Generalized quadrangles constructed from topological Laguerre planes

A note on Laguerre translations

“…We call L p a sister of L; see [9,Chapter 6]. In particular, if p is a point of L, then the sister L p with respect to p obtained in the fashion above has points the circles of L that pass through p, the points of L on the parallel class |p| of p but not p itself and the extra point ∞.…”

“…Equivalently, for every circle C and every tangent pencil B(p, D) with p not on C there is precisely one circle in the tangent pencil that touches C. For example, every ovoidal Laguerre plane over an oval that also is a dual oval, every 2n-dimensional Laguerre plane and every finite Laguerre plane of odd order has this property. With this notation we have the following, compare the proof of Theorem 3.4 in [9] and the remark following that Theorem.…”

Semi-biplanes on the cylinder

“…Also in this example we can see at a glance that this particular semi-biplane is self-dual. Proof The main tool in the proof of this result is Table 2 in [13] (see also [4], [5]). This table is the solution of the Apollonius problem for 2-dimensional MObius planes.…”

“…In [13] Schroth presents a unified approach to 2-dimensional circle planes. He gives, among other things, a method to construct 2-dimensional Laguerre planes from other 2-dimensional Laguerre planes, from 2-dimensional MObius planes, or from 2-dimensional Minkowski planes.…”

On 4-dimensional elation Laguerre planes admitting simple Lie groups of automorphisms

“…Furthermore, these automorphisms induce elations in the associated Lie geometry and one obtains an elation generalized quadrangle; cf. [26) for generalized quadrangles and their relation to Laguerre planes and the others types of circle planes. For 4-dimensional Laguerre planes, the collection ~ of all automorphisms in the kernel T that fix no circle plus the identity is a (closed) normal subgroup of T.…”