# Higher generation by subgroups

## Idea

(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 $G$ and a family, $ℋ$, of subgroups of $G$, consider the nerve of the covering of the set of elements of $G$ by the cosets $Hg$, for $H\in ℋ$. Although this is a Čech nerve (and thus we have an established notation for nerves as in Čech method), we will denote it $N\left(ℌ\right)$ as in the source.

• The group $G$ is generated by the union of the subgroups $H$ in $ℋ$ if and only if $N\left(ℌ\right)$ is connected.

• The group $G$ is the coproduct of the subgroups, amalgamated along their intersections, if and only if $N\left(ℌ\right)$ is simply connected.

The point of the article is to explore the consequences of the higher connectivity of $N\left(ℌ\right)$, 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 $N\left(ℌ\right)$ influence the properties of $G$.

The techniques used include the bar resolution construction and homotopy colimits. There are applications to Tits systems and to buildings.

## The nerve of a family of subgroups

We start with a group, $G$, and a family, $ℋ=\left\{{H}_{i}\mid i\in I\right\}$ of subgroups of $G$. Each subgroup, $H$, determines a family of right cosets, ${H}_{g}$, which cover the set, $G$. Of course, these partition $G$, so there are no non-trivial intersections between them. If we use all the right cosets, ${H}_{\mathrm{ig}}$, for all the ${H}_{i}$ in $ℋ$, then, of course, we expect to get non-trivial intersections.

Let $ℌ={\coprod }_{i\in I}{H}_{i}\G=\left\{{H}_{i}g\mid {H}_{i}\in ℋ\right\},$ where the $g$ is more as an indicator of right cosets than strictly speaking an index. This is the family of all right cosets of subgroups in $ℋ$. This covers $G$ and we write $N\left(ℌ\right)$ for the corresponding simplicial complex, which is the nerve of this covering.

More formally:

###### Definition

Let $G$ be a group and $ℋ$ a family of subgroups of $G$. Let $ℌ$ denote the corresponding covering family of right cosets, ${H}_{\mathrm{ig}}$, ${H}_{i}\in ℋ$. (We will write $ℌ=ℌ\left(G,ℋ\right)$ or even $ℌ=\left(G,ℋ\right)$, as a shorthand as well.) The nerve of $ℌ$ is the simplicial complex, $N\left(ℌ\right)$, whose vertices are the cosets, ${H}_{\mathrm{ig}}$, $i\in I$, and where a non-empty finite family, $\left\{{H}_{\mathrm{ig}}{}_{i}{\right\}}_{i\in J}$, is a simplex if it has non-empty intersection.

###### Examples
1. If $ℋ$ consists just of one subgroup, $H$, then $ℌ$ is just the set of cosets, $H\G$ and $N\left(ℌ\right)$ is 0-dimensional, consisting just of 0-simplices / vertices.

2. If $ℋ=\left\{{H}_{1},{H}_{2}\right\}$, (and ${H}_{1}$ and ${H}_{2}$ are not equal!), then any right ${H}_{1}$ coset, ${H}_{1}g$, will intersect some of the right ${H}_{2}$-cosets, for instance, ${H}_{1}g\cap {H}_{2}g$ always contains $g$. The nerve, $N\left(ℌ\right)$, is a bipartite graph, considered as a simplicial complex. (The number of edges will depend on the sizes of ${H}_{1}$, ${H}_{2}$ and ${H}_{1}\cap {H}_{2}$, 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:

• ${S}_{3}\equiv \left(a,b:{a}^{3}={b}^{2}=\left(\mathrm{ab}{\right)}^{2}=1\right)$, (so $a$ denotes, say, the 3-cycle $\left(123\right)$ and $b$, a transposition $\left(12\right)$).

* Take ${H}_{1}=⟨a⟩=\left\{1,\left(123\right),\left(132\right)\right\}$, yielding two cosets ${H}_{1}$ and ${H}_{1}b$.

* Similarly take ${H}_{2}=⟨b⟩=\left\{1,\left(12\right)\right\}$ giving cosets ${H}_{2}$, ${H}_{2}a$ and ${H}_{2}{a}^{2}$.

The covering of ${S}_{3}$ is then $ℌ=\left\{{H}_{1},{H}_{1}b,{H}_{2},{H}_{2}a,{H}_{2}{a}^{2}\right\}$, and the nerve is a complete bipartite graph? on $2+3$-vertices.

## $n$-generating families

###### Definition

A family, $ℋ$, of subgroups of $G$ is called $n$-generating if the nerve, $N\left(ℌ\right)$, of the corresponding coset covering is (n-1)-connected, i.e., ${\pi }_{i}N\left(ℌ\right)=0$ for $i.

###### Proposition

There are isomorphisms:

(a) ${\pi }_{0}N\left(ℌ\right)\cong G/⟨\bigcup {H}_{j}⟩;$

(b) ${\pi }_{1}N\left(ℌ\right)\cong \mathrm{Ker}\left(\bigsqcup _{\cap }ℋ\to G\right).$

###### Examples
1. Keeping in the case where $ℋ=\left\{{H}_{1},{H}_{2}\right\}$, so two subgroups of $G$ then we have that $\bigsqcup _{\cap }ℋ\to G$ is an isomorphism if and only if $N\left(ℌ\right)$ is a tree. This gives one of the basic types of a graph of groups?. In more generality, if $ℋ=\left\{{H}_{1},\dots ,{H}_{n}\right\}$ then there is a complex of groups associated with $N\left(ℌ\right)$.

2. A more complex family of examples of the above situation occurs in algebraic K-theory. This has the general linear group, $G{\ell }_{n}\left(R\right)$, of invertible $n×n$ matrices together with a family of subgroups corresponding to lower triangular matrices, …. but with some subtleties involved.

• Let $R$ be an associative ring, and now let $\sigma$ be a partial order on $\left\{1,\dots ,n\right\}$. If $i$ is less that $j$ in the partial order $\sigma$, it is convenient to write $i\stackrel{\sigma }{<}j$. (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 $\sigma$.) We will write $\mathrm{PO}\left(n\right)$ for the set of partial orders of $\left\{1,\dots ,n\right\}$.

### Definition

We say an $n×n$ matrix, $A=\left({a}_{\mathrm{ij}}\right)$ is $\sigma$-triangular if, when $i\stackrel{\sigma }{\le }j$ is false, ${a}_{\mathrm{ij}}=0$, and all diagonal entries, ${a}_{\mathrm{ii}}$ are $1$.

We let ${T}_{n}^{\sigma }\left(R\right)$ be the subgroup of $G{\ell }_{n}\left(R\right)$ formed by the $\sigma$-triangular matrices and then look at all such subgroups for all $n$, considering the stable general linear group $G\ell \left(R\right)$ as the colimit of the nested sequence of all the $G{\ell }_{n}\left(R\right)$, take $G=G\ell \left(R\right)$. Considering the family, $ℋ$, of all the ${T}_{n}^{\sigma }\left(R\right)$, form the corresponding nerve, $N\left(ℌ\right)$. This space has the same homotopy type as the Volodin model for algebraic K-theory, since it is the Čech nerve of the covering $ℌ$, whilst the Vietoris nerve of that covering is the Volodin model.

## References

• H. Abels and S. Holz, Higher generation by subgroups, J. Alg, 160, (1993), 311–341.
Revised on May 12, 2013 15:53:31 by Tim Porter (95.147.236.160)