Advertisement

Mathematische Zeitschrift

, Volume 278, Issue 3–4, pp 649–661 | Cite as

Fixed point theorem for reflexive Banach spaces and uniformly convex non positively curved metric spaces

  • Izhar Oppenheim
Article

Abstract

This article generalizes the work of Ballmann and Światkowski to the case of Reflexive Banach spaces and uniformly convex Busemann spaces, thus giving a new fixed point criterion for groups acting on simplicial complexes.

Keywords

Fixed point theorem Busemann space Reflexive Banach space Property (T) 

Mathematics Subject Classification (2010)

20F65 

Notes

Acknowledgments

I would like to thank Uri Bader for many discussions about property (T) and fixed point properties and for communicating the work of Nowak which motivated this article.

References

  1. 1.
    Ballmann, W., Świątkowski, J.: On \(L^2\)-cohomology and property (T) for automorphism groups of polyhedral cell complexes. Geom. Funct. Anal. 7(4), 615–645 (1997). doi:  10.1007/s000390050022. http://dx.doi.org/10.1007/s000390050022
  2. 2.
    Bekka, B., de la Harpe, P., Valette, A.: Kazhdan’s property (T), New Mathematical Monographs, vol. 11. Cambridge University Press, Cambridge (2008). doi:  10.1017/CBO9780511542749. http://dx.doi.org/10.1017/CBO9780511542749
  3. 3.
    Bourdon, M.: Un théorème de point fixe sur les espaces \(L^p\). Publ. Mat 56, 375–392 (2012). doi:  10.5565/PUBLMAT5621205. http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?handle=euclid.pm/1340127810&view=body&content-type=pdf_1
  4. 4.
    Dymara, J., Januszkiewicz, T.: New Kazhdan groups. Geom. Dedicata 80(1–3), pp. 311–317 (2000). doi:  10.1023/A:1005255003263. http://dx.doi.org/10.1023/A:1005255003263
  5. 5.
    Fleming, R.J., Jamison, J.E.: Isometries on Banach Spaces: Function Spaces, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics. Chapman & Hall/CRC, Boca Raton, FL (2003)Google Scholar
  6. 6.
    Gelander, T., Karlsson, A., Margulis, G.A.: Superrigidity, generalized harmonic maps and uniformly convex spaces. Geom. Funct. Anal. 17(5), pp. 1524–1550 (2008). doi:  10.1007/s00039-007-0639-2. http://dx.doi.org/10.1007/s00039-007-0639-2
  7. 7.
    Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Monographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker Inc., New York (1984)Google Scholar
  8. 8.
    Gromov, M.: Random walk in random groups. Geom. Funct. Anal. 13(1), pp. 73–146 (2003). doi:  10.1007/s000390300002. http://dx.doi.org/10.1007/s000390300002
  9. 9.
    Izeki, H., Nayatani, S.: Combinatorial harmonic maps and discrete-group actions on Hadamard spaces. Geom. Dedicata 114, pp. 147–188 (2005). doi:  10.1007/s10711-004-1843-y. http://dx.doi.org/10.1007/s10711-004-1843-y
  10. 10.
    Kohlenbach, U., Leuştean, L.: Asymptotically nonexpansive mappings in uniformly convex hyperbolic spaces. J. Eur. Math. Soc. (JEMS) 12(1), 71–92 (2010). doi:  10.4171/JEMS/190. http://dx.doi.org/10.4171/JEMS/190
  11. 11.
    Reich, S., Shafrir, I.: Nonexpansive iterations in hyperbolic spaces. Nonlinear Anal. 15(6), pp. 537–558 (1990). doi:  10.1016/0362-546X(90)90058-O. http://dx.doi.org/10.1016/0362-546X(90)90058-O
  12. 12.
    Żuk, A.: La propriété (T) de Kazhdan pour les groupes agissant sur les polyèdres. C. R. Acad. Sci. Paris Sér. I Math. 323(5), pp. 453–458 (1996)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of MathematicsOhio State UniversityColumbusUSA

Personalised recommendations