A note on Laguerre translations

Abstract

Using the correspondence between 2n-dimensional Laguerre planes and compact antiregular generalized quadrangles with parameter n = 1, 2 we show that almost each automorphism of such a Laguerre plane that induces a translation in the derived affine plane at a point p is a Laguerre translation, that is, fixes all points on the parallel class of p. The same is true for ovoidal Laguerre planes over ovals that also are dual ovals.

More on Kleinewillinghöfer Types of Flat Laguerre Planes

“…In case of flat Laguerre planes it suffices to require that an automorphism induces a translation in a non-vertical direction in the derived affine plane at a fixed point p in order to obtain a (G, B(p, C))-translation, that is, all the points of G = |p| are automatically fixed; see [18].…”