Comparing the Floyd and ideal boundaries of a metric space

Buckley, Stephen; Kokkendorff, Simon

doi:10.1090/S0002-9947-08-04580-7

Comparing the Floyd and ideal boundaries of a metric space
HTML articles powered by AMS MathViewer

by Stephen M. Buckley and Simon L. Kokkendorff PDF

Trans. Amer. Math. Soc. 361 (2009), 715-734 Request permission

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

Zoltán M. Balogh and Stephen M. Buckley, Geometric characterizations of Gromov hyperbolicity, Invent. Math. 153 (2003), no. 2, 261–301. MR 1992014, DOI 10.1007/s00222-003-0287-6
Zoltán M. Balogh and Stephen M. Buckley, Sphericalization and flattening, Conform. Geom. Dyn. 9 (2005), 76–101. MR 2179368, DOI 10.1090/S1088-4173-05-00124-4
Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486, DOI 10.1007/978-3-662-12494-9
Mario Bonk, Juha Heinonen, and Pekka Koskela, Uniformizing Gromov hyperbolic spaces, Astérisque 270 (2001), viii+99. MR 1829896
M. Bonk and O. Schramm, Embeddings of Gromov hyperbolic spaces, Geom. Funct. Anal. 10 (2000), no. 2, 266–306. MR 1771428, DOI 10.1007/s000390050009
James W. Cannon and William P. Thurston, Group invariant Peano curves, Geom. Topol. 11 (2007), 1315–1355. MR 2326947, DOI 10.2140/gt.2007.11.1315
M. Coornaert, T. Delzant, and A. Papadopoulos, Géométrie et théorie des groupes, Lecture Notes in Mathematics, vol. 1441, Springer-Verlag, Berlin, 1990 (French). Les groupes hyperboliques de Gromov. [Gromov hyperbolic groups]; With an English summary. MR 1075994, DOI 10.1007/BFb0084913
William J. Floyd, Group completions and limit sets of Kleinian groups, Invent. Math. 57 (1980), no. 3, 205–218. MR 568933, DOI 10.1007/BF01418926
É. Ghys and P. de la Harpe (eds.), Sur les groupes hyperboliques d’après Mikhael Gromov, Progress in Mathematics, vol. 83, Birkhäuser Boston, Inc., Boston, MA, 1990 (French). Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988. MR 1086648, DOI 10.1007/978-1-4684-9167-8
M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR 919829, DOI 10.1007/978-1-4613-9586-7_{3}
David A. Herron, Quasiconformal deformations and volume growth, Proc. London Math. Soc. (3) 92 (2006), no. 1, 161–199. MR 2192388, DOI 10.1017/S0024611505015509
Anders Karlsson, Free subgroups of groups with nontrivial Floyd boundary, Comm. Algebra 31 (2003), no. 11, 5361–5376. MR 2005231, DOI 10.1081/AGB-120023961
Anders Karlsson, Boundaries and random walks on finitely generated infinite groups, Ark. Mat. 41 (2003), no. 2, 295–306. MR 2011923, DOI 10.1007/BF02390817
Curtis T. McMullen, Local connectivity, Kleinian groups and geodesics on the blowup of the torus, Invent. Math. 146 (2001), no. 1, 35–91. MR 1859018, DOI 10.1007/PL00005809
Pekka Tukia, A remark on a paper by Floyd, Holomorphic functions and moduli, Vol. II (Berkeley, CA, 1986) Math. Sci. Res. Inst. Publ., vol. 11, Springer, New York, 1988, pp. 165–172. MR 955838, DOI 10.1007/978-1-4613-9611-6_{1}1
Jussi Väisälä, Gromov hyperbolic spaces, Expo. Math. 23 (2005), no. 3, 187–231. MR 2164775, DOI 10.1016/j.exmath.2005.01.010