A family of flat Minkowski planes admitting 3-dimensional simple groups of automorphisms

Abstract

In this paper we construct a new family of flat Minkowski planes of group dimension 3. These planes share the positive half with the classical real Minkowski plane and admit simple groups of automorphisms isomorphic to PSL 2 ðRÞ acting diagonally on the torus. We further determine the full automorphism groups and the Klein-Kroll types of these flat Minkowski planes. 2000 Mathematics Subject Classification. MSC 2000: 51H15, 51B20 Brought to you by | University of Iowa Libraries Authenticated Download Date | 6/10/15 4:09 AM 2 The incidence structures M (k) We construct a flat Minkowski plane MðkÞ by replacing the negative half of the classical flat Minkowski plane by the images of the generating circle C k ¼ fðx; Àxjxj kÀ1 Þ j x A Rg U fðy; yÞg under the group S ¼ fðx; yÞ 7 ! ðdðxÞ; dðyÞÞ j d A PSL 2 ðRÞg: Gü nter F. Steinke 320Brought to you by |

Three-dimensional connected groups of automorphisms of toroidal circle planes

“…Besides Case 2 which is fully determined, there are examples of flat Minkowski planes of group dimension 3 for Cases 1 and 5. Two families of flat Minkowski planes admitting 3-dimensional groups fixing no points but fixing and acting transitively on a circle were constructed by Steinke [17] and [18]. An Artzy-Groh plane M AG (f, g) (cf.…”

On the Klein–Kroll types of flat Minkowski planes

“…Each of these planes has group dimension 3 and admits as a group of automorphisms; see [17], Theorem 3.5.…”