Available in electronic format
Available in print format		 Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

Comparing the Floyd and ideal boundaries of a metric space

Author(s): Stephen M. Buckley; Simon L. Kokkendorff
Journal: Trans. Amer. Math. Soc. 361 (2009), 715-734.
MSC (2000): Primary 54D35; Secondary 30F45, 53A30
Posted: September 29, 2008
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: We discuss and compare the notions of ideal boundaries, Floyd boundaries and Gromov boundaries of metric spaces. The three types of boundaries at infinity are compared in the general setting of unbounded length spaces as well as in the special cases of CAT(0) and Gromov hyperbolic spaces. Gromov boundaries, usually defined only for Gromov hyperbolic spaces, are extended to arbitrary metric spaces.

References:

[BB1]
Z. Balogh and S.M. Buckley, Geometric characterizations of Gromov hyperbolicity, Invent. Math. 153 (2003), 261-301. MR 1992014 (2004i:30042)

[BB2]
Z. Balogh and S.M. Buckley, Sphericalization and flattening, Conform. Geom. Dyn. 9 (2005), 76-101. MR 2179368 (2006k:30046)

[BH]
M. Bridson and A. Haefliger, Metric spaces of non-positive curvature, Springer-Verlag, Berlin, 1999. MR 1744486 (2000k:53038)

[BHK]
M. Bonk, J. Heinonen, and P. Koskela, Uniformizing Gromov hyperbolic spaces, Astérisque 270 (2001). MR 1829896 (2003b:30024)

[BS]
M. Bonk and O. Schramm, Embeddings of Gromov hyperbolic spaces, Geom. Funct. Anal. 10 (2000), 266-306. MR 1771428 (2001g:53077)

[CT]
J.W. Cannon and W.P. Thurston, Group invariant Peano curves, Geom. Topol. 11 (2007), 1315-1355. MR 2326947 (2008i:57016)

[CDP]
M. Coornaert, T. Delzant, and A. Papadopoulos, Géometrie et théories des groupes, Lecture Notes in Mathematics 1441, Springer, Berlin, 1990. MR 1075994 (92f:57003)

[F]
W.J. Floyd, Group completions and limit sets of Kleinian groups, Invent. Math. 57 (1980), 205-218. MR 568933 (81e:57002)

[GH]
E. Ghys and P. de la Harpe (Eds.), Sur les groupes hyperboliques d'aprés Mikhael Gromov, Progress in Math. 38, Birkhäuser, Boston, 1990. MR 1086648 (92f:53050)

[G]
M. Gromov, Hyperbolic Groups, Essays in Group Theory, S. Gersten, Editor, MSRI Publication, Springer-Verlag, 1987, pp. 75-265. MR 919829 (89e:20070)

[H]
D. Herron, Quasiconformal deformations and volume growth, Proc. London Math. Soc. (3) 92 (2006), 161-199. MR 2192388 (2006m:30035)

[K1]
A. Karlsson, Free subgroups of groups with nontrivial Floyd boundary, Comm. Algebra 31 (2003), 5361-5376. MR 2005231 (2004g:20059)

[K2]
A. Karlsson, Boundaries and random walks on finitely generated infinite groups, Ark. Mat. 41 (2003), 295-306. MR 2011923 (2004j:60095)

[M]
C.T. McMullen, Local connectivity, Kleinian groups and geodesics on the blowup of the torus, Invent. Math. 146 (2001), 35-91. MR 1859018 (2004e:30068)

[T]
P. Tukia, A remark on a paper by Floyd, Holomorphic functions and moduli, Vol. II (Berkeley, CA, 1986), Math. Sci. Res. Inst. Publ., 11, Springer, New York, 1988, pp. 165-172. MR 955838 (90b:30054)

[V]
J. Väisälä, Gromov hyperbolic spaces, Expo. Math. 23 (2005), 187-231. MR 2164775 (2006j:53055)

Similar Articles:

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 54D35, 30F45, 53A30

Retrieve articles in all Journals with MSC (2000): 54D35, 30F45, 53A30

Additional Information:

Stephen M. Buckley
Affiliation: Department of Mathematics, National University of Ireland, Maynooth, Co. Kildare, Ireland
Email: Stephen.Buckley@may.ie

Simon L. Kokkendorff
Affiliation: Department of Mathematics, Technical University of Denmark, Building 303, 2800 Kgs. Lyngby, Denmark
Address at time of publication: National Survey and Cadastre, Rentemestervej 8, 2400 Copenhagen, Denmark
Email: s.l.kokkendorff@mat.dtu.dk, simlk@kms.dk

DOI: 10.1090/S0002-9947-08-04580-7
PII: S 0002-9947(08)04580-7
Keywords: Ideal boundary, Floyd boundary, conformal distortion, Gromov hyperbolicity, Gromov boundary, CAT(0)-spaces
Received by editor(s): January 15, 2006
Received by editor(s) in revised form: November 3, 2006
Posted: September 29, 2008
Additional Notes: The first author was partially supported, and the second author fully supported, by Enterprise Ireland
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2009, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google