Achtdimensionale lokalkompakte Translationsebenen mit mindestens 17- dimensionaler Kollineationsgruppe. (Eight-dimensional locally compact translation planes with at least 17-dimensionally collineation group)
Let \({\mathcal P}\) be a (locally) compact connected projective plane of finite topological dimension; only dimensions 2,4,8, and 16 are possible by a deep result of \textit{R. Löwen} [J. Reine Angew. Math. 343, 108-122 (1983; Zbl 0524.57011)]. The most interesting planes are those with large automorphism group; largeness may be measured by the topological dimension of the group \(\Sigma\) of all continuous collineations of \({\mathcal P}\). By results of \textit{H. Salzmann} [yet unpublished, cp. his paper in Math. Z. 176, 345-357 (1981; Zbl 0465.51004)], if dim \({\mathcal P}=8\) and dim \(\Sigma\geq 17\), then \({\mathcal P}\) is a translation plane or a dual translation plane or one of the Hughes planes described in \textit{H. Salzmann} [loc. cit.]. The paper under review concludes Hähl's classification, announced in the author's paper in Math. Z. 159, 259-294 (1978; Zbl 0381.51009), of all locally compact translation planes \({\mathcal P}\) of dimension 8 with dim \(\Sigma\geq 17\). The result may be summarized as follows. If dim \(\Sigma\geq 19\), then \({\mathcal P}\) is the desarguesian plane over the quaternions \({\mathbb{H}}\); there are several (explicitly known) families of nondesarguesian planes with dim \(\Sigma\in \{17,18\}\), see (b)-(f) below. We remark that there are analogous results of Salzmann for the other possible dimensions of \({\mathcal P}\), again leading to translation planes, as well as classifications of these translation planes, see \textit{D. Betten} [Math. Z. 154, 125-141 (1977; Zbl 0335.50016), Geom. Dedicata 16, 179-193 (1984; Zbl 0547.51012)] and the author [loc. cit.; Arch. Math. 48, 267-276 (1987); ''Eine Kennzeichnung der Oktavenebene'', Indag. Math., to appear; ''16-dimensionale lokalkompakte Translationsebenen mit SU(4,\({\mathbb{C}})\) als Kollineationsgruppe'', Geom. Dedicata, to appear]. The author contructs a large class of quasifields by modifying the multiplication of \({\mathbb{H}}:\) let \(h\in {\mathbb{H}}\) and let \(\Phi\) be a topological transversal for the cosets of \({\mathbb{R}}^{\perp}h\) in \({\mathbb{H}}\) with 0,1\(\in \Phi\). Then the multiplication \((ah+b)\circ x = axh + xb\) with \(a\in {\mathbb{R}}^{\perp}\), \(b\in \Phi\) defines a topological quasifield Q(h,\(\Phi)\). It is a division algebra (i.e. it satisfies both distributive laws) if and only if \(\Phi ={\mathbb{R}}\). These division algebras are closely related to the nearfields constructed by \textit{F. Kalscheuer} [Abh. Math. Semin. Univ. Hamb. 13, 413-435 (1940; Zbl 0023.00602)], the connexion being given by derivation in the sense of calculus (i.e. the set of all multiplications \(x\to s\circ x\), \(s\in {\mathbb{H}}\), is the Lie-algebra of the multiplicative group of one of Kalscheuer's nearfields). The author determines the collineation groups of the translation planes coordinatized by the quasifields Q(h,\(\Phi)\). In particular, he proves Theorem (4.4): The plane over Q(h,\(\Phi)\) has an automorphism group of dimension at least 17 if and only if \(\Phi ={\mathbb{R}}^+\cup \{0\}\cup g{\mathbb{R}}^+\), i.e. \(\Phi\) consists of two linear pieces. The classification mentioned above proceeds as follows. Let \({\mathcal P}\) be a locally compact translation plane of dimension 8 with automorphism group \(\Sigma\) satisfying dim \(\Sigma\geq 17\). By earlier work of the author [Math. Z. 159, 259-294 (1978; Zbl 0381.51009), Result. Math. 2, 62-87 (1979; Zbl 0437.51011), Arch. Math. 34, 231-242 (1980; Zbl 0443.51010), Monatsh. Math. 90, 207-218 (1980; Zbl 0443.51011), J. Reine Angew. Math. 330, 76-92 (1982; Zbl 0476.51010), Abh. Math. Semin. Univ. Hamburg 53, 84-102 (1983; Zbl 0528.51003), Monatsh. Math. 97, 23-45 (1984; Zbl 0526.51013)] it remains to consider the following constellation: \({\mathcal P}\) has an affine flag \(p\in S\) such that the shears with axis S form a group of dimension at least 3, and the stabilizer \(\Sigma_{p,S}\) contains a subgroup isomorphic to Spin(3) acting as \(S0_ 3({\mathbb{R}})\) on the pencil through p. By Theorem (2.3), this constellation occurs if and only if \({\mathcal P}\) is coordinatized by one of the quasifields Q(h,\(\Phi)\). Together with (4.4) this gives the classification theorem: The locally compact translation planes of dimension 8 with \(\dim \Sigma \geq 17\) are precisely the following ones: (a) the desarguesian plane over \({\mathbb{H}}\) (here dim \(\Sigma\) \(=35)\), (b) the planes over the division algebras Q(h,\({\mathbb{R}})\), or the dual or transposed planes (here dim \(\Sigma\) \(=18)\), (c) the planes from Theorem (4.4), (d) the planes over Kalscheuer's nearfields, (e) the planes over the division algebras \(D_{\alpha}\) from the author, Geom. Dedicata 4, 333-361 (1975; Zbl 0325.50014), (f) the planes over Rees' division algebras [cp. \textit{D. Rees}, Proc. Cambridge Philos. Soc. 46, 1-18 (1950; Zbl 0034.307)]. The examples (c)-(f) all have dim \(\Sigma\) \(=17.\) In a separate section, the author puts his results on division algebras with large automorphism groups (see the reference in (d) above) in the more general context of the present classification.
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