We construct two families of topological Laguerre planes with a 2-dimensional point set and with an at least 3-dimensional automorphism group. Circles of these planes will be graphs in (R w or) × R of the formwhere a, s, u, v e R and f is a suitably chosen 'generating function'. As special cases, our class of planes comprises(1) all locally compact ovoidal planes over the real numbers;(2) the nonovoidal planes of Hartmann [-6];(3) some of the nonovoidal planes of Artzy and Groh [1];(4) some of the planes of Steinke [13];(5) all locally compact 2-dimensional planes with an at least 5-dimensional automorphism group; (6) all locally compact 2-dimensional planes with a 4-dimensional automorphism group fixing a point; (7) planes whose full automorphism group is 3-dimensional and fixes all parallel classes of points.The fact that the planes (2) and (3) are topological planes has not been stated in the literature. Assertions (5) and (6) will be proved in our subsequent paper [7]. Steinke[13] asks whether planes of type (7) exist. The present work is based on the second author's doctoral dissertation [9].For a precise definition of topological Laguerre planes, we refer the reader to [3] and give here only a rough description. A Laguerre plane 8= (P,N,J) consists of a point set P, a set N of circles c~_ P, and an equivalence relation J (parallelism) on P. The class of p e P is denoted ip ~ J.The axioms say mainly that three pairwise nonparallel points lie on a unique circle and that for pE cN iq there is a unique circle d through the point q touching the circle c in the point p. Here, touching means that c = d or c~ d = {p}. Moreover, each class ip meets each circle in a unique point. The plane is called topological if the operations established by these axioms, including intersection of circles, are continuous with respect to given topologies on P, q¢, J. Locally compact 2-dimensional topological Laguerre planes can be characterized by the properties that P is homeomorphic to a cylinder S a x R, each ce N is a topological circle c -$1, and each class ip is closed in P and homeomorphic to R (see [4, 3.10]). The only other finite dimension where locally compact connected Laguerre planes exist is four [2].

Let P be a sphere or a circular cylinder or a one-sheeted hyperboloid in real 3-space. 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 [2]. 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 [11]. The authors wish to thank G. F. Steinke for helpful comments on a first draught of this paper.Prerequisites. For basic information on topological Laguerre planes, consult [4]. A locally compact 2-dimensional Laguerre plane 8 = (P, c~,3t) Geometriae Dedicata 23 (1987), 87-96.

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