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.

Three-dimensional connected groups of automorphisms of toroidal circle planes

Schenkel proved that the automorphism group of a flat Minkowski plane is a Lie group of dimension at most 6 and described planes whose automorphism group has dimension at least 4 or one of whose kernels has dimension 3. We extend these results to the case of toroidal circle planes.

On automorphism groups of toroidal circle planes

Using suitable convex functions, we construct a new family of flat Minkowski planes whose automorphism groups are at least 3-dimensional. These planes admit groups of automorphisms isomorphic to the direct product of $${\mathbb {R}}$$ and the connected component of the affine group on $${\mathbb {R}}$$ . We also determine isomorphism classes, automorphisms and possible Klein–Kroll types for our examples. 

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A family of flat Minkowski planes over convex functions

Using suitable convex functions, we construct a new family of flat Minkowski planes whose automorphism groups are at least $3$-dimensional. These planes admit groups of automorphisms isomorphic to the direct product of $\mathbb{R}$ and the connected component of the affine group on $\mathbb{R}$. We also determine isomorphism classes, automorphisms and possible Klein-Kroll types for our examples.

Duy Ho

Papers

Three-dimensional connected groups of automorphisms of toroidal circle planes

On automorphism groups of toroidal circle planes

On automorphism groups of toroidal circle planes

A family of flat Minkowski planes over convex functions

Three-dimensional connected groups of automorphisms of toroidal circle planes