Collineations of \(V\)-spaces (Q1591229)

An \(\mathbb R^{n}\)-space \((P,{\mathcal L})\) is a linear incidence geometry such that the point space \(P\) is a topological space homeomorphic to \(\mathbb R^{n}\) and each element of the line space \({\mathcal L}\) is a closed subset of \(P\) homeomorphic to \(\mathbb R\); furthermore the geometric operation assigning to two points the unique line containing them is continuous, if \({\mathcal L}\) carries the topology of Hausdorff--convergence. The situations for \(n=2\) and for \(n>2\) are quite different: In the two-dimensional case joining is automatically continuous and the space is stable, i.e., the set of pairs of lines with a common point is open in \({\mathcal L}^{2}\). Given a ``well-behaved'' \(\mathbb R^{2}\)-space \((\{0\}\times\mathbb R^{2},{\mathcal N})\) all shift maps and all rotations with a vertical axis are applied to the line space \({\mathcal N}\) to obtain a system \({\mathcal L}\) of subsets of \(\mathbb R^{3}\). Then \({\mathcal V}:=(\mathbb R^{3},{\mathcal L})\) is an \(\mathbb R^{3}\)-space. Such spaces are called \(V\)-spaces and were introduced in [\textit{D.~Betten}, Result. Math. 12, 37-61 (1987; Zbl 0631.51006)]. The group \(\Gamma_{{\mathcal V}}\) of all collineations of \({\mathcal V}\) is a Lie group with dimension at least \(4\) and at most \(12\) in which case \({\mathcal V}\) is an affine space. The author proves that \({\mathcal V}\) is an affine space if \(\dim\Gamma_{{\mathcal V}}\geq 9\) extending a result of [\textit{D.~Betten}, Atti Semin. Mat. Fis. Univ. Modena 34, 173-180 (1986; Zbl 0623.51009)] about planar \(\mathbb R^{3}\)-spaces. Let \(f:\mathbb R\to\mathbb R\) be a continuous strictly convex symmetric function with minimum \(0\) at \(0\) such that \(\lim_{\pm\infty}f-l=\infty\) for every linear function \(l:\mathbb R\to\mathbb R\). Then \(f\) defines an \(\mathbb R^{2}\)-space (namely a shift-plane [\textit{H.~Salzmann} et al., `Compact projective planes,' de Gruyter-Verlag, Berlin (1996; Zbl 0851.51003)]) and therefore a \(V\)-space \({\mathcal V}_{f}\). The author proves that \(f=(x\mapsto c|x|^{p})\) with \(c>0\) and \(p>1\) if and only if \(\dim\Gamma_{{\mathcal V}_{f}}\geq 5\), and that \(p=2\) if and only if \(\dim\Gamma_{{\mathcal V}_{f}}\geq 6\) in which case \({\mathcal V}_{f}\) is an affine space.