Reflection groups and \(K\)-loops
There is a vast literature on reflection groups -- that is, groups generated (or containing a subgroup generated) by reflections in lines -- as a language in which to axiomatize 2-dimensional geometry [\textit{F. Bachmann}, `Aufbau der Geometrie aus dem Spiegelungsbegriff', 2. Auflage (1973; Zbl 0254.50001) and `Ebene Spiegelungsgeometrie' (1989; Zbl 0681.51001), \textit{R. Lingenberg}, `Metric planes and metric vector spaces' (1979; Zbl 0419.51001)]. By contrast, very little has been written on geometries determined by groups generated by reflections in points [\textit{K. Prazmowski}, Mitt. Math. Semin. Giessen 193, 107 pp. (1989; Zbl 0684.51002), \textit{C. F. Manara} and \textit{M. Marchi}, Rend. Sci. Mat. Appl. A 125, No. 2, 203-217 (1991; Zbl 0798.51017)], \textit{H. Hotje, M. Marchi} and \textit{S. Pianta}, Discrete Math. 129, No. 1-3, 139-147 (1994; Zbl 0804.51018)], and it refers to the (generalized) Euclidean case only. The paper under review studies the geometry that can be associated to reflection groups \(\Gamma\) containing a subset \({\mathcal D}\) of point- reflections, that satisfy the axioms (1) ``The product of the reflections in three collinear points is a reflection in a point'' and (2) ``Every pair of points has a unique midpoint'' (the `line' determined by the `points' (`point' = point-reflection in that point) \(a\) and \(b\) is defined as the set of all points \(x \in {\mathcal D}\) such that \(xab\) is involutory; let \(\mathfrak G\) be the set of all lines in \(\mathcal D\); a `midpoint' of \(a\), \(b\) is a point \(m\) such that \(b = mam\)). It is shown that, by fixing a point \(o \in {\mathcal D}\), one can define an operation \(+ : {\mathcal D} \times {\mathcal D} \rightarrow {\mathcal D}\) by \(a + b := a'oboa'\), where \(a' \in {\mathcal D}\) is the midpoint of \(o\) and \(a\) (i.e. \(a = a'oa'\)), which turns \(({\mathcal D},+)\) into a \(K\)-loop (a structure introduced by the first author [Abh. Math. Semin. Univ. Hamburg 32, 191- 206 (1968; Zbl 0162.241)] and revived, in a different context, by \textit{A. A. Ungar} [Found. Phys. Lett. 1, No. 1, 57-89 (1988) and Result. Math. 17, No. 1/2, 149-168 (1990; Zbl 0699.20055)]. One can also consider, in a most natural way, \(({\mathcal D}, {\mathfrak G})\) as an incidence space. The authors study several properties of \(\mathcal D\) and show that the motion group \(\Gamma\) of an absolute space (or any dimension) with \(\mathcal D\) as the set of reflections in points satisfies (1) and (2), as well as a third axiom, which could not be shown to be independent of (1) and (2). The associated incidence space \(({\mathcal D},{\mathfrak G})\) of a metric- Euclidean absolute space is a line, and that of an ordinary (i.e. not metric-Euclidean) absolute space is isomorphic to the incidence space of that absolute space.
scientific article; zbMATH DE number 739184
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