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

Abstract

We determine, up to isomorphism, all flat (or 2-dimensional) Laguerre planes that admit 4-dimensional groups of automorphisms that fix a parallel class.Mathematics Subject Classification. Primary 51H15; Secondary 51B15.

A new family of 2-dimensional Laguerre planes that admit PSL2(ℝ) × ℝ as a group of automorphisms

“…The collection of all automorphisms of a 2-dimensional Laguerre plane ᏸ forms a group with respect to composition, the automorphism group of ᏸ. This group is a Lie group of dimension at most 7 with respect to the compact-open topology; see [Steinke 1986]. We call the dimension of the group dimension of ᏸ.…”

“…It was shown that such planes must be classical. The 2-dimensional Laguerre planes admitting 4-dimensional groups of automorphisms that fix a parallel class were completely determined in [Steinke 2015]. These planes are covered by the families of Laguerre planes of generalized shear type, Laguerre planes of translation type and Laguerre planes of shift type; see [Steinke 2015, Corollary 3.5] for details and references to the various types of Laguerre planes.…”

“…In [Polster and Steinke 2004], 2-dimensional Laguerre planes were considered and their so-called Kleinewillinghöfer types were investigated, that is, the Kleinewillinghöfer types with respect to the full automorphism group. The classification of those types that can occur in 2-dimensional Laguerre planes is almost complete except for two open cases; see [Steinke 2012] and the references to models of various types given there.…”

“…In 1988], semiclassical Laguerre planes were introduced. These are 2-dimensional Laguerre planes which are composed of two classical half-planes.…”