Partial Geometries of Rank n

Abstract

Publisher Summary This chapter discusses the partial geometries of rank n. In the theory of Buekenhout diagrams and their geometries, some partial geometries arise very naturally: (1) the circle geometries, (2) generalized quadrangles, (3) linear spaces and their duals were quickly recognized as being important. Extensions of projective geometries and polar spaces were studied because of their connections with the sporadic simple groups, and for polar spaces in particular. Extensions in a more general setting have been around for a long time—for example inversive planes or the geometries associated with the Mathieu groups. A natural generalization of this was to consider extensions of partial geometries, where an interesting theory has developed. Another step takes us to “partial geometries of rank n ” that includes most of the geometries already studied and belonging to some Buekenhout diagram. The chapter describes some of these ideas. It describes partial geometries of rank 2, and highlights the results about extended partial geometries. It also discusses other classes of rank 3 partial geometries. Many of these geometries concerns either finding their automorphism groups, or characterizing with an automorphism group transitive on maximal flags.