Browsing by Author "Stroppel, Markus"

Now showing 1 - 17 of 17

Open Access
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.
Open Access
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.
Open Access
A characterization of quaternion planes
(1990) Stroppel, Markus
The eight-dimensional planes admitting SL2H as a group of automorphisms are determined.
Open Access
Compact groups of automorphisms of stable planes
(1994) Stroppel, Markus
It is shown that compact groups of automorphisms of stable planes are either elliptic motion groups, acting in the usual way, or their dimension is bounded by the dimension of a point stabilizer in the elliptic motion group. Examples show that this bound is sharp.
Open Access
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.
Open Access
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.
Open Access
Formen und Kräfte : ein mathematisch-physikalischer Gang zur Kunst auf dem Campus Vaihingen
(Stuttgart : Fakultät 8 - Mathematik und Physik, Universität Stuttgart, 2022) Stroppel, Markus; Scheffler, Marc; Engstler, Katja Stefanie; Engstler, Katja Stefanie (Konzept und Gestaltung)
Der Rundgang erläutert und interpretiert einzelne Objekte und künstlerische Elemente der Lernstraße auf dem Campus Vaihingen aus mathematischer und physikalischer Sicht für die interessierte Allgemeinheit, aber auch für Schülerinnen und Schüler und für Studierende.
Open Access
Locally compact Hughes planes
(1994) Stroppel, Markus
Among 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.
Open Access
A note on Hilbert and Beltrami systems
(1993) Stroppel, Markus
Hilbert and Beltrami (line-) systems were introduced by H. Mohrmann, Math. Ann. 85 (1922) p.177-183. These systems give examples of non-desarguesian affine planes, in fact, the earliest known examples are of this type. We describe a construction for "generalized Beltrami systems", and show that every such system defines a topological affine plane with point set R 2. Since our construction uses only the topological structure of R 2-planes, it is possible to iterate this process. As an application, we obtain an embeddability theorem for a class of two-dimensional stable planes, including Strambach's exceptional SL2R-plane.
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).
Open Access
Quasi-perspectivities in stable planes
(1993) Stroppel, Markus
Stable planes are a generalization of compact connected projective planes. The possible configurations of fixed points for quasi-perspectivities are determined (extending results of R. Baer), and restrictions to the structure of finite quasiperspective groups as well as bounds for the dimension of quasi-perspective groups are derived.
Open Access
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.
Open Access
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.
Open Access
Slanted symplectic quadrangles
(1994) Grundhöfer, Theo; Joswig, Michael; Stroppel, Markus
By "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.
Open Access
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.
Open Access
Stable planes
(1994) Stroppel, Markus
Stable 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.
Open Access
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.