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Gromov hyperbolicity of the $j_G$ and ${\tilde\jmath}_G$ metrics

Author: Peter A. Hästö
Journal: Proc. Amer. Math. Soc. 134 (2006), 1137-1142
MSC (2000): Primary 30F45; Secondary 53C23, 30C99
Published electronically: August 29, 2005
MathSciNet review: 2196049
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Abstract | References | Similar Articles | Additional Information

Abstract: In this note it is shown that the ${\tilde\jmath}_G$ metric is always Gromov hyperbolic, but that the $j_G$ metric is Gromov hyperbolic if and only if $G$ has exactly one boundary point. As a corollary we get a new proof for the fact that the quasihyperbolic metric is Gromov hyperbolic in uniform domains.

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  • 1. Y. Benoist: Convexes hyperboliques et fonctions quasisymétriques (French) [Hyperbolic convex sets and quasisymmetric functions], Publ. Math. Inst. Hautes Études Sci. 97 (2003), 181-237. MR 2010741
  • 2. M. Bonk, J. Heinonen and P. Koskela: Uniformizing Gromov hyperbolic spaces, Astérisque 270 (2001). MR 1829896 (2003b:30024)
  • 3. M. Bonk and O. Schramm: Embeddings of Gromov hyperbolic spaces, Geom. Funct. Anal. 10 (2000), no. 2, 266-306. MR 1771428 (2001g:53077)
  • 4. D. Burago, Y. Burago and S. Ivanov: A course in metric geometry, Graduate Studies Math. 33, AMS, Providence, RI, 2001. MR 1835418 (2002e:53053)
  • 5. M. Coornaert, T. Delzant and A. Papadopoulos: Géométrie et théorie des groupes, Lecture Notes in Math. 1441, Springer-Verlag, Berlin-Heidelberg-New York, 1990. MR 1075994 (92f:57003)
  • 6. F. W. Gehring and B. G. Osgood: Uniform domains and the quasihyperbolic metric, J. Anal. Math. 36 (1979), 50-74. MR 0581801 (81k:30023)
  • 7. E. Ghys and P. de la Harpe (eds.): Sur les groupes hyperboliques d'après Mikhael Gromov, Progress in Mathematics 83, Birkhäuser, Boston, MA, 1990. MR 1086648 (92f:53050)
  • 8. M. Gromov: Hyperbolic groups, in Essays in group theory (S. M. Gersten, ed.), MSRI Publ. 8 (1987), 75-263. MR 0919829 (89e:20070)
  • 9. A. Karlsson and G. A. Noskov: The Hilbert metric and Gromov hyperbolicity, Enseign. Math. (2) 48 (2002), no. 1-2, 73-89. MR 1923418 (2003f:53061)
  • 10. J. Väisälä: Gromov hyperbolic spaces, preprint (2004). [Available at fi/$\sim$jvaisala/preprints.html]
  • 11. J. Väisälä: Hyperbolic and uniform domains in Banach spaces, preprint (2004). [Available at$\sim$jvaisala/preprints.html]
  • 12. M. Vuorinen: Conformal invariants and quasiregular mappings, J. Anal. Math. 45 (1985), 69-115. MR 0833408 (87k:30034)
  • 13. M. Vuorinen: Conformal Geometry and Quasiregular Mappings, Lecture Notes in Math. 1319, Springer-Verlag, Berlin-Heidelberg-New York, 1988. MR 0950174 (89k:30021)

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Additional Information

Peter A. Hästö
Affiliation: Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491 Trondheim, Norway
Address at time of publication: Department of Mathematics and Statistics, P.O. Box 68, FIN-00014 University of Helsinki, Finland

Keywords: Gromov hyperbolic, $j_G$ metric, ${\tilde\jmath}_G$ metric, quasihyperbolic metric
Received by editor(s): March 3, 2004
Received by editor(s) in revised form: November 3, 2004
Published electronically: August 29, 2005
Additional Notes: The author was supported in part by a Gehring-Finland Post-doctoral Fellowship and by the Finnish Academy of Science and Letters.
Communicated by: Juha M. Heinonen
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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