Three-dimensional connected groups of automorphisms of toroidal circle planes (Q2663688)

A toroidal circle plane \(\mathbb T\) consists of the point space \(\mathbb{S}^1 \times\mathbb{S}^1\) and a collection of circles, which are subsets homeomorphic to \(\mathbb{S}^1\), such that any three non-parallel points are on a unique circle. Here, two points \((x_1, y_1), (x_2, y_2) \in\mathbb{S}^1 \times \mathbb{S}^1\) are parallel if either \(x_1 = x_2\) or \(y_1 = y_2\). If it is also satisfied that for any circle \(C\) and any two non-parallel points \(p \in C\) and \(q \not\in C\) there is a unique circle through \(q\) which intersects \(C\) only at the point \(p\) then \(\mathbb{T}\) is called a flat Minkowski plane. The authors investigate toroidal circle planes admitting a connected three-dimensional group of automorphisms. They show that there are exactly five possible cases for such a group and its action on the plane. For three of these cases, examples are known, and in one case even a complete classification is available. In the remaining two cases, it is unknown whether examples exist.