Point-regular geometries
Tóm tắt
We study incidence structures with a sharply point transitve group of automorphisms (i.e., incidence groups), and state conditions for the possibility to reconstruct the lines from information about the action of the group. This treatment generalizes the familiar description of translation planes. It also includes H. Groh's characterization [Abh. Math. Sem. Hamburg48 (1979) 171–202, Theorem 5.2] of the real affine plane and the real half plane. In a second part, we generalize H. Groh's topological construction ofarc planes [loc. cit.] to the discrete case (and arbitrary groups with partitions), and prove that the regular action that was used in the construction yields also an action on the new set of lines (that is, an action by collineations). Finally, we consider possible applications to stable planes.
Tài liệu tham khảo
J. André,Über nicht-Desarguessche Ebenen mit transitiver Translationsgruppe, Math. Z.60 (1954), 156–186.
J. André,Über Parallelstrukturen, Teil I und II, Math. Z.76 (1961), 85–102, 155–163.
H. Freudenthal,Einige Sätze über topologische Gruppen, Ann. of Math.37 (1936), 46–56.
H. Groh,R 2 -planes with 2-dimensional point transitive automorphism group, Abh. Math. Sem. Univ. Hamburg48 (1979), 171–202.
T. Grundhöfer, R. Löwen,Linear topological geometries, Handbook of Incidence Geometry (F. Buekenhout, ed.), Elsevier, Amsterdam, 1995, Chap. 23.
A. Hohti,Another alternative proof for Effros' theorem, Topology Proc.12 (1987), 295–298.
H. Karzel,Bericht über projektive Inzidenzgruppen, Jahresbericht d. DMV 67 (1964), 58–92.
N. Knarr,4-dimensionale projektive Ebenen mit groβer abelscher Kollineations-gruppe, J. Geom.31 (1988), 114–124.
R. Löwen,Vierdimensionale stabile Ebenen, Geom. Dedicata5 (1976), 239–294.
R. Löwen,Topology and dimension of stable planes: On a conjecture of H. Freudenthal, J. Reine Angew. Math.343 (1983), 108–122.
R. Löwen,Compact spreads and compact translation planes over locally compact fields, J. Geom.36 (1989), 110–116.
I. Pieper-Seier,Über ebene affine Inzidenzgruppen, Geom. Dedicata22 (1987), 1–20.
P. Plaumann, K. Strambach,Partitionen Liescher und algebraischer Gruppen, Forum Math.2 (1990), 523–578.
P. Plaumann, K. Strambach,Partitionen einfach zusammenhängender auflösbarer Liegruppen, Seminar Sophus Lie, Darmstadt Erlangen Greifswald Leipzig1 (1991), 101–106.
H. Salzmann, D. Betten, T. Grundhöfer, H. Hähl, R. Löwen, M. Stroppel,Compact protective planes, De Gruyter, Berlin, 1995.
I.Schellhammer,Einige Klassen von projektiven Ebenen, Thesis, Tübingen, 1981.
P. Sperner, Vierdimensionale ℂ*·SU2ℂ-Ebenen, Geom. Dedicata34 (1990), 301–312.
M. Stroppel,Reconstruction of incidence geometries from groups of automorphisms, Arch. Math. (Basel)58 (1992), 621–624.
M. Stroppel,A categorical glimpse at the reconstruction of geometries, Geom. Dedicata46 (1993), 47–60.
M.Stroppel,Actions of Locally Compact Groups on Stable Planes, Preprint Nr.1626, Fachbereich Mathematik der Technischen Hochschule Darmstadt, 1994.
M. Suzuki,On the finite group with a complete partition, J. Math. Soc. Jap.2 (1950), 165–185.