Hyperbolic Manifolds and Discrete Groups

Kapovich, Michael

doi:10.1007/978-0-8176-4913-5

Cited by 311 publications

(360 citation statements)

References 208 publications

(341 reference statements)

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“…Basic references of this section are [2], [10], and [6]. The set ω 0 is contained properly in this set.…”

Section: Ultralimit Of Metric Spacesmentioning

confidence: 99%

The energy of equivariant maps and a fixed-point property for Busemann nonpositive curvature spaces

Tanaka

2011

Trans. Amer. Math. Soc.

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Abstract. For an isometric action of a finitely generated group on the ultralimit of a sequence of global Busemann nonpositive curvature spaces, we state a sufficient condition for the existence of a fixed point of the action in terms of the energy of equivariant maps from the group into the space. Furthermore, we show that this energy condition holds for every isometric action of a finitely generated group on any global Busemann nonpositive curvature space in a family which is stable under ultralimit, whenever each of these actions has a fixed point.We also discuss the existence of a fixed point of affine isometric actions of a finitely generated group on a uniformly convex, uniformly smooth Banach space in terms of the energy of equivariant maps.

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“…Basic references of this section are [2], [10], and [6]. The set ω 0 is contained properly in this set.…”

Section: Ultralimit Of Metric Spacesmentioning

confidence: 99%

The energy of equivariant maps and a fixed-point property for Busemann nonpositive curvature spaces

Tanaka

2011

Trans. Amer. Math. Soc.

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“…A system of meridians α on M is a collection of pairwise nonisotopic and pairwise disjoint meridians α i bounding discs D i such that M \ (∪N (D i )) is a disjoint union of trivial I-bundles over closed surfaces, where N (D i ) is a regular neighborhood of D i . By Thurston's Hyperbolization Theorem (see Kapovich [13]) and Marden ([21]), we can fix σ a convex cocompact representation of π 1 (M ) such that N σ = N σ ∪ ∂ C N σ is homeomorphic to M , where ∂ C N σ is the conformal boundary of N σ . Let Λ(σ) be the limit set of σ(Γ).…”

Section: Compression Bodiesmentioning

confidence: 99%

Separable-stable representations of a compression body

Ким

Lee

2016

Topology and its Applications

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“…The ends of N are in one-to-one correspondence with the components of ∂C − P, where C is a relative compact core and P is the intersection of C with the noncompact components of N thin( ) (for a precise definition of ends see [22], Section 4.23). Any hyperbolic 3-manifold N with finitely generated fundamental group has finitely many ends.…”

Section: Ends Of Hyperbolic Manifoldsmentioning

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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Hyperbolic Manifolds and Discrete Groups

Cited by 311 publications

References 208 publications

The energy of equivariant maps and a fixed-point property for Busemann nonpositive curvature spaces

The energy of equivariant maps and a fixed-point property for Busemann nonpositive curvature spaces

Separable-stable representations of a compression body

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

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