A note on finite semifield planes that admit affine homologies

Let be a translation plane of order n that admits an abelian group of order n in its translation complement. If admits an affine homology then it is a semifield plane.     

Zur Konstruktion geschlitzter Inzidenzgruppen

In the first part a derivation method for incidence-groups is developed. The connections to the usual derivations of near-rings are shown in part 2 and examples are constructed. We characterize those finite slit incidence-groups with special affine kernel which are derivations of abelian incidence-groups. Applying these results to special classes of finite incidence-groups we show that there are 2-sided non-abelian finite incidence-groups with abelian affine kernel. The class of derivations of abelian finite incidence-groups contains all splitting a-2-sided incidence-groups with abelian kernel and no splitting kernel-2-sided non-2-sided incidence-group with abelian kernel. In the last part the a-2-sided incidence-groups are algebraically described.The results were partly communicated at the Conference on Geometry in March 1971 at the University of Waterloo, Ontario, Canada.     

Triangular actions on

Every locally trivial action of the additive group of complex numbers on four-dimensional complex affine space that is given by a triangular derivation is conjugate to a translation. A criterion for a proper action on complex affine -space to be locally trivial is given, along with an example showing that the hypotheses of the criterion are sharp.

     

Harmonic spaces associated with non-symmetric translation invariant Dirichlet forms

Continuing earlier work on construction of harmonic spaces from translation invariant Dirichlet spaces defined on locally compact abelian groups, it is shown that the potential kernel for a non-symmetric translation invariant Dirichlet form on a locally compact abelian group under the extra assumptions that

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 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.     

One-Factorizations of Complete Graphs with a Doubly Transitive Automorphism Group

It is shown that a 1-factorization of K_n with a doubly transitiveautomorphism group on vertices is either the affine line-parallelismof AG(d, 2), or one of three ‘sporadic’ exampleswith n = 6, 12 or 28. The full automorphism groups are respectivelyAGL (d, 2) (the holomorph of an elementary abelian group oforder 2^d), PGL(2,5), PSL(2,11) and PL(2,8).     

Sur la hauteur de Faltings des variétés abéliennes à multiplication complexe

Motivated by a result of Bost, we use the relationship between Faltings' heights of abelian varieties with complex multiplication and logarithmic derivatives of Artin L-functions at s=0 to investigate these heights. In particular, we prove that the height of an elliptic curve with complex multiplication by Q-d is bounded from below by an effective affine function of log d.     

Simple abelian quadruple systems

We generalize a construction of simple cyclic 3-designs due to Köhler (1981) to that of simple abelian 3-designs. We prove that for any abelian group A of order , there exists a simple 3-(v,4,3) design with A?Aut(A) as an automorphism group.     

Affine Barbilian structures

W.Leissner has characterized, by geometric axioms, the affine Barbilian planes over a Z-ring (i.e, a ring with 1 such that ab=1 ba=1) [10].The aim of the present paper is to characterize correspondingly the affine Barbilian planes over an arbitrary ring with 1. First we shall deal with the translation Barbilian planes, which generalize Leissner's parallelodromic planes [11]. The paper concludes with a study of the kernel of the translation Barbilian plane.Here, the terms affine Barbilian structure and affine Barbilian plane are used in a more general sense than in [10] and [11]. Also, the definitions of translation and parallelodromy are slightly different from those in [10] and [11], insomuch that the invariance of the non-neighbour relation is not postulated any more, this being a consequence in a translation Barbilian plane.In H.J.Arnold's geometry of rings, for any two distinct points, there exists a smallest line incident with them [1]. This property, assumed only for the non-neighbour pairs of points, will replace the usual postulate that two non-neighbour points are incident with exactly one line. Thus, ideas of D.Barbilian [2] and H. J.Arnold [1] are combined with methods of affine ring-geometry due to J.Hjelmslev [5], [6], W.Klingenberg [7], [8], [9], H.Lüneburg [12], W.Benz [3], [4], W.Leissner [10], [11], and others. Many parts of the proofs in [10] and [11] could be used here almost unchanged, under relaxed assumptions.     

Affine,quasi‐affine and co‐affine frames on local fields of positive characteristic

The concept of quasi‐affine frame in Euclidean spaces was introduced to obtain translation invariance of the discrete wavelet transform. We extend this concept to a local field K of positive characteristic. We show that the affine system generated by a finite number of functions is an affine frame if and only if the corresponding quasi‐affine system is a quasi‐affine frame. In such a case the exact frame bounds are equal. This result is obtained by using the properties of an operator associated with two such affine systems. We characterize the translation invariance of such an operator. A related concept is that of co‐affine system. We show that there do not exist any co‐affine frame in .     

