Characterizations of Miquelian Laguerre planes
The Lenz-Barlotti classes of projective planes classify these planes according to the configuration of point-line pairs \((p,L)\) for which the group of central collineation \(C(p,L)\) with centre \(p\) and axis \(L\) is linearly transitive, that is, transitive on \(M \backslash \{p,M \cap L\}\) for each line \(M \neq L\) through \(p\). In the Desarguesian projective planes \(C(p,L)\) is linearly transitive for every possible point-line pair \((p,L)\). However, the linear transitivity of \(C(p,L)\) for much fewer point-line pairs suffices in order to characterize the Desarguesian planes. In a similar fashion \textit{R. Kleinewillinghöfer} [`Eine Klassifikation der Laguerre-Ebenen', Diss. Darmstadt (1979; Zbl 0436.51007) and Arch. Math. 34, 469-480 (1980; Zbl 0457.51010)] obtained a classification of Laguerre planes with respect to central automorphisms and \textit{E. Hartmann} [J. Geom. 18, 9-27 (1982; Zbl 0504.51008)] gave, among other results, characterizations of Miquelian Laguerre planes by the circular transitivity of certain groups of central automorphisms of the Laguerre plane. In the paper under review the author deals with two such characterizations, one improving on one given by \textit{E. Hartmann} [loc. cit.], the other answering a conjecture in the same paper in the affirmative. The first result deals with \(L\)-translations of a Laguerre plane \(L\), that is, automorphisms of \(L\) that fix one parallel class \(P\) pointwise and one tangent pencil of circles through \(P\) elementwise. It is shown that a Laguerre plane \(L\) is Miquelian of characteristic \(\neq 2\) if and only if each group of \(L\)-translations is circularly transitive (that is, the \(L\) is of Kleinewillinghöfer type at least \(K)\) and at least one of these groups does not contain an involution. Secondly, \(L\)- dilatations are considered; these are automorphisms of \(L\) that fix a point \(P\) and a circle \(z\) through \(P\) and induce an homology with axis a parallel class \(e\) in the derived plane at \(P\). The author proves that a Laguerre plane is Miquelian if and only if each group of \(L\)-dilatations is circularly transitive. (This was conjectured by E. Hartmann and proved in odd characteristic under the weaker hypothesis that the respective groups are circularly transitive for one fixed parallel class \(e_ 0\) and all points on \(e_ 0\) but under the additional assumption that each such group contains an involution.) In the proofs of both theorems a derived affine plane is coordinatised by a distributive quasifield or a skewfield, \(K\), respectively. Circles not passing through the point of derivation are described by parabolic curves in the derived affine plane. Using the circular transitivity of the respective groups, functional equations for these parabolic curves are obtained and skilfully combined to infer that \(K\) must be a field and parabolic curves must be usual parabolae. This gives the familiar description of the Miquelin Laguerre plane over the field \(K\).
scientific article; zbMATH DE number 553941