Laguerre planes of even order and translation ovals

Translation Laguerre planes of even order are represented in high dimensional projective space over GF(2) by a collection of subspaces that satisfies a very simple condition.This research was supported for the respective authors by a grant from the University of Canterbury and by a Feodor Lynen Fellowship.     

More on Kleinewillinghöfer Types of Flat Laguerre Planes

Polster and Steinke [Result. Math., 46 (2004), 103–122] determined the possible Kleinewillingh?fer types of flat Laguerre planes. These types reflect transitivity properties of groups of certain central automorphisms. We exclude three more types from the list given there with respect to Laguerre homotheties. This yields a complete determination of all possible single types with respect to Laguerre homotheties that can occur in flat Laguerre planes. Building on results by M?urer and Hartmann to characterize ovoidal or miquelian Laguerre planes we further characterize certain flat Laguerre planes in terms of their Kleinewillingh?fer types. Received: January 16, 2007. Revised: July 26, 2007.     

Eight and seven point degenerations of Miquel's Theorem in Laguerre planes

For M?bius planes Schaeffer [9] has proved that all seven point degenerations of Miquel's Theorem characterize miquelian M?bius planes. For Laguerre planes we have several degenerations of Miquel's Theorem with eight and seven points. We prove that all except one of these degenerations characterize miquelian Laguerre planes. The remaining degeneration characterizes elation Laguerre planes. Received 14 September 2001; revised 1 September 2002.     

Half-ovoidal flat Laguerre planes

We classify the 2-dimensional Laguerre planes which can be obtained by gluing together halves of ovoidal Laguerre planes along two parallel classes.This research was supported by a Feodor Lynen fellowship and a DAAD fellowship.     

Homologies and elations in compact,connected projective planes

In a compact, connected topological projective plane, let Ω be a closed Lie subgroup of the group of all axial collineations with a fixed axis A. We compare the set З\A consisting of the centres of all non-identical homologies in Ω to orbits of the group $\text{Ω}{}_{\mathrm{[A]}}$ of all elations contained in Ω and of its connected component $\text{θ = (Ω}{}_{\mathrm{[A]}}\text{)}{}^1$. It is shown that З\A is the union of at most countably many θ-orbits; moreover, З\A turns out to be a single θ-orbit whenever the connected component of Ω contains non-identical homologies. This result is analogous to a well-known theorem of André for finite planes. It has numerous consequences for the structure of collineation groups of compact, connected projective planes.     

Three-dimensional quadrangles and flat Laguerre planes

Every three-dimensional generalized quadrangle can be constructed from flat Laguerre planes.Dedicated to Prof. H. Salzmann on his 60th birthday     

4-Dimensional Elation Laguerre Planes Admitting Non-Solvable Automorphism Groups

 This paper concerns 4-dimensional (topological locally compact connected) elation Laguerre planes that admit non-solvable automorphism groups. It is shown that such a plane is either semi-classical or a single plane admitting the group SL(2, ). Various characterizations of this single Laguerre plane are obtained. Received October 17 2000; in revised form April 23 2001 Published online August 5, 2002     

Two-transitive orbits in finite projective planes

The problem of classifying finite projective planes of order n with an automorphism group G and a point orbit on which G acts two-transitively is investigated in considerable detail, under the assumption that has length at last n. Combining old and new results a rather satisfying classification is obtained, even though some cases for orbit lengths n and n + 1 remain unsolved.     

The homeomorphism type of unital-like submanifolds in smooth projective planes

We prove that a compact, connected submanifold of the point space of a smooth projective plane is homeomorphic to a sphere provided that certain intersection properties with lines are satisfied. As an application, we show that the set of absolute points of a smooth polarity in a smooth projective plane of dimension 2l is empty or homeomorphic to a sphere of dimension 2l - 1 or .Received: 13 September 2002     

A completion problem for finite affine planes

A partial affine plane (PAP) of ordern is ann ²-setS of points together with a collection ofn-subsets ofS called lines such that any two lines meet in at most one point. We obtain conditions under which a PAP with nearlyn ²+n lines can be completed to an affine plane by adding lines. In particular, we make use of Bruck’s completion condition for nets to show that certain PAP’s with at leastn ²+n−√n can be completed and that forn≠3 any PAP withn ²+n−2 lines can be completed.     

Coordinate rings of topological Klingenberg planes I: The affine perspective

Although the coordinate ternary field of a topological affine plane is topological, the converse does not hold. However, an affine plane is topological precisely when its coordinate biternary fields are topological. We extend this result to topological biternary rings and their topological affine Klingenberg planes. Then we examine the locally compact situation. Finally, following the ideas of Knarr and Weigand, we show that in certain circumstances, the continuity of the ternary operators is sufficient to ensure that the biternary ring is topological. This facilitates the construction of locally compact, locally connected affine Klingenberg planes.Dedicated to Professor Dr. Helmut Salzmann on his 65th birthday     

Symmetric planes with non-classical tangent translation planes

We construct symmetric planes associated with an arbitrary locally compact connected nearfield . If is a proper nearfield, i.e. {;;}, then the tangent translation plane of this symmetric plane is not classical. All previously known examples of symmetric planes have classical tangent translation planes.Herrn Professor Dr. H. Salzmann zum 65. Geburtstag gewidmet     

A class of d-dimensional flag transitive translation planes

Journal of Geometry -     

Shear planes

A shear plane is a 2n-dimensional stable plane admitting a quasi-perspective collineation group which is a vector group of the same dimension 2n and fixes no point. We show that all of these planes can be derived from a special kind of partial spreads by a construction analogous to the construction of (punctured) dual translation planes from compact spreads. Finally we give a criterion (and examples) for shear planes which are not isomorphic to an open subplane of a topological projective plane.     

Two-dimensional flag-transitive planes revisited

This paper shows that the odd order two-dimensional flag-transitive planes constructed by Kantor-Suetake constitute the same family of planes as those constructed by Baker-Ebert. Moreover, for orders satisfying a modest number theoretical assumption this family consists of all possible such planes of that order. In particular, it is shown that the number of isomorphism classes of (non-Desarguesian) two-dimensional flag-transitive affine planes of order q ² is precisely (q–1)/2 when q is an odd prime and precisely (q–1)/2e when q=p ^e is an odd prime power with exponent e that is a power of 2. An enumeration is given in other cases that uses the Möbius inversion formula.This work was partially supported by NSA grant MDA 904-95-H-1013.This work was partially supported by NSA grant MDA 904-94-H-2033.     

The collineation groups of the translation planes of order 25

Czerwinski and Oakden have classified all the translation planes of order 25. In this paper the author gives the order and orbit lengths on the line at infinity of the collineation groups of these planes.     

Unitals in the regular nearfield planes

Classes of parabolic unitals in the regular nearfield planes of odd square order are enumerated and classified. These unitals correspond to certain Buekenhout-Metz unitals in the classical plane. Their collineation groups are determined and the unitals are sorted by projective equivalence.