Characterizing complete 𝐶𝐴𝑇(𝜅)-spaces, 𝜅<0, with geodesic boundary

Foertsch, Thomas; Radke, Katrin

doi:10.1090/S0002-9947-2010-05144-X

Characterizing complete $\operatorname {CAT}(\kappa )$-spaces, $\kappa <0$, with geodesic boundary
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by Thomas Foertsch and Katrin Radke

Trans. Amer. Math. Soc. 363 (2011), 75-93

DOI: https://doi.org/10.1090/S0002-9947-2010-05144-X

Published electronically: August 27, 2010

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Abstract:

We investigate the Bourdon and Hamenstädt boundaries of complete $\operatorname {CAT}(\kappa )$-spaces, $\kappa <0$, and characterize those with geodesic Hamenstädt boundary up to isometry.

References

Stephanie B. Alexander and Richard L. Bishop, Warped products of Hadamard spaces, Manuscripta Math. 96 (1998), no. 4, 487–505. MR 1639844, DOI 10.1007/s002290050078
S. B. Alexander and R. L. Bishop, Curvature bounds for warped products of metric spaces, Geom. Funct. Anal. 14 (2004), no. 6, 1143–1181. MR 2135163, DOI 10.1007/s00039-004-0487-2
Mario Bonk and Thomas Foertsch, Asymptotic upper curvature bounds in coarse geometry, Math. Z. 253 (2006), no. 4, 753–785. MR 2221098, DOI 10.1007/s00209-005-0931-5
M. Bonk & T. Foertsch, Asymptotic lower curvature bounds in coarse geometry, preprint 2007
R. L. Bishop and B. O’Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 (1969), 1–49. MR 251664, DOI 10.1090/S0002-9947-1969-0251664-4
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
Marc Bourdon, Structure conforme au bord et flot géodésique d’un $\textrm {CAT}(-1)$-espace, Enseign. Math. (2) 41 (1995), no. 1-2, 63–102 (French, with English and French summaries). MR 1341941
Sergei Buyalo and Viktor Schroeder, Elements of asymptotic geometry, EMS Monographs in Mathematics, European Mathematical Society (EMS), Zürich, 2007. MR 2327160, DOI 10.4171/036
Chien-Hsiung Chen, Warped products of metric spaces of curvature bounded from above, Trans. Amer. Math. Soc. 351 (1999), no. 12, 4727–4740. MR 1466944, DOI 10.1090/S0002-9947-99-02154-6
Thomas Foertsch, Alexander Lytchak, and Viktor Schroeder, Nonpositive curvature and the Ptolemy inequality, Int. Math. Res. Not. IMRN 22 (2007), Art. ID rnm100, 15. MR 2376212, DOI 10.1093/imrn/rnm100
Thomas Foertsch and Viktor Schroeder, A product construction for hyperbolic metric spaces, Illinois J. Math. 49 (2005), no. 3, 793–810. MR 2210259
T. Foertsch & V. Schroeder, Hyperbolicity, $\operatorname {CAT}(-1)$-spaces and the Ptolemy Inequality, to appear in Math. Ann.
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}
Ursula Hamenstädt, A new description of the Bowen-Margulis measure, Ergodic Theory Dynam. Systems 9 (1989), no. 3, 455–464. MR 1016663, DOI 10.1017/S0143385700005095
Urs Lang and Thilo Schlichenmaier, Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions, Int. Math. Res. Not. 58 (2005), 3625–3655. MR 2200122, DOI 10.1155/IMRN.2005.3625