Truncations of inductively minimal geometries (Q1394810)

Inductively minimal geometries were introducted by \textit{F. Buekenhout, M. Dehon} and the author [in Mostly finite geometries (Iowa City, 1996), Lecture Notes in Pure and Appl. Math. 190, 185-190 (1997; Zbl 0889.51015)]. These geometries form an infinite family of incidence geometries on which finite groups act flag-transitively. \textit{F. Buekenhout} and the author [Bull. Belg. Math. Soc. Simon Stevin 5, 213-219 (1998; Zbl 0928.51009)] investigated some properties of inductively minimal geometries, among them the properties RWPRI and (IP)\(_2\) which they showed each inductively minimal geometry possesses. In the paper under review the author studies the behavior of inductively minimal geometries under truncations, which is a well-known method to create new geometries from known ones. However, in general, the truncation of a geometry does not inherit all the properties of this geometry. The author succeeds in characterising those truncations of inductively minimal geometries that satisfy RWPRI and (IP)\(_2\) and determines all rank 2 residues in such truncations. He shows that a \(J\)-truncation of an inductively minimal pair \((\Gamma,G)\) with basic diagram \((I,\sim)\) satisfies (IP)\(_2\) if and only if the subgraph induced on \(J\subseteq I\) is connected such that for every maximal clique \(M\) of \((I,\sim)\) one has that \(J\) contains either none, one or all the vertices of \(M\). In this case, the basic diagram of the truncation \(\Gamma^J\) is the graph induced on \(J\) by \((I,\sim)\). If \(i\sim j\) is an edge of the basic diagram of \(\Gamma^J\), then a residue of type \(\{i,j\}\) is isomorphic to a certain subset geometry \textbf{SsG}(\(k,l\)). Furthermore, showing that no vertex \(j\in J\) can be the middle of a connected component of \((I\setminus J)\cup\{j\}\), the author obtains that a truncation of an inductively minimal geometry which satisfies (IP)\(_2\) also satisfies RWPRI.