|
Gromov hyperbolicity of the  and  metrics
Author(s):
Peter
A.
Hästö
Journal:
Proc. Amer. Math. Soc.
134
(2006),
1137-1142.
MSC (2000):
Primary 30F45;
Secondary 53C23, 30C99
Posted:
August 29, 2005
Retrieve article in:
PDF DVI PostScript
Abstract |
References |
Similar articles |
Additional information
Abstract:
In this note it is shown that the  metric is always Gromov hyperbolic, but that the  metric is Gromov hyperbolic if and only if  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.
References:
-
- 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 http://www.helsinki. fi/
 jvaisala/preprints.html] - 11.
- J. Väisälä: Hyperbolic and uniform domains in Banach spaces, preprint (2004). [Available at http://www.helsinki.fi/
 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)
Similar Articles:
Retrieve articles in Proceedings of the American Mathematical Society
with MSC
(2000):
30F45,
53C23, 30C99
Retrieve articles in all Journals with MSC
(2000):
30F45,
53C23, 30C99
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
Email:
peter.hasto@helsinki.fi
DOI:
10.1090/S0002-9939-05-08053-6
PII:
S 0002-9939(05)08053-6
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.
Posted:
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
Copyright of article:
Copyright
2005,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
Comments: webmaster@ams.org