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Advances in Geometry

Managing Editor: Grundhöfer, Theo / Joswig, Michael

Editorial Board: Bamberg, John / Bannai, Eiichi / Cavalieri, Renzo / Coskun, Izzet / Duzaar, Frank / Eberlein, Patrick / Gentili, Graziano / Henk, Martin / Kantor, William M. / Korchmaros, Gabor / Kreuzer, Alexander / Lagarias, Jeffrey C. / Leistner, Thomas / Löwen, Rainer / Ono, Kaoru / Ratcliffe, John G. / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard

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Three-dimensional connected groups of automorphisms of toroidal circle planes

Brendan Creutz / Duy Ho / Günter F. Steinke
Published Online: 2019-09-11 | DOI: https://doi.org/10.1515/advgeom-2019-0031


We contribute to the classification of toroidal circle planes and flat Minkowski planes possessing three-dimensional connected groups of automorphisms. When such a group is an almost simple Lie group, we show that it is isomorphic to PSL(2, ℝ). Using this result, we describe a framework for the full classification based on the action of the group on the point set.

Keywords: Circle plane; Minkowski plane; automorphism group

MSC 2010: 51H15; 51B20


  • [1]

    R. Artzy, H. Groh, Laguerre and Minkowski planes produced by dilatations. J. Geom. 26 (1986), 1–20. MR837764 Zbl 0598.51004CrossrefGoogle Scholar

  • [2]

    W. Benz, Vorlesungen über Geometrie der Algebren. Springer 1973. MR0353137 Zbl 0258.50024Google Scholar

  • [3]

    L. E. J. Brouwer, Die Theorie der endlichen kontinuierlichen Gruppen, unabhängig von den Axiomen von Lie. Math. Ann. 67 (1909), 246–267. MR1511528 JFM 40.0194.01CrossrefGoogle Scholar

  • [4]

    B. Creutz, D. Ho, G. F. Steinke, On automorphism groups of toroidal circle planes. J. Geom. 109 (2018), Art. 15, 13 pages. MR3763105 Zbl 1396.51006Web of ScienceGoogle Scholar

  • [5]

    E. Ghys, Groups acting on the circle. Enseign. Math. (2) 47 (2001), 329–407. MR1876932 Zbl 1044.37033Google Scholar

  • [6]

    H. Groh, ℝ2-planes with 2-dimensional point transitive automorphism group. Abh. Math. Sem. Univ. Hamburg 48 (1979), 171–202. MR537455 Zbl 0415.51007CrossrefGoogle Scholar

  • [7]

    H. Groh, R2-planes with point transitive 3-dimensional collineation group. Nederl. Akad. Wetensch. Indag. Math. 44 (1982), 173–282. MR662653Google Scholar

  • [8]

    H. Groh, M. F. Lippert, H.-J. Pohl, ℝ2-planes with 3-dimensional automorphism group fixing precisely a line. J. Geom. 21 (1983), 66–96. MR731530 Zbl 0528.51004CrossrefGoogle Scholar

  • [9]

    D. Ho, On the classification of toroidal circle planes. PhD thesis, University of Canterbury, 2017.Google Scholar

  • [10]

    H. Karzel, Circle geometry and its application to code theory. In: Geometries, codes and cryptography (Udine, 1989), volume 313 of CISM Courses and Lect., 25–75, Springer 1990. MR1140927 Zbl 0707.51016Google Scholar

  • [11]

    A. Navas, Groups of circle diffeomorphisms. University of Chicago Press, Chicago, IL 2011. MR2809110 Zbl 1236.37002Google Scholar

  • [12]

    A. L. Onishchik, E. B. Vinberg, Foundations of Lie theory. In: Lie groups and Lie algebras, I, volume 20 of Encyclopaedia Math. Sci., 1–94, 231–235, Springer 1993. MR1306738 Zbl 0781.22003Google Scholar

  • [13]

    B. Polster, Toroidal circle planes that are not Minkowski planes. J. Geom. 63 (1998), 154–167. MR1651572 Zbl 0928.51011CrossrefGoogle Scholar

  • [14]

    B. Polster, G. Steinke, Geometries on surfaces, volume 84 of Encyclopedia of Mathematics and its Applications. Cambridge Univ. Press 2001. MR1889925 Zbl 0995.51004Google Scholar

  • [15]

    H. Salzmann, D. Betten, T. Grundhöfer, H. Hähl, R. Löwen, M. Stroppel, Compact projective planes, volume 21 of De Gruyter Expositions in Mathematics. De Gruyter 1995. MR1384300 Zbl 0851.51003Google Scholar

  • [16]

    H. Salzmann, T. Grundhöfer, H. Hähl, R. Löwen, The classical fields, volume 112 of Encyclopedia of Mathematics and its Applications. Cambridge Univ. Press 2007. MR2357231 Zbl 1173.00006Google Scholar

  • [17]

    H. R. Salzmann, Topological planes. Advances in Math. 2 (1967), 1–60. MR0220135 Zbl 0153.21601CrossrefGoogle Scholar

  • [18]

    A. Schenkel, Topologische Minkowski-Ebenen. PhD thesis, Universität Erlangen-Nürnberg, 1980.Google Scholar

  • [19]

    G. F. Steinke, A family of flat Minkowski planes admitting 3-dimensional simple groups of automorphisms. Adv. Geom. 4 (2004), 319–340. MR2071809 Zbl 1067.51008Google Scholar

  • [20]

    G. F. Steinke, Modified classical flat Minkowski planes. Adv. Geom. 17 (2017), 379–396. MR3680361 Zbl 1387.51015Web of ScienceGoogle Scholar

About the article

Received: 2018-09-14

Revised: 2018-12-27

Published Online: 2019-09-11

Communicated by: R. Löwen

Funding: The second author was supported by a University of Canterbury Doctoral Scholarship.

Citation Information: Advances in Geometry, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2019-0031.

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