On toroidal circle planes with groups of automorphisms fixing exactly one point (Q6181311)

Toroidal circle planes are geometries on the torus \(\mathbb{S}^1\times \mathbb{S}^1\) that generalize 2-dimensional Minkowski planes in that all axioms of a Minkowski plane are satisfied except possibly the axiom of touching. In his thesis, on Chapter 7 of which the current paper is based, the author [On the classification of toroidal circle planes. Ph.D thesis, University of Canterbury (2017)] carried out a systematic investigation of toroidal circle planes \(\mathbb{T}\). He found that the automorphism group \(\Gamma\) of \(\mathbb{T}\) is a Lie group of dimension at most 6, and if \(\dim \Gamma \ge 4\), then \(\mathbb{T}\) is a Minkowski plane. He further studied some cases where \(\dim \Gamma=3\). The author [J. Geom. 113, No. 2, Paper No. 26, 25 p. (2022; Zbl 1487.51003)] obtained the so-called strongly hyperbolic Minkowski planes, which admit the 3-dimensional group \(\Phi_\infty=\{ (x,y) \mapsto (x+b, ay+c) \mid a,b,c\in \mathbb{R}, a>0\}\) as a group of automorphisms. In the paper under review the author characterizes strongly hyperbolic planes among toroidal circle planes by the existence of a group of automorphisms isomorphic to \(\Phi_\infty\). In a similar way the so-called Artzy-Groh planes introduced by \textit{R. Artzy} and \textit{H. Groh} [J. Geom. 26, 1--20 (1986; Zbl 0598.51004)] are characterized as toroidal circle planes that admit a group of automorphisms isomorphic to \(\Phi_1 =\{ (x,y) \mapsto (ax+b, ay+c) \mid a,b,c\in \mathbb{R}, a>0\}\). Finally, it is shown that the first examples of toroidal circle planes that are not Minkowski planes constructed by \textit{B. Polster} [J. Geom. 63, No. 1--2, 154--167 (1998; Zbl 0928.51011)] have 2-dimensional automorphism groups.