Projektive Darstellung der Moulton-Ebenen

The Moulton planes can be characterized as 2-dimensional topological projective planes having a 4-dimensional collineation group, which fixes exactly one nonincident point-line-pair aw. We give a representation of these geometries on the real protective plane such that a and W coincide with the origin and the line of infinity. This representation shows that the collineation groups of nonisomorphic Moulton planes act differently, although they are isomorphic as topological groups.     

Four-dimensional compact projective planes with two fixed points

We determine all 4-dimensional compact projective planes with a solvable 6-dimensional collineation group fixing two distinct points, and acting transitively on the affine pencils through the fixed points. These planes form a 2-parameter family, and one exceptional member of this family is the dual of the exceptional translation plane with 8-dimensional collineation group.     

Die Oktavenebene als Translationsebene mit großer Kollineationsgruppe

It is shown that the affine plane over the Cayley numbers is the only 16-dimensional locally compact topological translation plane having a collineation group of dimension at least 41. This (hitherto unpublished) result is one of the ingredients of H. Salzmann's characterizations of the Cayley plane among general compact projective planes by the size of its collineation group.The proof involves various case studies of the possibilities for the structure and size of collineation groups of 16-dimensional locally compact translation planes. At the same time, these case studies are important steps for a classification program aiming at the explicit determination of all such translation planes having a collineation group of dimension at least 38.     

4-Dimensional compact projective planes with a 5-dimensional nilradical

The article is a contribution to the classification of all 4-dimensional flexible compact projective planes. We assume that the collineation group is a 6-dimensional solvable Lie group which fixes some flag. If, moreover, the nilradical of the collineation group is 5-dimensional, then we get 4 families of new planes which are neither translation planes nor shift planes.Meinem Lehrer H. Salzmann zum 65. Geburtstag am 3.11.1995 in Dankbarkeit gewidmet     

16-Dimensional Smooth Projective Planes with Large Collineation Groups

Smooth projective planes are projective planes defined on smooth manifolds (i.e. the set of points and the set of lines are smooth manifolds) such that the geometric operations of join and intersection are smooth. A systematic study of such planes and of their collineation groups can be found in previous works of the author. We prove in this paper that a 16-dimensional smooth projective plane which admits a collineation group of dimension d 39 is isomorphic to the octonion projective plane P₂ O. For topological compact projective planes this is true if d 41. Note that there are nonclassical topological planes with a collineation group of dimension 40.     

Polaritäten Ebener projektiver Ebenen

A projective plane is called flat if the spaces of points and lines are locally compact and 2-dimensional and the joining of points and the intersecting of lines are continuous. H. Salzmann studied planes of this type in [11]–[21]. Here polarities of such planes are considered. In II general properties of polarities of flat planes are discussed. For example, a polarity with absolute points has always an oval of absolute points. A flat projective plane with a cartesian ternary field K admits a polarity iff multiplication in K is commutative. In III the polarities of flat projective planes with a 3-dimensional collineation group are determined.     

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.     

Flat projective planes with 2-dimensional collineation group fixing at least two lines and more than two points

All flat projective planes whose collineation group contains a 2-dimensional subgroup fixing at least two lines and more than two points are classified. Furthermore, all isomorphism types of such planes are determined. This completes the classification of all flat projective planes admitting a 2-dimensional collineation group.     

Collineation groups irreducible on the components of a translation plane

We investigate finite translation planes of odd dimension over their kernels in which the translation complement induces on each component l a permutation group whose order is divisible by a p-primitive divisor. Using results of this investigation, we show that rank 3 affine planes of odd dimension over their kernels are either generalized André planes or semi-field planes. A similar result is given for translation planes having a collineation group which is doubly transitive on each affine line; besides the above two possibilities, there is a third possibility; the plane has order 27, the translation complement is doubly transitive on , and SL(2, 13) is contained in the translation complement.We also consider translation planes of odd dimension over their kernels which have a collineation group isomorphic to SL(2, w) with w prime to 5 and the characteristic, and having no affine perspectivity. We show that such planes have order 27, the prime power w=13, and the given group together with the translations forms a doubly transitive collineation group on {ie153-1}. This indicates quite strongly that the Hering translation plane of order 27 is unique with respect to the above properties.Both authors supported in part by NSF Grant No. MCS76-0661 A01.     

