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Boundaries of
hyperbolic metric spaces
Corran Webster and Adam Winchester |
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Vol. 221 (2005), No. 1, 147–158
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Abstract |
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We investigate the relationship between the metric boundary and the Gromov boundary of a hyperbolic metric space. We show that the Gromov boundary is a quotient of the metric boundary and the quotient map is continuous, and that therefore a word-hyperbolic group has an amenable action on the metric boundary of its Cayley graph. Furthermore, if the space is 0-hyperbolic, the boundaries agree, and as a consequence there are no non-Busemann points on the boundary of such spaces. These results have significance for the study of Lip-norms on group C∗-algebras. |
Keywords
hyperbolic space, hyperbolic group, Gromov boundary, Cayley
graph, group C∗-algebra, quantum metric space
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Mathematical Subject Classification 2000
Primary: 20F65
Secondary: 46L87, 53C23
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Milestones
Received: 16 January 2004
Accepted: 30 September 2004
Published: 1 September 2005
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Authors |
| Corran Webster | |
| Department of Mathematical
Sciences University of Nevada Las Vegas Las Vegas, NV 89154 |
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| www.nevada.edu/~cwebster/ | |
| Adam Winchester | |
| Mathematics Department University of California, Los Angeles Los Angeles, CA 90095-1555 |
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