Three-dimensional connected groups of automorphisms of toroidal circle planes

Abstract

We contribute to the classification of toroidal circle planes and flat Minkowski planes possessing three-dimensional connected groups of automorphisms. When such a group is an almost simple Lie group, we show that it is isomorphic to PSL(2, R). Using this result, we describe a framework for the full classification based on the action of the group on the point set.Lemma 4.2. Assume Σ fixes at least one parallel class. Without loss of generality, let π be the fixed (+)-parallel class. Then exactly one of the following occurs.

A family of flat Minkowski planes over convex functions

“…[Sch80]. The current open case of interest is when n " 3, in which a list of possible connected groups of automorphisms of M was presented in [CHS20].…”