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Conformal Geometry and Dynamics

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The Horofunction boundary of the Heisenberg Group: The Carnot-Carathéodory metric

Authors: Tom Klein and Andrew Nicas
Journal: Conform. Geom. Dyn. 14 (2010), 269-295
MSC (2010): Primary 20F65, 22E25, 53C23, 53C70
Published electronically: November 17, 2010
MathSciNet review: 2738530
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Abstract: We find the horofunction boundary of the $ (2n+1)$-dimensional Heisenberg group with the Carnot-Carathéodory distance and show that it is homeomorphic to a $ 2n$-dimensional disk and that the Busemann points correspond to the $ (2n-1)$-sphere boundary of this disk. We also show that the compactified Heisenberg group is homeomorphic to a $ (2n+1)$-dimensional sphere. As an application, we find the group of isometries of the Carnot-Carathéodory distance.

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Additional Information

Tom Klein
Affiliation: 325 Island Drive, Apt 6, Madison, Wisconsin 53705

Andrew Nicas
Affiliation: Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K1

Keywords: Heisenberg group, Carnot-Carathéodory distance, horofunction boundary, Busemann points, isometries
Received by editor(s): February 26, 2010
Published electronically: November 17, 2010
Additional Notes: The second author was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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