Elementary constructions of locally compact affine planes

In this paper we describe several elementary constructions of 4-, 8- and 16-dimensional locally compact affine planes. The new planes share many properties with the classical ones and are very easy to handle. Among the new planes we find translation planes, planes that are constructed by gluing together two halves of different translation planes, 4-dimensional shift planes, etc. We discuss various applications of our constructions, e.g. the construction of 8- and 16-dimensional affine planes with a point-transitive collineation group which are neither translation planes nor dual translation planes, the proof that a 2-dimensional affine plane that can be coordinatized by a linear ternary field with continuous ternary operation can be embedded in 4-, 8- and 16-dimensional planes, the construction of 4-dimensional non-classical planes that admit at the same time orthogonal and non-orthogonal polarities. We also consider which of our planes have tangent translation planes in all their points. In a final section we generalize the Knarr-Weigand criterion for topological ternary fields.This research was supported by a Feodor Lynen fellowship.     

Collineation groups of translation planes admitting hyperbolic Buekenhout or parabolic Buekenhout-Metz unitals

It is shown that for every semifield spread in PG(3,q) and for every parabolic Buekenhout-Metz unital, there is a collineation group of the associated translation plane that acts transitively and regularly on the affine points of the parabolic unital. Conversely, any spread admitting such a group is shown to be a semifield spread. For hyperbolic Buekenhout unitals, various collineation groups of translation planes admitting such unitals and the associated planes are determined.     

4-dimensional compact projective planes with a 7-dimensional collineation group

The classification of 4-dimensional compact projective planes having a 7-dimensional collineation group is completed. Besides one single shift plane all such planes are either translation planes or dual translation planes.Dedicated to H. R. Salzmann on his 60th birthday     

4-Dimensionale projektive Ebenen mit grosser abelscher Kollineationsgruppe

We show that a 4-dimensional connected abelian group can act in exactly five different ways as a collineation group of a compact 4-dimensional projective plane. Furthermore the complex projective plane is characterized as the only compact 4-dimensional projective plane which admits two different 4-dimensional abelian collineation groups.

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Herrn Professor Dr. Eelmut Karzel zum 60. Geburtstag     

A Semifield Plane of Odd Order Admitting an Autotopism Subgroup Isomorphic to <Emphasis Type="Italic">A</Emphasis><Subscript>5</Subscript>

We develop an approach to constructing and classifying semifield projective planes with the use of a spread set. The famous conjecture is discussed on the solvability of the full collineation group of a finite semifield nondesarguesian plane. We construct a matrix representation of a spread set of a semifield plane of odd order admitting an autotopism subgroup isomorphic to the alternating group A₅ and find a series of semifield planes of odd order not admitting A₅.     

Two-transitive groups on a hyperbolic unital

A classification given previously of all projective translation planes of order q² that admit a collineation group G admitting a two-transitive orbit of q+1 points is applied to show that the only projective translation planes of order q² admitting a hyperbolic unital acting two-transitively on a secant are the Desarguesian planes and the unital is a Buekenhout hyperbolic unital.     

A class of non-Desargusian planes

A new class of non-Desargusian planes of order q², where q is a power of an odd prime, is constructed. These planes have the interesting property that they all admit a collineation group of order (q² ? 1).     

Large 2-transitive arcs

The projective planes of order n with a collineation group acting 2-transitively on an arc of length v, with n>v?n/2, are investigated and several new examples are provided.     

Translation planes of order q 2 admitting a two-transitive orbit of length q + 1 on the line at infinity

A classification is given of all translation planes of order q ² that admit a collineation group G admitting a two-transitive orbit of q + 1 points on the line at infinity.     

Smooth Hughes planes are classical

We prove that the only compact projective Hughes planes which are smooth projective planes are the classical planes over the complex numbers \Bbb C \Bbb C , the quaternions \Bbb H \Bbb H , and the Caley numbers \Bbb O \Bbb O . As a by-product this shows that an 8-dimensional smooth projective plane which admits a collineation group of dimension d 3 17d \geq 17 is isomorphic to the quaternion projective plane P ₂\Bbb H {\cal P _2\Bbb H }. For topological compact projective planes this is true if d 3 19d \geq 19, and this bound is sharp.     

Multiplicative Loops of 2-Dimensional Topological Quasifields

We determine the algebraic structure of the multiplicative loops for locally compact 2-dimensional topological connected quasifields. In particular, our attention turns to multiplicative loops which have either a normal subloop of positive dimension or which contain a 1-dimensional compact subgroup. In the last section, we determine explicitly the quasifields which coordinatize locally compact translation planes of dimension 4 admitting an at least 7-dimensional Lie group as collineation group.