Universität Stuttgart
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Item Open Access Embedding a non-embeddable stable plane(1993) Stroppel, MarkusIn [4], K. Strambach describes a 2-dimensional stable plane R admitting Σ=SL2 R as a group of automorphisms such that there exists no Σ-equivarient embedding into a 2-dimensional projective plane. R. Löwen [3] has given a 4-dimensional analogue C, admitting Δ=SL2Copf. He posed the question whether there are embeddings of Strambach's plane R into C. We show that such embeddings exist, in fact we determine all -Σ-equivariant embeddings of 2-dimensional stable planes admitting Σ as atransitive group of automorphisms.Item Open Access Endomorphisms of stable planes(1992) Stroppel, MarkusEndomorphisms of stable planes are introduced, and it is shown that these are injective, locally constant or collapsed. Examples are studied, and it is shown that there are stable planes admitting "substantially more" endomorphisms than automorphisms.Item Open Access Stable planes(1994) Stroppel, MarkusStable planes are a special kind of topological linear spaces. In particular, there is a 'planarity condition' that excludes spaces of geometrical dimension greater than 2. Embeddability problems are posed and answered, and an outline of the classification program is given.Item Open Access Solvable groups of automorphisms of stable planes(1992) Stroppel, MarkusAn interesting problem in the foundations of geometry is the following question: What is the impact of the interplay of topological assumptions and homogeneity? One possibility to make this (rather philosophical) question treatable for a mathematician is the classification project for stable planes. We shall briefly outline the necessary definitions and basic (though occasionally deep) results. In section 2, we treat solvable groups of automorphisms of stable planes. This may serve as an example how the project works. Note, however, that the case of (semi-)simple groups of automorphisms is where Lie structure theory shows its full strength, cf. the final Remark.Item Open Access Planar groups of automorphisms of stable planes(1992) Stroppel, Markus(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).Item Open Access Locally compact Hughes planes(1994) Stroppel, MarkusAmong the eight-dimensional stable planes, the compact connected generalized Hughes planes and the geometries induced on the outer points are characterized by the property that these planes admit an effective action of the group SL3C.Item Open Access Achtdimensionale stabile Ebenen mit quasieinfacher Automorphismengruppe(1991) Stroppel, MarkusIn der vorliegenden Arbeit wird das systematische Studium achtdimensionaler stabiler Ebenen mit großer Automorphismengruppe begonnen. Stabile Ebenen stellen eine Verallgemeinerung kompakter zusammenhängender projektiver Ebenen dar. Viele Methoden und manche Ergebnisse lassen sich übertragen. Die genaue Problemstellung und ihr Hintergrund sind in der Einleitung skizziert, dort werden auch die wichtigsten Ergebnisse dieser Arbeit genannt.Item Open Access Quaternion hermitian planes(1993) Stroppel, MarkusThe quaternion hermitian planes are defined, and are characterized by certain groups of automorphisms. For this purpose, characterizations of locally compact connected translation planes (in the context of stable planes) and compact connected projective desarguesian planes are given.Item Open Access A categorical glimpse at the reconstruction of geometries(1993) Stroppel, MarkusThe author's 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.Item Open Access Slanted symplectic quadrangles(1994) Grundhöfer, Theo; Joswig, Michael; Stroppel, MarkusBy "slanting" symplectic quadrangles W(F) over fields F, 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.