This paper makes some preliminary observations towards an extension of current work on graphs defined on groups to simplicial complexes. I define a variety of simplicial complexes on a group which are preserved by automorphisms of the group, and in many cases have a relation to familiar graphs on the group. The ones which seem to reach deepest into the graph structure are two forms of independence complex, and some results on the class of groups for which these two complexes coincide are given. Other examples are treated more briefly.