Journal of Geometry

, 109:15 | Cite as

On automorphism groups of toroidal circle planes

  • Brendan Creutz
  • Duy Ho
  • Günter F. Steinke


Schenkel proved that the automorphism group of a flat Minkowski plane is a Lie group of dimension at most 6 and described planes whose automorphism group has dimension at least 4 or one of whose kernels has dimension 3. We extend these results to the case of toroidal circle planes.



The authors would like to thank the referee for the careful reading and helpful suggestions. The second author was supported by a UC Doctoral Scholarship.


  1. 1.
    Arens, R.: Topologies for homeomorphism groups. Am. J. Math. 68, 593–610 (1946)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Dugundji, J.: Topology. Allyn and Bacon Inc., Boston (1966)zbMATHGoogle Scholar
  3. 3.
    Hartmann, E.: Beispiele nicht einbettbarer reeller Minkowski–Ebenen. Geom. Dedicata 10(1–4), 155–159 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ho, D.: On the classification of toroidal circle planes. PhD thesis, University of Canterbury (2017)Google Scholar
  5. 5.
    Munkres, J.R.: Topology: a first course. Prentice-Hall Inc., Englewood Cliffs (1975)zbMATHGoogle Scholar
  6. 6.
    Polster, B.: Toroidal circle planes that are not Minkowski planes. J. Geom. 63, 154–167 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Polster, B., Steinke, G.F.: Geometries on Surfaces. Encyclopedia of Mathematics and its Applications, vol. 84. Cambridge University Press, Cambridge (2001)zbMATHGoogle Scholar
  8. 8.
    Salzmann, H.: Topological planes. Adv. Math. 2, 1–60 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Salzmann, H., et al.: Compact Projective Planes. De Gruyter Expositions in Mathematics, 21. Walter de Gruyter & Co., Berlin (1995)CrossRefGoogle Scholar
  10. 10.
    Schenkel, A.: Topologische Minkowski–Ebenen. Dissertation, Erlangen-Nürnberg (1980)Google Scholar
  11. 11.
    Steinke, G.F.: The automorphism group of locally compact connected topological Benz planes. Geom. Dedicata 16, 351–357 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Steinke, G.F.: Some Minkowski planes with 3-dimensional automorphism group. J. Geom. 25(1), 88–100 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Steinke, G.F.: A family of flat Minkowski planes admitting 3-dimensional simple groups of automorphisms. Adv. Geom. 4(3), 319–340 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Steinke, G.F.: Modified classical flat Minkowski planes. Adv. Geom. 17(3), 379–396 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Strambach, K.: Sphärische Kreisebenen. Math. Z. 113, 266–292 (1970)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of CanterburyChristchurchNew Zealand

Personalised recommendations