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Polarities of Schellhammer planes. (English) Zbl 1229.51013

A projective plane \({\mathcal P} = (P,{\mathcal L})\) with point set \(P\) and line set \({\mathcal L}\) is defined by the authors to be a Schellhammer plane if there exists a subgroup \(S\) of \(\text{Aut}({\mathcal{P}})\), which fixes a line \(\infty \) and a non incident point \(o\) and which acts sharply transitively on \(P \setminus (\{o\}\cup \{p\in P : p \in \infty\})\) and on \({\mathcal L} \setminus (\{L\in {\mathcal L} : o \in L \}\cup\{\infty\} )\). \(S\) is called a Schellhammer group on \({\mathcal P}\). If only a point \(a\) and a line \(A\) are chosen conveniently, then via sharp transitivity lines and points are labeled by elements of \(S\), \(L(s):=s(A)\), \(s\) identified with \(s(a)\), the affine point row of \(A\) then given by \(L(1)=\{ t\in S : t(a)\in A\}\).
The authors consider compact connected Schellhammer planes \({\mathcal P}\) where \(S\) is a locally compact, \(\sigma\)-compact topological group acting continuously on \(P\). This implies for instance (1.1) (see the paper) that \(S\) is homeomorphic to \(\mathbb{R}^{n}\setminus \{0\}\) where \(n=\dim P\). Polarities of non-Desarguesian Moulton planes normalize some Schellhammer group, and all Schellhammer groups form a single conjugacy class (1.5) (see the paper). Two-dimensional Schellhammer planes with \(S=\mathbb{C}^{\times}\) have been constructed by I. Schellhammer [Diploma thesis, University of Tübingen (1981)], her method is sketched in (2.1) of the paper, and its relation to Betten’s model of the Moulton plane, and Sperner’s planes are given, cf. D. Betten [J. Geometry 2, 107–114 (1972; Zbl 0236.50016)] and P. Sperner [Geom. Dedicata 34, No. 3, 301–312 (1990; Zbl 0702.51011)]. A reformulation (2.6) in the paper of Sperner’s construction of 4-dimensional planes shows its similarity to Schellhammer’s planes (2.1). Furthermore, \(\mathbb{H}^{\times}\) acts as a Schellhammer group on these planes.
The main part of the paper is devoted to the construction and investigation of polarities in Schellhammer planes \({\mathcal P}\): An involutorial automorphism \(\gamma\) of a Schellhammer group \(S\) is called suitable by the authors if \(\gamma\iota (L(1)) = L(1)\), where \(\iota : S\to S\) denotes inversion. Such involutions \(\gamma\) give rise to a polarity \(J_{\gamma}\) of \({\mathcal P}\) interchanging \(0\) and \(\infty\) by defining \(J_{\gamma}(s):=L(\gamma(s))\). In commutative Schellhammer groups \(\iota\) itself is suitable. The polarities \(J_{\gamma}\) and their cosets are studied in some detail; in a 2-dimensional Schellhammer plane, for instance, \(\mathbb{C}^{\times} J_{\gamma}\) consists of polarities (4.5) (see the paper). The conjugacy classes of polarities are determined for Schellhammer’s examples as well as for P. Sperner’s 4-dimensional planes. The sets of absolute points are given in either case, in the Sperner case the set of absolute points is homeomorphic to a sphere or it is empty (6.9) (see the paper).
The authors point out that this provides examples of elliptic polarities for non-Moufang compact projective planes for the first time. Hitherto, polarities of compact connected topological projective planes have been examined by H. Salzmann [Abh. Math. Semin. Univ. Hamb. 28, 250–261 (1965; Zbl 0167.49001); Abh. Math. Semin. Univ. Hamb. 29, 212–216 (1965; Zbl 0139.37804)], Th. Bedürftig [J. Geometry 5, 39–66 (1974; Zbl 0288.50022)], B. Polster [Abh. Math. Semin. Univ. Hamb. 66, 113–129 (1996; Zbl 0883.51005)], S. Immervoll [Result. Math. 39, No. 3–4, 218–229 (2001; Zbl 1017.51014)], and M. Stroppel [Arch. Math. 83, 171–182 (2004; Zbl 1070.51003); Monatsh. Math. 144, 317–328 (2005; Zbl 1073.51006)], cf. also §18 in [H. Salzmann et al., Compact projective planes. With an introduction to octonion geometry. De Gruyter Expositions in Mathematics. 21. Berlin: de Gruyter (1996; Zbl 0851.51003)] for a comprehensive exposition on polarities in the octonion plane.

MSC:

51H10 Topological linear incidence structures
57S20 Noncompact Lie groups of transformations
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