Interplay between interior and boundary geometry in Gromov hyperbolic spaces. (English) Zbl 1206.53047
The author shows that two visual and geodesic Gromov hyperbolic metric spaces are roughly isometric if and only if their boundaries at infinity, equipped with suitable quasimetrics, are bi-Lipschitz quasi-Möbius equivalent. This result can be regarded as an extension theorem for bi-Lipschitz mappings.
Reviewer: Peibiao Zhao (Nanjing)
MSC:
| 53C23 | Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces |
| 54C20 | Extension of maps |
| 20F67 | Hyperbolic groups and nonpositively curved groups |
References:
| [1] | Bonk M., Schramm O.: Embeddings of Gromov hyperbolic spaces. Geom. Funct. Anal. 10, 266–306 (2000) · Zbl 0972.53021 · doi:10.1007/s000390050009 |
| [2] | Bridson M., Haefliger A.: Metric Spaces of Non-Positive Curvature. Springer, Berlin (1999) · Zbl 0988.53001 |
| [3] | Buyalo S., Schroeder V.: Elements of Asymptotic Geometry. EMS Monographs in Mathematics, European Mathematical Society, Zurich (2007) · Zbl 1125.53036 |
| [4] | Frink A.: Distance functions and the metrization problem. Bull. Am. Math. Soc. 43, 133–142 (1937) · Zbl 0016.08205 · doi:10.1090/S0002-9904-1937-06509-8 |
| [5] | Meyer, J.: Uniformly perfect boundaries of Gromov hyperbolic spaces. PhD thesis, University of Zurich (2009) |
| [6] | Paulin F.: Un groupe hyperbolique est déterminé par son bord. J. Lond. Math. Soc. 54(2), 50–74 (1996) · Zbl 0854.20050 |
| [7] | Schroeder, V.: An introduction to asymptotic geometry. To appear in IRMA Lectures in Mathematics and Theoretical Physics (2009) · Zbl 1251.53025 |
| [8] | Trotsenko D., Väisälä J.: Upper sets and quasisymmetric maps. Ann. Acad. Sci. Fenn. 24, 465–488 (1999) · Zbl 0945.30021 |
| [9] | Tukia P., Väisälä J.: Quasisymmetric embeddings of metric spaces. Ann. Acad. Sci. Fenn. Ser. A I Math. 5, 97–114 (1980) · Zbl 0403.54005 |
| [10] | Väisälä, J.: Quasimöbius maps. J. Anal. Math. 44, 218–234 (1984/1985) · Zbl 0593.30022 |
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