The geometries of all plane sections of these sets P are the classical examples of topological circle planes. They are known as the real miquelian M6bius, Laguerre, and Minkowski plane, respectively. In the last two cases, there are also complex analogues. For any locally compact connected circle geometry, the topological dimension of the point set is either infinite (no examples are known for that) or 2 or 4 as in the classical examples . We deal here with locally compact 2-dimensional planes exclusively.
The automorphism group F of a locally compact 2-or 4-dimensional circle plane g is a Lie group [14] whose dimension d is considered as a measure of the degree of similarity between 8 and its classical counterpart. The real miquelian M6bius plane has F 1 = PSL 2 C and d = 6. According to Strambach, all other M6bius planes have d ~ 3. There is a vast number of M6bius planes with d = 3, completely classified by Strambach [16], [17]. Generalizing the traditional description by 'cycles' and 'spears' of the classical Laguerre plane (d= 7), Groh [6] constructs 2-dimensional Laguerre planes from each of these M6bius planes. The resulting planes appear to have d ~< 3, like all other 2-dimensional Laguerre planes known to date [7], [8] except the ovoidal planes. An ovoidal plane consists of the plane sections of a cylinder with an oval cross section and has d >/4, cf. [10], [5]. Other examples with d >-4 were given for the first time in our preceding paper[9] and by Steinke[15]. They are obtained by a kind of 'deformation' from a familiar description of the classical plane by parabolae in ~2. The aim of the present paper is to show that the examples given in [9] exhaust all possibilities for d >/5 and, when F fixes a point, even for d = 4. Only the classical plane has d > 5, and only the ovoidal planes over 'skew parabolae' have d = 5. For 2-dimensional Minkowski planes, similar results were obtained by Schenkel [-13]. There, d = 6 for the classical plane and d-K< 4 otherwise. Planes with a 4-dimensional point set and a large group are studied by F6rtsch [-3] and Steinke (unpublished).
The present work is based on the second author's doctoral dissertation .