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Remarks on the \(n\)-dimensional real Möbius geometry. (Bemerkungen zur \(n\)-dimensionalen reellen Möbiusgeometrie.) (German) Zbl 0736.51001

The author considers the real \(n\)-dimensional Möbius geometry \(M^ n={\mathbb{R}}^n\cup\{\infty\}\) with blocks being the hyperplanes of \(\mathbb{R}^n\) extended by \(\infty\) and the hyperspheres of \(\mathbb{R}^n\); a more homogeneous model of this geometry is the \(n\)-dimensional sphere \(S^n\) embedded in \(\mathbb{R}^{n+1}\) with the intersections of hyperplanes of \(\mathbb{R}^{n+1}\) that meet \(S^n\) in at least two points. It is well-known that each automorphism of the \(n\)-dimensional Möbius geometry is a finite composition of reflections in hyperspheres or hyperplanes and that, in the homogeneous model, each automorphism is induced by a collineation of the surrounding \((n+1)\)-dimensional projective space leaving \(S^n\) invariant; e.g. A. F. Beardon [The geometry of discrete groups. New York etc.: Springer-Verlag (1983; Zbl 0528.30001)] or in a more general algebraic-geometric setting H. Mäurer [Math. Z. 98, 355–386 (1967; Zbl 0152.19006)]. Here the author gives a new proof of the former result and more precisely determines the maximal number of reflections in hyperspheres or hyperplanes that are needed to generate an automorphism.

MSC:

51B10 Möbius geometries
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References:

[1] Alan F. Beardon: The Geometry of Discrete Groups. Springer-Verlag New York-Heidelberg- Berlin 1983. · Zbl 0528.30001
[2] H. S. M. Coxeter: Unvergängliche Geometrie. Birkhäuser Verlag Basel und Stuttgart 1963. · Zbl 0108.16201
[3] Z. Jankovský: Zur Möbiusschen Geometrie im n-dimensionakn Raum. Fritz Hohenberg Gedächtniskolloquium Seggauberg, 30. 4.-6. 5. 1989, Tagungsbericht.
[4] H. Schaal: Lineare Algebra und Analytische Geometrie Band II. 2. Auflage. Vieweg Verlag Braunschweig 1980. · Zbl 0508.15001
[5] H. Schaal: Zur perspektiven Zerlegung und Fixpunktbestimmung der Affinitäten von \(A^n (K)\). Archiv d. Math. 38, 116-123 (1982). · Zbl 0469.51019
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