Three-dimensional connected groups of automorphisms of toroidal circle planes

TL;DR

This paper classifies toroidal circle planes with three-dimensional automorphism groups, showing that almost simple Lie groups acting on these planes are isomorphic to PSL(2,R), and provides a framework for full classification.

Contribution

It establishes that three-dimensional automorphism groups are isomorphic to PSL(2,R) in certain cases and offers a new framework for classifying these geometric structures.

Findings

Automorphism group is isomorphic to PSL(2,R) for certain planes.

Framework for classifying toroidal circle planes based on group action.

Enhanced understanding of symmetry groups in geometric plane structures.

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 $\text{PSL}(2,\mathbb{R})$. Using this result, we describe a framework for the full classification based on the action of the group on the point set.