Point-regular geometries
The author continues his investigation in Geom. Dedicata 46, No. 1, 47-60 (1993; Zbl 0783.51002) of how geometries can be reconstructed from their automorphism groups. In the paper under review he considers incidence structures \((P,{\mathcal L})\) that admit sharply point transitive groups \(G\) of automorphisms. In this case, \(P\) is identified with \(G\) and \(G\) acts on itself by the right regular action. For a set \({\mathbf S}\) of subgroups of \(G\) two systems of subsets of \(G\) are defined as follows. \({\mathcal L}^{\mathbf S}=\bigcup_{H\in{\mathbf S}}G/H\) is the union of all left coset spaces \(G/H\) for \(H\) in \({\mathbf S}\). Taking right cosets \(H\setminus G\) instead he likewise defines \(^{\mathbf S} {\mathcal L}=\cup_{H\in{\mathbf S}}H\setminus G\). In the latter case, \({\mathbf S}\) consists of the stabilizers of representatives for the \(G\)-orbits on \(^{\mathbf S}{\mathcal L}\) and the whole system of lines can be thought of as moving around some standard lines; this generalizes the familiar construction of translation planes. Various properties of these two systems are investigated in the first part of the paper. Whereas the group \(G\) acts on \(^{\mathbf S} {\mathcal L}\) in a natural way, this happens to be so for \({\mathcal L}^{\mathbf S}\) if and only if \({\mathbf S}\) is invariant under conjugation (and then \(^{\mathbf S} {\mathcal L}={\mathcal L}^{\mathbf S}\)). One obtains a linear space in both situations if and only if \({\mathbf S}\) forms a group partition of \(G\), that is, every non-identity element of \(G\) is in exactly one subgroup contained in \({\mathbf S}\). Furthermore, \((G,^{\mathbf S}{\mathcal L})\) always is a sketched geometry. In fact, it is shown that every point-regular sketched geometry can be represented in this form for some collection \({\mathbf S}\) of subgroups. In the second part the author applies his method to topological planes where the point set \(P\) is a topological manifold. In this case, the regular action of a locally compact, \(\sigma\)-compact group \(G\) on \(P\) is equivalent to the right regular action and \(G\) and \(P\) are homeomorphic. In particular, the author looks at the construction of arc planes by \textit{H. Groh} [Abh. Math. Semin. Univ. Hamburg 48, 171-202 (1979; Zbl 0415.51007)] and at stable planes. He generalizes a result of \textit{H. Groh} [loc. cit.] in characterising arcs \(A\) in \((G,^{\mathbf S} {\mathcal L})\), \({\mathbf S}\) a group partition, for which \((G,^{\mathbf S} {\mathcal L}(A))\) is a linear space where \(^{\mathbf S} {\mathcal L}(A)\) is obtained from \(^{\mathbf S} {\mathcal L}\) by deleting all lines, i.e., cosets \(Hg\) for \(H\in{\mathbf S}\) and \(g\in G\), for which \(AA^{-1}\) intersects \(H\) in more than the identity element (this gives a set of `slopes' for \(A\)) and adding all orbits of \(A\) under \(G\). Finally the author collects examples of point-regular actions on stable planes to illustrate how his construction may be used in the study of such planes.
scientific article; zbMATH DE number 1053898
scientific article