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

Abstract

ABSTRACT. This paper concerns 4-dimensional ( topological locally compact connected) elation Laguerre planes that admit large automorphism groups. In particular, it is shown that such a plane is classical if its automorphism group is at least 11-dimensional. Furthermore, the elation Laguerre planes admitting simple Lie groups of automorphisms are investigated and various characterizations of the classical complex Laguerre plane and the semi-classical Laguerre planes are obtained.

Flat Laguerre planes admitting 4-dimensional groups of automorphisms that fix a parallel class

“…All flat Laguerre planes that admit 3-dimensional groups of automorphisms in the kernel were determined in [16]. In order to state the result we need some definitions.…”

“…With this notation the following Laguerre planes whose automorphism groups are at least 4-dimensional and whose kernels are at least 3-dimensional were found in [16]. Theorem 3.1.…”

A Characterization of Certain Elation Laguerre Planes in Terms of Kleinewillinghöfer Types

“…A 4-dimensional Laguerre plane L is an elation Laguerre plane if and only if the collection of all automorphisms in T that fix no circle, plus the identity, which is a closed normal subgroup of T , acts transitively on the set of circles, see [22]. 2n-dimensional elation Laguerre planes can be characterized in terms of transitivity properties or the dimension of T ; see [22], [24,Theorem 2.7], and [16,Section 5.4.2].…”

“…In that respect elation Laguerre planes are the closest thing one can get to miquelian Laguerre planes. For a 4-dimensional Laguerre plane to be an elation Laguerre plane it suffices that the kernel is large enough or is transitive on the set of circles; see [22] or [24,Theorem 2.7]. Furthermore, the first examples of non-classical 4-dimensional Laguerre planes found were elation Laguerre planes.…”

A family of 2-dimensional Laguerre planes of generalised shear type

“…In [15] it was shown that these planes comprise, up to isomorphisms, all 2-dimensional Laguerre planes whose kernels are at least 3-dimensional. More precisely, these planes are obtained for h = id, k = 1, or / = g, k = 1, or / = g, h = id.…”