Weakly discrete motion groups of a pseudoeuclidean plane (Q1365026)

A discontinuous motion group \(G\) of a pseudoeuclidean (Minkowski) plane (the real affine plane endowed with the quadratic form \(x^2-y^2\)) is said to be (8\('\))-discrete if there is a nontrivial orbit \(G(P)\) and a Minkowskian disk \(U_r(X)\) of radius \(r\) with centre \(X\) which contains only a finite number of elements of \(G(P)\). As in the Euclidean case, the (8\('\))-discrete motion groups are divided into rosette, frieze, and wallpaper groups, if their translation subgroups are zero-, one-, and two-dimensional, respectively. The author provides a classification of these three types of groups (the notion of equivalence of groups being defined by: \(G_1\) and \(G_2\) are equivalent if and only if there is an affine transformation \(\varphi\) that preserves both the space-like and the time-like reflections and glide-reflections of the pseudoeuclidean plane such that \(G_1 = \varphi\circ G\circ \varphi^{-1}\)), and proves the following characterization theorems: (1) \(G\) is a rosette group if and only if \(G(X) = \{X\}\) for some point \(X\); (2) \(G\) is a wallpaper group if and only if its translation subgroup is generated by two non-parallel translations.