Abstract
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.
Acknowledgements
The authors would like to thank the referee for the careful reading and helpful suggestions, especially the comments on Lemma 3.2.
Communicated by: R. Löwen
Funding: The second author was supported by a University of Canterbury Doctoral Scholarship.
References
[1] R. Artzy, H. Groh, Laguerre and Minkowski planes produced by dilatations. J. Geom. 26 (1986), 1–20. MR837764 Zbl 0598.5100410.1007/BF01221003Search in Google Scholar
[2] W. Benz, Vorlesungen über Geometrie der Algebren. Springer 1973. MR0353137 Zbl 0258.5002410.1007/978-3-642-88670-6Search in Google 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 40.0194.0110.1007/BF01450182Search in Google 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 1396.5100610.1007/s00022-018-0420-8Search in Google Scholar
[5] E. Ghys, Groups acting on the circle. Enseign. Math. (2) 47 (2001), 329–407. MR1876932 Zbl 1044.37033Search in Google Scholar
[6] H. Groh, ℝ2-planes with 2-dimensional point transitive automorphism group. Abh. Math. Sem. Univ. Hamburg48 (1979), 171–202. MR537455 Zbl 0415.5100710.1007/BF02941299Search in Google Scholar
[7] H. Groh, R2-planes with point transitive 3-dimensional collineation group. Nederl. Akad. Wetensch. Indag. Math. 44 (1982), 173–282. MR66265310.1016/S1385-7258(82)80009-7Search in Google 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 0528.5100410.1007/BF01918132Search in Google Scholar
[9] D. Ho, On the classification of toroidal circle planes. PhD thesis, University of Canterbury, 2017.Search in 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 0707.5101610.1007/978-3-7091-2838-1_2Search in Google Scholar
[11] A. Navas, Groups of circle diffeomorphisms. University of Chicago Press, Chicago, IL 2011. MR2809110 Zbl 1236.3700210.7208/chicago/9780226569505.001.0001Search in Google 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 0781.2200310.1007/978-3-642-57999-8Search in Google Scholar
[13] B. Polster, Toroidal circle planes that are not Minkowski planes. J. Geom. 63 (1998), 154–167. MR1651572 Zbl 0928.5101110.1007/BF01221246Search in Google Scholar
[14] B. Polster, G. Steinke, Geometries on surfaces, volume 84 of Encyclopedia of Mathematics and its Applications. Cambridge Univ. Press 2001. MR1889925 Zbl 0995.5100410.1017/CBO9780511549656Search in Google 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 0851.5100310.1515/9783110876833Search in Google 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 1173.0000610.1017/CBO9780511721502Search in Google Scholar
[17] H. R. Salzmann, Topological planes. Advances in Math. 2 (1967), 1–60. MR0220135 Zbl 0153.2160110.1016/S0001-8708(67)80002-1Search in Google Scholar
[18] A. Schenkel, Topologische Minkowski-Ebenen. PhD thesis, Universität Erlangen-Nürnberg, 1980.Search in 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 1067.5100810.1515/advg.2004.019Search in Google Scholar
[20] G. F. Steinke, Modified classical flat Minkowski planes. Adv. Geom. 17 (2017), 379–396. MR3680361 Zbl 1387.5101510.1515/advgeom-2017-0026Search in Google Scholar
© 2020 Walter de Gruyter GmbH, Berlin/Boston
