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

Klein, Tom; Nicas, Andrew

doi:10.1090/S1088-4173-2010-00217-1

The Horofunction boundary of the Heisenberg Group: The Carnot-Carathéodory metric
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by Tom Klein and Andrew Nicas

Conform. Geom. Dyn. 14 (2010), 269-295

DOI: https://doi.org/10.1090/S1088-4173-2010-00217-1

Published electronically: November 17, 2010

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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.

References

Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486, DOI 10.1007/978-3-662-12494-9
Ovidiu Calin, Der-Chen Chang, and Peter Greiner, Geometric mechanics on the Heisenberg group, Bull. Inst. Math. Acad. Sinica 33 (2005), no. 3, 185–252. MR 2165285
Luca Capogna, Donatella Danielli, Scott D. Pauls, and Jeremy T. Tyson, An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem, Progress in Mathematics, vol. 259, Birkhäuser Verlag, Basel, 2007. MR 2312336, DOI 10.1007/978-3-7643-8133-2
M. Gromov, Hyperbolic manifolds, groups and actions, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978) Ann. of Math. Stud., vol. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 183–213. MR 624814
Ursula Hamenstädt, Some regularity theorems for Carnot-Carathéodory metrics, J. Differential Geom. 32 (1990), no. 3, 819–850. MR 1078163
Iwao Kishimoto, Geodesics and isometries of Carnot groups, J. Math. Kyoto Univ. 43 (2003), no. 3, 509–522. MR 2028665, DOI 10.1215/kjm/1250283693
Tom Klein and Andrew Nicas, The horofunction boundary of the Heisenberg group, Pacific J. Math. 242 (2009), no. 2, 299–310. MR 2546714, DOI 10.2140/pjm.2009.242.299
Adam Korányi, Geometric properties of Heisenberg-type groups, Adv. in Math. 56 (1985), no. 1, 28–38. MR 782541, DOI 10.1016/0001-8708(85)90083-0
Marc A. Rieffel, Group $C^*$-algebras as compact quantum metric spaces, Doc. Math. 7 (2002), 605–651. MR 2015055
Cormac Walsh, The horofunction boundary of finite-dimensional normed spaces, Math. Proc. Cambridge Philos. Soc. 142 (2007), no. 3, 497–507. MR 2329698, DOI 10.1017/S0305004107000096