A new family of locally compact 4-dimensional translation planes admitting a 7-dimensional collineation group

Abstract

The present paper contains the construction of a family of 4-dimensional translation planes with a 7-dimensional collineation group. The planes of this family do not appear in the comprehensive classification [2, 1, 3]. We furthermore determine the isomorphism types of the new planes.

Multiplicative Loops of 2-Dimensional Topological Quasifields

“…Using 2-dimensional spreads, Betten in [3], [4], [5], [6], [7], [8], see also [13] and [15], has classified all locally compact 4-dimensional translation planes which admit an at least 7-dimensional collineation group. His normalized 2dimensional spreads are images of sharply transitive sections σ ′ :…”

“…where l = 1 k ln β, l 1 = 1 k ln(−β) (cf. [15], Proposition 4.1, p. 6, and [6], Satz 3, pp. 422-423).…”

“…The classification of topological translation planes A was accomplished by reconstructing the spreads corresponding to A from the translation complement which is the stabilizer of a point in the collineation group of A. In this way all planes A having an at least 7-dimensional collineation group have been determined ( [3]- [8], [15]).…”

“…Locally compact connected topological non-desarguesian translation planes have been a popular subject of geometrical research since the seventies of the last century ( [18], [2]- [9], [13], [15]). These planes are coordinatized by locally compact quasifields Q such that the kernel of Q is either the field R of real numbers or the field C of complex numbers (cf.…”