(This is partially an account of the paper by Abels and Holz (1993) listed in the references below. It is augmented with some simple examples and discussion.)
For a group and a family, , of subgroups of , consider the nerve of the covering of the set of elements of by the cosets , for . Although this is a Čech nerve (and thus we have an established notation for nerves as in Čech method), we will denote it as in the source.
The group is generated by the union of the subgroups in if and only if is connected.
The group is the coproduct of the subgroups, amalgamated along their intersections, if and only if is simply connected.
The point of the article is to explore the consequences of the higher connectivity of , corresponding to ‘higher generation by the subgroups’, but, from the nPOV, perhaps the real questions are to ask in what way does the homotopy type of influence the properties of .
The techniques used include the bar resolution construction and homotopy colimits. There are applications to Tits systems and to buildings.
We start with a group, , and a family, of subgroups of . Each subgroup, , determines a family of right cosets, , which cover the set, . Of course, these partition , so there are no non-trivial intersections between them. If we use all the right cosets, , for all the in , then, of course, we expect to get non-trivial intersections.
Let where the is more as an indicator of right cosets than strictly speaking an index. This is the family of all right cosets of subgroups in . This covers and we write for the corresponding simplicial complex, which is the nerve of this covering.
Let be a group and a family of subgroups of . Let denote the corresponding covering family of right cosets, , . (We will write or even , as a shorthand as well.) The nerve of is the simplicial complex, , whose vertices are the cosets, , , and where a non-empty finite family, , is a simplex if it has non-empty intersection.
If consists just of one subgroup, , then is just the set of cosets, and is 0-dimensional, consisting just of 0-simplices / vertices.
If , (and and are not equal!), then any right coset, , will intersect some of the right -cosets, for instance, always contains . The nerve, , is a bipartite graph, considered as a simplicial complex. (The number of edges will depend on the sizes of , and , etc.) It is just a graphical way of illustrating the intersections of the cosets, a sort of intersection diagram.
As a specific very simple example, consider:
Take , yielding two cosets and .
Similarly take giving cosets , and .
The covering of is then , and the nerve is a complete bipartite graph on -vertices.
A family, , of subgroups of is called -generating if the nerve, , of the corresponding coset covering is (n-1)-connected, i.e., for .
Rephrasing and extending comments made earlier, we have
There are isomorphisms:
Keeping in the case where , so two subgroups of then we have that is an isomorphism if and only if is a tree. This gives one of the basic types of a graph of groups?. In more generality, if then there is a complex of groups associated with .
A more complex family of examples of the above situation occurs in algebraic K-theory. This has the general linear group, , of invertible matrices together with a family of subgroups corresponding to lower triangular matrices, …. but with some subtleties involved.
Let be an associative ring, and now let be a partial order on . If is less that in the partial order , it is convenient to write . (Note that this means that some of the elements may only be related to themselves and hence are really not playing a role in such a .) We will write for the set of partial orders of .
We say an matrix, is -triangular if, when is false, , and all diagonal entries, are .
We let be the subgroup of formed by the -triangular matrices and then look at all such subgroups for all , considering the stable general linear group as the colimit of the nested sequence of all the , take . Considering the family, , of all the , form the corresponding nerve, . This space has the same homotopy type as the Volodin model for algebraic K-theory, since it is the Čech nerve of the covering , whilst the Vietoris nerve of that covering is the Volodin model.