Characterization of contact structures which are chain geometries (Q1266495)

A \textit{partial linear space} is an incidence structure \((P,G)\) such that any two points \(p,q \in P\) lie on at most one line \(g \in G\). A \textit{partial parallel structure} \((P,G,| |)\) is a partial linear space \((P,G)\) with an equivalence relation \(| | \) (parallelity). Finally, a partial parallel structure is called a \textit{partial affine space}, if it can be embedded into an (ordinary) affine space. A \textit{contact structure} \((P,K,(\beta_p)_{p\in P})\) is an incidence structure \((P,K)\), where any \(k \in K\) is called a \textit{chain}, such that any \(\beta_p\) is an equivalence relation (contact relation) on the set \(K_p\) of chains that pass through \(p\). Let \(P_p := \{q \in P\;|\;\exists K \in K_p : q \in K\}\). Moreover, a contact structure \((P,K,(\beta_p)_{p\in P})\) has to satisfy three axioms: (1) any three distinct points lie on exactly one chain, (2) if two chains \(j,k \in K_p\) are in contact, then \(j \cap k = \{p\}\), (3) for any chain \(j \in K\), any point \(p\) on \(j\), and any point \(q \in P_p\) there is exactly one chain \(k\) through \(p\) and \(q\) with \(j \beta_p k\). It is evident from the definition that the derived incidence structures \(\Sigma_p := (P_p, K_p, | |)\), the so-called residues, are partial parallel structures. If every residue is even a partial affine space, the contact structure is then called a \textit{chain space}. For a beautiful introduction into the theory of chain spaces see the author's article `Chain geometries' in the Handbook of incidence geometry, 781-842 (1995; Zbl 0829.51003). The classical chain spaces are the Moebius, the Laguerre and the Minkowski geometries over the real numbers i.e. the geometry of plane sections of the sphere, the cylinder, and the hyperboloid. More classical examples can be obtained by considering a \(K\)-algebra \(R\) over some commutative field \(K\). Let \(P := \{ R(a,b)\;|\;\exists\;c,d \in R: \left ( \begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \right) \in \text{ GL}_2R\}\), \(k := \{ R(s,t)\;|\;s,t \in K^2\setminus\{(0,0)\}\}\), and \(K := \{ k^\gamma\;|\;\gamma \in \text{ PGL}_2R\}\). Such chain spaces \(\Sigma(K,R)\) are called \textit{chain geometries}. It turns out that any chain geometry can be characterized as being a chain space whose automorphism group \(\Pi\) satisfies certain (transitivity) properties. This nice characterization has been proved by the author [Geom. Dedicata 36, No. 2/3, 315-327 (1990; Zbl 0716.51014) and ibid. 41, No. 1, 101-102 (1992; Zbl 0749.51008)]. In the paper under review this result is generalized by substituting the assumption of a chain space by the weaker concept of a contact space.