow1,Bow4]). Subsets of this sort are termed "blocks" in [Mj1,Mj2], though in this paper, we use the terminology from [Bow1] to avoid a clash with the term "block" as used in [Mi].…”

Section: Introductionmentioning

confidence: 99%

“…Such a band arises whenever the end invariants of M have large subsurface projection distance in Φ, see [Mi,BrCM,Bow2]. They behave intrinsically like product manifolds of lower complexity.…”

Section: Introductionmentioning

confidence: 99%

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Quasiconvexity of Bands in Hyperbolic 3-Manifolds

Bowditch

2013

J. Aust. Math. Soc.

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Section: Statement Of Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Quasiconvexity of Bands in Hyperbolic 3-Manifolds

Bowditch

2013

J. Aust. Math. Soc.

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“…In particular, λ must be an ending lamination for ρ . By the ending lamination theorem ( [6]), possibly after conjugating, ρ k | π 1 (S i ) must converge to ρ| π 1 (S i ) . On the other factors of π 1 (M ), by construction ρ k converges to ρ.…”

Section: Examples Of Separable-stable Points On ∂Ah(m )mentioning

confidence: 99%

Dynamics on the PSL(2,ℂ)–character variety of a compression body

Lee

2014

Algebr. Geom. Topol.

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Let M be a nontrivial compression body without toroidal boundary components. Let X (M ) be the PSL(2, C)-character variety of π 1 (M ). We examine the dynamics of the action of Out(π 1 (M )) on X (M ), and in particular, we find an open set on which the action is properly discontinuous that is strictly larger than the interior of the deformation space of marked hyperbolic 3-manifolds homotopy equivalent to M .In this paper we use the deformation theory of hyperbolic 3-manifolds to study the dynamics of Out(π 1 (M )) on the PSL(2, C)-character variety of π 1 (M ) when M is a nontrivial compression body without toroidal boundary components. In particular, we find a domain of discontinuity for the action that is strictly larger than the previously known domain of discontinuity.The study of Out(π 1 (M )) acting on character varieties or representation varieties is a blooming field of study. One motivation comes from the classical result that the mapping class group of a closed oriented surface S of genus at least two acts properly discontinuously on T (S) the Teichmüller space of S. Teichmüller space T (S) is a component of the representation variety Hom(π 1 (S), PSL(2, R))/ PSL(2, R) and together with T (S) the Teichmüller space of S with the opposite orientation, form the set of discrete and faithful representations. The group Out(π 1 (S)) acts properly discontinuously on T (S) T (S) and Goldman conjectured that the action on the remaining components is ergodic. The so-called higher Teichmüller spaces, which are analogies of Teichmüller space for higher rank Lie groups, also form domains of discontinuity (see, for example, [25] , [43], [20]).A compression body is the boundary connect sum of a 3-ball, a collection of I-bundles over closed surfaces and a handlebody where the other * Partially supported by NSF RTG grant DMS 0602191 1 arXiv:1307.0859v1 [math.GT] 2 Jul 2013 components are connected to the 3-ball along disjoint discs. The PSL(2, C)-character variety of π 1 (M ) isthe quotient of Hom(π 1 (M ), PSL(2, C)) from geometric invariant theory.The group Out(π 1 (M )) acts on X (M ) in the following way: an outer au-is AH(M ) the space of conjugacy classes of discrete and faithful representations of π 1 (M ) into PSL(2, C). It can also be thought of as the space of marked hyperbolic 3-manifolds homotopy equivalent to M. Using the parametrization of the interior of AH(M ) (see [14] Chapter 7 for more details on this parametrization), it is well known that this action is properly discontinuous on the interior of AH(M ). In this paper we find a domain of discontinuity containing the interior of AH(M ) as well as some but not all points on ∂AH(M ) when M is a nontrivial hyperbolizable compression body without toroidal boundary components. Namely, we prove the following. In proving the theorem we show that pinching a Masur domain curve or lamination on the boundary component are points in this domain of discontinuity.Canary-Storm ([16]) showed that whenever M has an primitive essential annulus the action of Ou...

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“…The theorem was conjectured by Thurston in 1982 and it was proved in several steps by Brock, Canary, and Minsky, cf. [6] and [21]. The proof uses in an essential way the large-scale geometry of Teichmüller space.…”

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confidence: 99%

Book Review: Teichmüller theory and applications to geometry, topology, and dynamics. Volume 1: Teichmüller theory; Teichmüller theory and applications to geometry, topology, and dynamics. Volume 2: Surface homeomorphisms and rational functions

Papadopoulos¹

2018

Bull. Amer. Math. Soc.

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The classification of Kleinian surface groups, II: The Ending Lamination Conjecture

Cited by 190 publications

References 69 publications

Quasiconvexity of Bands in Hyperbolic 3-Manifolds

Quasiconvexity of Bands in Hyperbolic 3-Manifolds

Dynamics on the PSL(2,ℂ)–character variety of a compression body

Book Review: Teichmüller theory and applications to geometry, topology, and dynamics. Volume 1: Teichmüller theory; Teichmüller theory and applications to geometry, topology, and dynamics. Volume 2: Surface homeomorphisms and rational functions

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