An extension of the Masur domain

Lecuire, Cyril

doi:10.1017/cbo9781139106993.004

Cited by 19 publications

(19 citation statements)

References 12 publications

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“…This criterion is a common generalization of Thurston's Double Limit Theorem [55] and Relative Boundedness Theorem [56, Thm 3.1]. It was generalized to manifolds with compressible boundary by Kleineidam-Souto [32] and Lecuire [35,Thm. 6.6].…”

Section: Neighborhood Systems For Quasiconformally Rigid Pointsmentioning

confidence: 99%

Local topology in deformation spaces of hyperbolic 3-manifolds II

et al. 2019

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Section: Neighborhood Systems For Quasiconformally Rigid Pointsmentioning

confidence: 99%

Local topology in deformation spaces of hyperbolic 3-manifolds II

et al. 2019

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“…If T is an incompressible boundary component of N and λ is a filling lamination on T, then λ can be realized as the pleating locus of a pleated surface homotopic to the inclusion of T in N . When T is a compressible boundary component of N and λ ⊂ T is the support of a doubly incompressible lamination, then λ can be realized as the pleating locus of a pleated surface homotopic to the inclusion of T in N ( [26], Theorem 5.1).…”

Section: Definitionmentioning

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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“…For example, it is an ingredient of the proof of the Bers-Sullivan-Thurston Density Conjecture, which states that the geometrically finite (or equivalently structurally stable) representations are dense in the deformation space of any group. (This proof, whose outline was in place since the work of Thurston and Bonahon, required for its completion also the work of Kleineidam-Souto [80], Lecuire [84], Kim-Lecuire-Ohshika [71], and NamaziSouto. An alternate treatment was given by Rees [114].…”

Section: Thurston's Ending Lamination Conjecture Is Then the Followinmentioning

confidence: 99%

Curve complexes, surfaces and 3-manifolds

Minsky¹

2007

Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006

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Abstract.A survey of the role of the complex of curves in recent work on 3-manifolds and mapping class groups. Mathematics Subject Classification (2000). Primary 57M50; Secondary 30F40, 30F60.Keywords. Kleinian groups, mapping class group, complex of curves. Disjoint curves in surfacesThe complex of curves, a combinatorial object associated to a surface, has been of interest recently in low-dimensional topology and geometry. In the study of mapping class groups of surfaces, it has shed some light on relative hyperbolicity properties, and more generally on the coarse geometry of these groups, and of the Teichmüller spaces (parameter spaces of Riemann surfaces) on which they act. In the setting of hyperbolic 3-manifolds, the complex of curves has been used in the classification theory, particularly the solution of Thurston's Ending Lamination Conjecture, and generally in the attempt to relate more concretely the topology of a 3-manifold to its geometric structure. This paper will attempt to survey some of these developments. Of necessity we will focus on those aspects with which the author is most familiar, thus leaving out a lot of interesting topology and geometry.We will begin with a leisurely discussion of the natural ways in which simple closed curves (i.e. embedded circles), and particularly the relation of disjointness, occur in low-dimensional topology. After this, in §2 we will lay out the basic definitions and theorems about curve complexes. In §3 we will describe work with H. Masur on the inductive structure of curve complexes and its relation to the geometry of mapping class groups. In §4 we will go into more detail about mapping class groups and outline some recent work with J. Behrstock on their asymptotic cones and the BrockFarb rank conjecture. In §5 we will describe work with J. Brock and R. Canary on Thurston's Ending Lamination Conjecture. In §6 we will lay out some thoughts on the hyperbolic geometry of Heegaard splittings, an area in which our knowledge is still rather incomplete.

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An extension of the Masur domain

Cited by 19 publications

References 12 publications

Local topology in deformation spaces of hyperbolic 3-manifolds II

Local topology in deformation spaces of hyperbolic 3-manifolds II

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

Curve complexes, surfaces and 3-manifolds

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