A 4-dimensional Kleinian group

Bowditch, Brian H.; Mess, Geoffrey

doi:10.1090/s0002-9947-1994-1240944-6

Cited by 26 publications

(25 citation statements)

References 20 publications

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“…The material in this section is based on Thurston's bending deformation, and the treatment of this which is given in [1]. We begin with a lemma about gluing convex manifolds.…”

Section: Constructing Immersed Surfaces By Gluing Convex Manifoldsmentioning

confidence: 99%

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Counting immersed surfaces in hyperbolic 3–manifolds

Masters

2005

Algebr. Geom. Topol.

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“…The material in this section is based on Thurston's bending deformation, and the treatment of this which is given in [1]. We begin with a lemma about gluing convex manifolds.…”

Section: Constructing Immersed Surfaces By Gluing Convex Manifoldsmentioning

confidence: 99%

“…This is a well-known construction, and the treatment here is similar to that of [1]. We will need the following definitions and notation.…”

Section: Example: Reflection Orbifoldsmentioning

confidence: 99%

Counting immersed surfaces in hyperbolic 3–manifolds

Masters

2005

Algebr. Geom. Topol.

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“…On the other hand, it was shown in [4] that a finitely generated but infinitely presented group can also appear as a subgroup of a cocompact group in SO(1, 4). (1) β 1 (F L ) contains infinitely many conjugacy classes of maximal parabolic subgroups,…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

The boundary of the deformation space of the fundamental group of some hyperbolic 3–manifolds fibering over the circle

Potyagailo

1998

Geometry &Amp; Topology Monographs

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By using Thurston's bending construction we obtain a sequence of faithful discrete representations ρ n of the fundamental group of a closed hyperbolic 3-manifold fibering over the circle into the isometry group Iso H 4 of the hyperbolic space H 4 . The algebraic limit of ρ n contains a finitely generated subgroup F whose 3-dimensional quotient Ω(F )/F has infinitely generated fundamental group, where Ω(F ) is the discontinuity domain of F acting on the sphere at infinity S 3 ∞ = ∂H 4 . Moreover F is isomorphic to the fundamental group of a closed surface and contains infinitely many conjugacy classes of maximal parabolic subgroups.

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“…Thus ρ(n + 1/2) > n + 10 n for all n and G(p, n + 1/2, n + 10 n ) is not isomorphic to π 1 (M ) for any n. Example 2.7. Bowditch and Mess [3] and Potyagailo [15] have shown that there are examples of complete hyperbolic manifolds such that ρ(r) = ∞ for all r ≥ 1. This is also pointed out by Sormani and Wei, see Example 4.1 in [23] .…”

Section: Nullhomotopy Radiusmentioning

confidence: 99%

Noncompact manifolds with nonnegative Ricci curvature

Wylie

2006

J Geom Anal

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Let (M, d) be a metric space. For 0 < r < R, let G(p, r, R) be the group obtained by considering all loops based at a point p ∈ M whose image is contained in the closed ball of radius r and identifying two loops if there is a homotopy betweeen them that is contained in the open ball of radius R. In this paper we study the asymptotic behavior of the G(p, r, R) groups of complete open manifolds of nonnegative Ricci curvature. We also find relationships between the G(p, r, R) groups and tangent cones at infinity of a metric space and show that any tangent cone at infinity of a complete open manifold of nonnegative Ricci curvature and small linear diameter growth is its own universal cover.

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A 4-dimensional Kleinian group

Abstract: Abstract.We give an example of a 4-dimensional Kleinian group which is finitely generated but not finitely presented, and is a subgroup of a cocompact Kleinian group.

Cited by 26 publications

References 20 publications

Counting immersed surfaces in hyperbolic 3–manifolds

Counting immersed surfaces in hyperbolic 3–manifolds

The boundary of the deformation space of the fundamental group of some hyperbolic 3–manifolds fibering over the circle

Noncompact manifolds with nonnegative Ricci curvature

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