Construction of Acylindrical Hyperbolic 3-Manifolds with Quasifuchsian Boundary

Zhang, Yongquan

doi:10.1080/10586458.2020.1718566

Cited by 5 publications

(7 citation statements)

References 19 publications

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“…We can deform Q by pushing closer or pulling apart Faces 3 and 5, fixing the dihedral angles, and obtain deformations of the orbifold. This gives [Zha,Thm. 1. , there exists a unique hyperbolic polyhedron Q(t) so that the hyperbolic distance between Faces 3 and 5 is cosh −1 (t).…”

Section: An Acylindrical Examplementioning

confidence: 93%

“…See [Zha,Figure 3] for some samples of the deformation. One way to explicitly construct M (t) is to take two copies of Q(t) and identify corresponding faces.…”

Section: An Acylindrical Examplementioning

confidence: 99%

“…Consistent with the notations there, let η be the simple closed geodesic in M (t) coming from two copies of the geodesic segment orthogonal to both Faces 3 and 5, and ξ be that from Faces 2 and 4. We have [Zha,Thm. 1.3]:…”

Section: An Acylindrical Examplementioning

confidence: 99%

“…In this section, we calculate the explicit value of t 0 predicted in the previous section. We refer to the calculations in [Zha,§6] freely.…”

Section: Computationsmentioning

confidence: 99%

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Existence of an exotic plane in an acylindrical 3-manifold

Zhang¹

2021

Preprint

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Let P be a geodesic plane in a convex cocompact, acylindrical hyperbolic 3-manifold M . Assume that P * = M * ∩ P is nonempty, where M * is the interior of the convex core of M . Does this condition imply that P is either closed or dense in M ? A positive answer would furnish an analogue of Ratner's theorem in the infinite volume setting.In [MMO2] it is shown that P * is either closed or dense in M * . Moreover, there are at most countably many planes with P * closed, and in all previously known examples, P was also closed in M .In this note we show more exotic behavior can occur: namely, we give an explicit example of a pair (M, P ) such that P * is closed in M * but P is not closed in M . In particular, the answer to the question above is no. Thus Ratner's theorem fails to generalize to planes in acylindrical 3-manifolds, without additional restrictions.

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Section: An Acylindrical Examplementioning

confidence: 93%

“…See [Zha,Figure 3] for some samples of the deformation. One way to explicitly construct M (t) is to take two copies of Q(t) and identify corresponding faces.…”

Section: An Acylindrical Examplementioning

confidence: 99%

Section: An Acylindrical Examplementioning

confidence: 99%

“…In this section, we calculate the explicit value of t 0 predicted in the previous section. We refer to the calculations in [Zha,§6] freely.…”

Section: Computationsmentioning

confidence: 99%

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Existence of an exotic plane in an acylindrical 3-manifold

Zhang¹

2021

Preprint

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“…We remark that this is a common strategy for producing hyperbolic 3-manifolds with desired properties, see e.g. [Thu2,§3.3], [PZ], [Fri], [Gas], and [Zha1].…”

Section: The Apollonian Orbifoldmentioning

confidence: 99%

Elementary planes in the Apollonian orbifold

Zhang¹

2021

Preprint

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In this paper, we study the topological behavior of elementary planes in the Apollonian orbifold M A , whose limit set is the classical Apollonian gasket. The existence of these elementary planes leads to the following failure of equidistribution: there exists a sequence of closed geodesic planes in M A limiting only on a finite union of closed geodesic planes. This contrasts with other acylindrical hyperbolic 3-manifolds analyzed in [MMO1, MMO2, BO].On the other hand, we show that certain rigidity still holds: the area of an elementary plane in M A is uniformly bounded above, and the union of all elementary planes is closed. This is achieved by obtaining a complete list of elementary planes in M A , indexed by their intersection with the convex core boundary. The key idea is to recover information on a closed geodesic plane in M A from its boundary data; requiring the plane to be elementary in turn puts restrictions on these data.

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Kleinian Groups from the Sphere at Infinity and Their Self-Joinings

Kim

2024

Handbook of Visual, Experimental and Computational Mathematics

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Construction of Acylindrical Hyperbolic 3-Manifolds with Quasifuchsian Boundary

Cited by 5 publications

References 19 publications

Existence of an exotic plane in an acylindrical 3-manifold

Existence of an exotic plane in an acylindrical 3-manifold

Elementary planes in the Apollonian orbifold

Kleinian Groups from the Sphere at Infinity and Their Self-Joinings

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