New geometries for finite groups and polytopes
In this rather interesting paper, the authors present two fairly general ways to create a new diagram geometry out of an old one, and they compare some connectivity properties of the old and new geometries. The first construction deals with geometries having a graph as some point-line truncation and having some additional properties such as (1) in the diagram the node corresponding to the points is only joined to the one corresponding to the lines, (2) if the two vertices (points) of an edge (line) are incident with the same element of type distinct from the type of lines, then that edge (as a line) is incident with that element. Then one replaces the set of lines (edges) by a copy of the points (vertices) and declares two points of different type to be incident when they are adjacent (hence distinct) in the graph. The rest of the new incidence relation is natural. This trick can be applied to a number of geometries with non-thick residues (ordinary \(n\)-gons and Petersen graphs, for instance). Applied to graphs themselves, one obtains self polar thick geometries, but not always partial linear spaces because the graph might contain 4-cycles. In fact, this construction goes both ways for self-polar geometries admitting a polarity without absolute points. Examples are the geometries obtained by deleting the absolute elements of a polarity of any rank 2 geometry (this way e.g. Desarguesian affine planes of even order minus a line pencil arise), or subgeometries thereof (the geometry induced by all exterior (interior) points and lines of a conic in Desarguesian projective planes of odd order; taking the interior ones in a plane of order 5, the Desargues configuration arising from the Petersen graph shows up as a special case -- it is mentioned as a sporadic example in the paper under review). The second construction is the generalization (or rather just putting in general context) of the reversed procedure related to the so-called oriflamme complex (producing a thick building of type \(D_n\) out of a non-thick building of type \(C_n)\). Here, roughly speaking, one replaces two prechosen types \(i\) and \(j\) by two other sets of elements: the first new type is the union of the types \(i\) and \(j\), the second type is the set of flags of type \(\{i,j\}\). This is strongly related to doubling geometries (or taking flag complexes). The author neglect to remark that this procedure also has a natural inverse. Anyway, this second construction, or at least its converse, is known and part of folklore. Finally, the authors give some tables of examples. One class of examples are geometries related to polytopes (they are thin, and that suited very well the purposes of the authors), another class are geometries related to sporadic simple groups (and they also tend to have a lot of non-thick residues such as Petersen graphs and triangles).
scientific article; zbMATH DE number 1563228