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Author: search.filters.author.Stroppel, MarkusSubject: search.filters.subject.510

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    Embedding a non-embeddable stable plane
    (1993) Stroppel, Markus
    In [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.
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    Endomorphisms of stable planes
    (1992) Stroppel, Markus
    Endomorphisms 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.
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    Solvable groups of automorphisms of stable planes
    (1992) Stroppel, Markus
    An 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.
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    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).
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    Achtdimensionale stabile Ebenen mit quasieinfacher Automorphismengruppe
    (1991) Stroppel, Markus
    In 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.
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    Quaternion hermitian planes
    (1993) Stroppel, Markus
    The 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.
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    A categorical glimpse at the reconstruction of geometries
    (1993) Stroppel, Markus
    The 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.
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    Reconstruction of incidence geometries from groups of automorphisms
    (1992) Stroppel, Markus
    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.
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    A characterization of quaternion planes
    (1990) Stroppel, Markus
    The eight-dimensional planes admitting SL2H as a group of automorphisms are determined.
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    Stable planes with large groups of automorphisms : the interplay of incidence, topology, and homogeneity
    (1993) Stroppel, Markus
    The theory of topological planes (or stable planes, to stress the importance of the stability axiom) originates from the foundations of geometry. In fact, a simultaneous axiomatic treatment of the "classical plane geometries" - the euclidean, hyperbolic and elliptic plane - has to combine incidence properties with topological (or ordering) properties as well as some assumptions that nowadays are conveniently stated by means of a group action (distance, or angles, among others). The use of topology instead of an ordering makes it also possible to include, e.g., the complex plane geometries. Of course, the theory will be substantial only if one imposes some conditions on the topologies involved. It turns out that the assumption of locally compactness in combination with connectedness singles out a very manageable class of topological planes. This class includes the planes whose point space is a two-dimensional manifold; i.e., the (topologically) nearest relatives of the classical plane geometries.