Until recently there have been relatively few articles in the finite group theoretic literature on representations of finite groups on simplicial complexes. However in the last few years that situation has begun to change. This paper discusses some of the activity in the area. We begin with a fairly general discussion intended to give a feeling for the kind of activity now going on. Since space is limited, we touch on only a few examples of such activity, and to provide focus, we eventually concentrate on p-group complexes of finite groups. In the end we concentrate even further on a particular problem in the area of p-group complexes: the question of when p-group complexes of finite groups are simply connected. There we go into more detail.

This volume is devoted to groups of Lie type and their geometries. The p-group complexes of a group G should be viewed as geometries for G. The p-group complexes of the groups of Lie type will be featured prominently here. In particular we will see that if G is of Lie type and characteristic p then the p-group complexes of G are homotopy equivalent to the building of G.

The term “simplicial complex” is used here to mean an abstract simplicial complex. Thus a simplicial complex K consists of a set K of objects called vertices together with a collection of finite subsets of K called simplices such that each subset of a simplex is a simplex.