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Geometriae Dedicata

, Volume 180, Issue 1, pp 49–68 | Cite as

Coarse median structures and homomorphisms from Kazhdan groups

  • Rudolf Zeidler
Original Paper
  • 178 Downloads

Abstract

We study Bowditch’s notion of a coarse median on a metric space and formally introduce the concept of a coarse median structure as an equivalence class of coarse medians up to closeness. We show that a group which possesses a uniformly left-invariant coarse median structure admits only finitely many conjugacy classes of homomorphisms from a given group with Kazhdan’s property (T). This is a common generalization of a theorem due to Paulin about the outer automorphism group of a hyperbolic group with property (T) as well as of a result of Behrstock–Druţu–Sapir on the mapping class groups of orientable surfaces. We discuss a metric approximation property of finite subsets in coarse median spaces extending the classical result on approximation of Gromov hyperbolic spaces by trees.

Keywords

Coarse median spaces Property (T) Outer automorphism group 

Mathematics Subject Classification (2010)

20F65 

Notes

Acknowledgments

The author would like to express his gratitude towards his Master thesis advisor, Goulnara Arzhantseva, for introducing him to this subject, sharing her knowledge, many helpful suggestions, and most of all, for encouraging him to write this paper. He wishes to thank Frédéric Paulin for giving an enlightening mini-course at the second Young Geometric Group Theory Meeting which stimulated the author’s work towards a generalization of the result concerning outer automorphisms of hyperbolic groups with Kazhdan’s property (T). The author also thanks Bogdan Nica and the anonymous referee for useful comments on the manuscript.

References

  1. 1.
    Ballmann, W., Świa̧tkowski, J.: On \(L^{2}\)-cohomology and property (T) for automorphism groups of polyhedral cell complexes. Geom. Funct. Anal. 4, 615–645 (1997). doi: 10.1007/s000390050022 CrossRefGoogle Scholar
  2. 2.
    Bandelt, H.-J., Hedlíková, J.: Median algebras. Discrete Math. 1, 1–30 (1983). doi: 10.1016/0012-365X(83)90173-5 CrossRefGoogle Scholar
  3. 3.
    Behrstock, J.A., Minsky, Y.N.: Centroids and the rapid decay property in mapping class groups. J. Lond. Math. Soc. 3(2), 765–784 (2011). doi: 10.1112/jlms/jdr027 CrossRefMathSciNetGoogle Scholar
  4. 4.
    Behrstock, J., Druţu, C., Sapir, M.: Median structures on asymptotic cones and homomorphisms into mapping class groups. Proc. Lond. Math. Soc. 3(3), 503–554 (2011). doi: 10.1112/plms/pdq025 CrossRefGoogle Scholar
  5. 5.
    Bowditch, B.H.: Coarse median spaces and groups. Pac. J. Math. 1, 53–93 (2013). doi: 10.2140/pjm.2013.261.53 CrossRefMathSciNetGoogle Scholar
  6. 6.
    Bowditch, B.H.: Embedding median algebras in products of trees. Geom. Dedic. (2014). doi: 10.1007/s10711-013-9874-x
  7. 7.
    Bowditch, B.H.: Invariance of coarse median spaces under relative hyperbolicity. Math. Proc. Camb. Philos. Soc. 1, 85–95 (2013). doi: 10.1017/S0305004112000382 CrossRefMathSciNetGoogle Scholar
  8. 8.
    Bowditch, B.H.: Notes on Gromov’s hyperbolicity criterion for path-metric spaces. In: Ghys, É., Haefliger, A., Verjovsky, A. (eds.) Group Theory from a Geometrical Viewpoint, pp. 64–167. World Science Publications, Singapore (1991)Google Scholar
  9. 9.
    Bowditch, B.H.: Some properties of median metric spaces. Groups Geom. Dyn. (2015, to appear). http://homepages.warwick.ac.uk/~masgak/preprints.html
  10. 10.
    Bridson, M.R., Haefliger, A.: Metric Spaces of Non-positive Curvature. Springer, Berlin (1999)CrossRefzbMATHGoogle Scholar
  11. 11.
    Chatterji, I., Drutu, C., Haglund, F.: Kazhdan and Haagerup properties from the median viewpoint. Adv. Math. 2, 882–921 (2010). doi: 10.1016/j.aim.2010.03.012 CrossRefMathSciNetGoogle Scholar
  12. 12.
    Chatterji, I., Drutu, C., Haglund, F.: Median spaces and spaces with measured walls (2007). http://infoscience.epfl.ch/record/126268
  13. 13.
    Chatterji, I., Niblo, G.: From wall spaces to CAT(0) cube complexes. Int. J. Algebra Comput. 5–6, 875–885 (2005). doi: 10.1142/S0218196705002669 CrossRefMathSciNetGoogle Scholar
  14. 14.
    Chepoi, V.: Graphs of some CAT(0) complexes. Adv. Appl. Math. 2, 125–179 (2000). doi: 10.1006/aama.1999.0677 CrossRefMathSciNetGoogle Scholar
  15. 15.
    Gromov, M.: Hyperbolic groups. In: Gersten, S. M. (ed.) Essays in Group Theory, pp. 75–263. Springer, New York (1987)Google Scholar
  16. 16.
    Nica, B.: Group Actions on Median Spaces. Master thesis, McGill University (2004)Google Scholar
  17. 17.
    Nica, B.: Group actions on median spaces (2008). arXiv: 0809.4099 [math.GR]
  18. 18.
    Paulin, F.: Outer automorphisms of hyperbolic groups and small actions on R-trees. In: Alperin, R. C. (ed.) Arboreal Group Theory, pp. 331–343. Springer, New York (1991). doi: 10.1007/978-1-4612-3142-4_12
  19. 19.
    Roller, M.: Poc Sets, Median Algebras and Group Actions. An Extended Study of Dunwoody’s Construction and Sageev’s Theorem. Habilitationsschrift, Regensberg (1998)Google Scholar
  20. 20.
    Wise, D.T.: Cubulating small cancellation groups. Geom. Funct. Anal. 1, 150–214 (2004). doi: 10.1007/s00039-004-0454-y CrossRefMathSciNetGoogle Scholar
  21. 21.
    Zeidler, R.: Coarse median structures on groups. Masterarbeit. Universität Wien (2013)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Mathematisches InstitutGeorg-August-Universität GöttingenGöttingenGermany

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