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The boundary of the Gieseking tree in hyperbolic three-space

✍ Scribed by R.C. Alperin; Warren Dicks; J. Porti

Publisher
Elsevier Science
Year
1999
Tongue
English
Weight
541 KB
Volume
93
Category
Article
ISSN
0166-8641

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✦ Synopsis

We give an elementary proof of the Cannon-Thurston Theorem in the case of the Gieseking manifold. We do not use Thurston's structure theory for Kleinian groups but simply calculate with two-by-two complex matrices. We work entirely on the boundary, using ends of trees, and obtain pictures of the regions which are successively filled in by the Peano curve of Cannon and Thurston.

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Let H be the three-dimensional hyperbolic space and let G be the identity component of the isometry group of H . It is known that some aspects of the dynamics of a rigid body in H contrast strongly with the Euclidean case, due to the lack of a subgroup of translations in G. We present the subject in