Non-realizability and ending laminations: Proof of the density conjecture

Namazi, Hossein; Souto, Juan

doi:10.1007/s11511-012-0088-0

Cited by 55 publications

(65 citation statements)

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“…j / actually represent ending laminations of ends of .M / 0 (Proposition 6.5). A similar result on the equivalence of being unrealisable and representing an ending lamination was given independently by Namazi and Souto [43] as we mentioned in the Introduction (Section 1).…”

Section: Unrealisable Laminations and Ending Laminationssupporting

confidence: 75%

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Realising end invariants by limits of minimally parabolic, geometrically finite groups

Ohshika

2011

Geom. Topol.

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We shall show that for a given homeomorphism type and a set of end invariants (including the parabolic locus) with necessary topological conditions which a topologically tame Kleinian group with that homeomorphism type must satisfy, there is an algebraic limit of minimally parabolic, geometrically finite Kleinian groups which has exactly that homeomorphism type and the given end invariants. This shows that the Bers-Sullivan-Thurston density conjecture follows from Marden's conjecture proved by Agol and Calegari-Gabai combined with Thurston's uniformisation theorem and the ending lamination conjecture proved by Minsky, partially collaborating with Masur, Brock and Canary.

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Section: Unrealisable Laminations and Ending Laminationssupporting

confidence: 75%

“…We note that Namazi and Souto also have given a proof of this latter step in [43], and have proved the Bers-Sullivan-Thurston density conjecture independently of our work.…”

Section: Introductionsupporting

confidence: 54%

Realising end invariants by limits of minimally parabolic, geometrically finite groups

Ohshika

2011

Geom. Topol.

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“…By Lemma 20, ρ cannot be separable-stable, as any simple closed curve on B is separable. Since purely hyperbolic points are dense in C − C ( [12], Lemma 4.2, [34], [35]), this completes the proof.…”

Section: Other Homeomorphism Types In Ah(m )mentioning

confidence: 56%

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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“…This was proven first by Y. Minsky [15] for representations of punctured torus groups, and, hence, in the Maskit slice, which suffices for our purposes. The general case is due to the work of many people, most notably, K. Bromberg [4], J. Brock and K. Bromberg [2], H. Namazi and J. Souto [18], and K. Ohshika [19]. We, thus, have: Proposition 7.…”

Section: Theorem 1 Chapter 2]mentioning

confidence: 94%

Discreteness is undecidable

Kapovich

2016

Int. J. Algebra Comput.

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Abstract. We prove that the discreteness problem for 2-generated nonelementary subgroups of SLp2, Cq is undecidable in the BSS computability model. This paper is motivated by the following basic question: Question 1. Let G be a connected Lie group and let A " pA 1 , . . . , A k q be a finite ordered subset of G. Is the discreteness problem for the subgroup Γ A :" xA 1 , . . . , A k y ă G decidable?This question, in the case of G " P SLp2, Cq, was raised, most recently, in the paper [8] by J. Gilman and L. Keen, who noted that "it is a challenging problem that has been investigated for more than a century and is still open." The decidability problem was solved in the case G " P SLp2, Rq by R. Riley [20] and, more efficiently, in the case of 2-generated subgroups, by J. Gilman and B. Maskit [9] and Gilman [6], (cf. [7] for a comparison of the two approaches).To make the general decidability question more precise one has to specify the model of computability. There are several computability models over the real numbers; we refer the reader to [1] and [21] for summaries of these and in-depth treatment of the BSS and the bitcomputability approaches respectively. In this paper we address decidability of the discreteness problem in the real-RAM or BSS (which stands for Blum-Shub-Smale) computability model as it is the closest in spirit to the papers by Gilman, Maskit and Keen mentioned above as well as Riley's work [20]. We will address decidability of the discreteness problem in the bitcomputabulity model in another paper, [13].Remark 2. We refer the reader to the paper by J. Gilman in [7], where several (semi)algorithms for the discreteness problem in P SLp2, Rq and P SLp2, Cq in different computability models, including the BSS model, are compared.Briefly, computations in the BSS model over the real numbers are performed by a BSS machine, which is an analogue of a Turing machine except that a BSS machine can store finite lists of real numbers and do elementary algebraic and order operation with real numbers: Such a machine can add, subtract, multiply and divide, as well as verify inequalities and equalities a ă b, a " b for real numbers. (BSS machines are also defined for computations in other rings, but, in this paper we will use only real numbers.) We refer to [1] for the details.A subset E Ă R n is BSS-semicomputable (or the membership problem for E is BSS-semidecidable) if E is the halting set of a BSS machine: There exists a BSS machine which, given

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Non-realizability and ending laminations: Proof of the density conjecture

Cited by 55 publications

References 51 publications

Realising end invariants by limits of minimally parabolic, geometrically finite groups

Realising end invariants by limits of minimally parabolic, geometrically finite groups

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

Discreteness is undecidable

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