The rhombidodecahedral tessellation of 3-space and a particular 15-vertex triangulation of the 3-dimensional torus

It is well known that the unique 7-vertex triangulation of the 2-dimensional torus S¹×S¹ is a consequence of the relationship between two hexagonal lattices in the euclidean plane: it is just the quotient of the triangular tessellation of the plane by a translation group. Each vertex star is a regular hexagon and the symmetry group of this triangulation is the affine group A(1,₇) in one dimension over ₇. In this paper we describe a particular 15-vertex triangulation of the 3-dimensional torus S¹×S¹×S¹ whose symmetry group is the affine group A(1,₁₅) and which is similarly related to two lattices in euclidean 3-space: it is just the quotient of a particular tessellation of 3-space by a translation group. Each vertex star happens to be a rhombidodecahedron, the dual of a (semiregular) cuboctahedron.     

Some remarks on symmetrical extensions of graphs

It is proved that results from a previous paper of the author on symmetrical 2-extensions of graphs can be extended to symmetrical p-extensions of graphs for any prime p. In particular, it is proved that, for any prime p, there are only finitely many symmetrical p-extensions of a locally finite graph with an abelian subgroup of finite index in its automorphism group. Some refinements of these results are also obtained. In addition, we consider the question on the possibility to represent symmetrical extensions of a d-dimensional grid (and similar graphs) in the d-dimensional affine Euclidean space in such a way that a corresponding vertex-transitive group of automorphisms of the extension is induced by some crystallographic group of motions of the space.     

On 4-dimensional Minkowski planes with 7-dimensional automorphism groups

This paper concerns 4-dimensional (topological locally compact connected) Minkowski planes that admit a 7-dimensional automorphism group . It is shown that such a plane is either classical or has a distinguished point that is fixed by the connected component of and that the derived affine plane at this point is a 4-dimensional translation plane with a 7-dimensional collineation group.This research was supported by a Feodor Lynen Fellowship.     

Varieties of Characters

Let G be a connected reductive affine algebraic group. In this short note we define the variety of G-characters of a finitely generated group Γ and show that the quotient of the G-character variety of Γ by the action of the trace preserving outer automorphisms of G normalizes the variety of G-characters when Γ is a free group, free abelian group, or a surface group.     

Sequences in abelian groups G of odd order without zero-sum subsequences of length exp(G)

We present a new construction for sequences in the finite abelian group without zero-sum subsequences of length n, for odd n. This construction improves the maximal known cardinality of such sequences for r > 4 and leads to simpler examples for r > 2. Moreover we explore a link to ternary affine caps and prove that the size of the second largest complete caps in AG(5, 3) is 42.      

Unitals in Topological Affine Translation Planes Need Not Be Strictly Convex

We study the relationship of two incidence geometric convexity notions, namely, ovoids in real affine spaces and compact unitals of codimension 1 in topological affine translation planes. In [3] we showed that every ovoid in a translation plane is a unital, and we asked if the converse is true. Here we introduce the notion of a shell, which is distinctly weaker than that of an ovoid and still implies the unital property if the translation plane is properly chosen (and the shell is not too degenerate). We give an explicit example of a shell that is not an ovoid. The question remains whether or not conversely, every compact unital of codimension 1 in a translation plane is a shell.     

Planar Functions and Planes of Lenz-Barlotti Class II

Planar functions were introduced by Dembowski and Ostrom [4] to describe projective planes possessing a collineation group with particular properties. Several classes of planar functions over a finite field are described, including a class whose associated affine planes are not translation planes or dual translation planes. This resolves in the negative a question posed in [4]. These planar functions define at least one such affine plane of order 3^e for every e 4 and their projective closures are of Lenz-Barlotti type II. All previously known planes of type II are obtained by derivation or lifting. At least when e is odd, the planes described here cannot be obtained in this manner.     

Über fastgeordnete affine Ebenen

An example of an affine plane is constructed, which cannot be ordered, but can be embedded in an ordered projective plane. There is no such an example, if the affine plane is a translation plane with kernel GF(2).

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Herrn Rafael Artzy zum 75. Geburtstag gewidmet