A new family of $2$-dimensional Laguerre planes that admit $\mathrm{PSL}_2(\mathbb R) \times\mathbb R$ as a group of automorphisms

Abstract

We construct a new family of $2$-dimensional Laguerre planes that differ from the classical real Laguerre plane only in the circles that meet a given circle in precisely two points. These planes share many properties with but are nonisomorphic to certain semiclassical Laguerre planes pasted along a circle in that they admit $4$-dimensional groups of automorphisms that contain ${PSL}_2\left(ℝ\right)$ and are of Kleinewillinghöfer type I.G.1.