Markus Stroppel

Reconstruction of incidence geometries from groups of automorphisms

Freudenthal describes a method to construct an incidence geometry from a group such that the given group acts transitively on the set of flags (incident point-line pairs) of the constructed geometry. This method can be found in [5], too. Here we give a useful generalization to geometries that are not flag-homogeneous. Such geometries occur quite naturally in the study of stable planes.

A categorical glimpse at the reconstruction of geometries

The authors reconstruction method [‘Reconstruction of incidence geometries from groups of automorphisms’,Arch. Math.58 (1992) 621–624] is put in a categorical setting, and generalized to geometries with an arbitrary number of ‘types’. The results amount to saying that the reconstruction process involves a pair of adjoint functors, and that the class of those geometries that are images under reconstruction forms a reflective subcategory.

Slanted symplectic quadrangles

By ‘slanting’ symplectic quadrangles W(F) over fieldsF, we obtain very simple examples of non-classical generalized quadrangles. We determine the collineation groups of these slanted quadrangles and their groups of projectivities. No slanted quadrangle is a topological quadrangle.

Polarities of shift planes

We construct polarities for arbitrary shift planes and develop criteria for conjugacy under the normalizer of the shift group. Under suitable assumptions (in particular, for finite or compact planes) we construct all shift groups on a given plane, and our constructions yield all conjugacy classes of polarities. We show that a translation plane admits an orthogonal polarity if, and only if, it is a shift plane. The corresponding planes are exactly those that can be coordinatized by commutative semifields. The orthogonal polarities form a single conjugacy class. Finally, we construct examples of compact connected shift planes with more conjugacy classes of polarities than the corresponding classical planes.

Planar groups of automorphisms of stable planes

Abstract(Semi-) planar groups of stable planes are introduced, and information about their size and their structure is derived. A special case are the stabilizers of quadrangles in compact connected projective planes (i.e. automorphism groups of locally compact connected ternary fields).

On Restrictions of Automorphism Groups of Compact Projective Planes to Subplanes

It is shown that the restriction of automorphisms of a compact projective plane to a closed Baer subplane or to an open subgeometry, respectively, is a quotient mapping (in contrast to restrictions to arbitrary subgeometries). In the proof, we investigate lineations in towers of Baer subplanes.

Locally compact groups, residual Lie groups, and varieties generated by Lie groups

Abstract The concept of approximating in various ways locally compact groups by Lie groups is surveyed with emphasis on pro-Lie groups and locally compact residual Lie groups. All members of the variety of Hausdorff groups generated by the class of all finite dimensional real Lie groups are residual Lie groups. Conversely, we show that every locally compact member of this variety is a pro-Lie group. For every locally compact residual Lie group we construct several better behaved residual Lie groups into which it is equidimensionally immersed. We use such a construction to prove that for a locally compact residual Lie group G the component factor group G G 0 is residually discrete.

Polarities and unitals in the Coulter---Matthews planes

For a class of finite shift planes introduced by Coulter and Matthews, we give a set of representatives for the isomorphism types, determine all automorphisms and describe all polarities explicitly. The planes in question are the only known examples of finite shift planes that are not translation planes. Each non-desarguesian Coulter–Matthews plane admits precisely two conjugacy classes of orthogonal polarities. In addition, each Coulter–Matthews plane of square order admits exactly one conjugacy class of unitary polarities. We prove that most of the corresponding unitals are not classical.

Groups that are Almost Homogeneous

We classify those groups whose automorphism group has at most three orbits. In other words, we classify those groups whose holomorph is a rank 3 permutation group.

Unitals over Composition Algebras

Abstract. We consider polar unitals in projective planes over composition algebras. We prove that the little projective group of such a unital is contained in the centralizer of the polarity defining the unital. This yields information about the full automorphism group of the unital; it is contained in the automorphism group of the normal subgroup generated by all translations of the unital. Over suitable ground fields (such as the real numbers) we obtain a complete description of the automorphism group of the unital in the sense that it coincides with the centralizer of the polarity. The action of a nilpotent regular normal subgroup is used to establish isomorphisms between unitals in spaces of different dimensions, and over different fields.