Chain geometry determined by the affine group (Q351150)

Let \(V\) be a vector space and let \(\mathcal F\) be a set of permutations of \(V\), identified with their graphs. Then the chain geometry \(\mathbf{M}^*(\mathcal{F})\) is the incidence structure with point set \(V\times V\) and chain set \(\mathcal F\). In this paper, the cases \(\mathcal{F}=\mathrm{GA}(V)\) (affine transformations) and \(\mathcal{F}=\mathrm{GL}(V)\) are studied. In particular, the automorphism groups of \(\mathbf{M}^*(\mathcal{F})\) are determined in these cases. For that, the incidence structures \(\mathbf{M}(\mathcal{F})=(V\times V, \mathcal{F}, \mathcal{L}^+, \mathcal{L}^-) \) with \(\mathcal{L}^+=\{\{u\}\times V:u\in V\}\) and \(\mathcal{L}^-=\{V\times \{u\}:u\in V\}\), and associated so-called reducts of the underlying affine or projective space are employed.