Chamber graphs of sporadic group geometries

Summary

Introduction

Early definitions of buildings, following the fundamental work of Tits [19], were couched in terms of simplicial complexes (see also [18, page 319]). More recently an alternative viewpoint, as expounded in [20] (or [11]), which brings chamber systems to the fore, has grown in importance.

Beginning with [1] a search commenced for geometric structures which would perform a similar service for the sporadic finite simple groups as buildings do for the finite simple groups of Lie type. That is, illuminate the internal structure of the sporadic finite simple groups and also place them in a wider context. This aim has yet (if ever?) to be realized. Nevertheless, many interesting and varied geometries have been unearthed in the past 15 or so years (see, for example, [2], [9], [10], [6]). Various aspects of these geometries have been studied—for example a great deal of effort has been expended on their point-line collinearity graphs (see, for example, [12], [13], [14], [17]). The associated chamber graph has received much less attention to date. This is surprising given that the building axioms and many of the concepts relating to buildings can be encoded in the chamber graph (of a building).

Here we survey some of the material in [15] and [16], where some tentative steps are taken in the study of chamber graphs of certain sporadic group geometries. Below we recall the definition of a geometry and a chamber system together with some related concepts, and notation sufficient for our purposes.