Topological pseudo-ovals, elation Laguerre planes, and elation generalized quadrangles
We introduce the notion of (topological) pseudo-ovals. They provide the key to a fuller understanding of Steinke's non-classical 4-dimensional Laguerre planes [\textit{G. F. Steinke}, `Eine Klassifikation 4- dimensionaler Laguerre-Ebenen mit großer Automorphismengruppe', Habil.-Sch. (1988); see also Forum Math. 2, 233-247 (1990; Zbl 0696.51008)] and of their relationships with other geometric structures. Roughly speaking, a pseudo-oval is a sufficiently large family of translation planes sitting on different \(2n\)-dimensional subspaes of one \(3n\)-dimensional vector space, such that the intersection of any two translation planes is a line in both of them. Finite analogues have been studied [see \textit{S. E. Payne} and \textit{J. A. Thas}, `Finite generalized quadrangles', Res. Notes Math., 110. (1984; Zbl 0551.05027), 8.7]. [There is a misprint in may paper: In the second matrix displayed on page 360, the third term of the bottom line should read \(-{1 \over 5} a(c + a^ 2)\)].
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