Changing topological type of compression bodies through cone manifolds

Generically, small deformations of cone manifold holonomy groups have wildly uncontrolled global geometry. We give a short concrete example showing that it is possible to deform complete hyperbolic metrics on a thickened genus \(2\) surface to complete hyperbolic metrics on the genus two handlebody...

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Bibliographic Details
Published in:arXiv.org 2024-11
Main Author: Elzenaar, Alex
Format: Article
Language:English
Subjects:
Curves
Deformation
Global geometry
Group theory
Manifolds (mathematics)
Online Access:Get full text
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Summary:Generically, small deformations of cone manifold holonomy groups have wildly uncontrolled global geometry. We give a short concrete example showing that it is possible to deform complete hyperbolic metrics on a thickened genus \(2\) surface to complete hyperbolic metrics on the genus two handlebody with a single unknotted cusp drilled out via cone manifolds of prescribed singular structure. In other words, there exists a method to construct smooth curves in the character variety of \( \pi_1(S_{2,0}) \) which join open sets parameterising discrete groups (quasi-conformal deformation spaces) through indiscrete groups where the indiscreteness arises in a very controlled, local, way: a cone angle increasing along a fixed singular locus.
ISSN:2331-8422