On affine motions of the generalized spaces of paths (Q677764)

Let \(A_{n,y}\) be a generalized space of paths as considered by \textit{M. S. Knebelman} [Am. J. Math. 51, 527-564 (1929; JFM 55.0407.02)], and suppose that the group of affine motions \(G_r\) of order \(r\) has orbits of dimension one. Then it can be shown that the Lie algebra \({\mathfrak L}_r\) of \(G_r\) is either Abelian or contains an Abelian ideal \({\mathfrak L}_{r-1}\) such that for each generator \(X\in{\mathfrak L}_r\), \(X\in {\mathfrak L}_{r-1}\), \(\text{ad} X\) preserves each one-dimensional subspace in \({\mathfrak L}_{r-1}\). The maximum order \(r\) of \(G_r\) is \(n+1\) and the corresponding \(A_{n,y}\) is projectively flat. Let now \(r\) be the maximum order of Abelian groups of affine motions. Then \(r=(n^2 +2n)/4\), \(n\geq 2\), if \(n\) is even, and \(r=n+3\), if \(n\) is odd. If, in particular, \(\dim A_{n,y}\) is 3, then \(r=3\).