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

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

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

The point of the article is to explore the consequences of the higher connectivity of $N(ℌ)$, 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 $N(ℌ)$ 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, $\mathscr{H}=\{H_i\mid i\in I\}$ 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 $\mathscr{H}$, then, of course, we expect to get non-trivial intersections.

Let $ℌ={\coprod }_{i\in I}{H}_i\backslash G=\{H_{i}g\mid H_i\in \mathscr{H}\},$ 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 $\mathscr{H}$. This covers $G$ and we write $N(ℌ)$ for the corresponding simplicial complex, which is the nerve of this covering.

More formally:

Definition

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

Examples

1. If $\mathscr{H}$ consists just of one subgroup, $H$, then $ℌ$ is just the set of cosets, $H\backslash G$ and $N(ℌ)$ is 0-dimensional, consisting just of 0-simplices / vertices.

2. If $\mathscr{H}=\{H_1,H_2\}$, (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, $N(ℌ)$, 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:

• $S_3\equiv (a,b:a^3=b^2=(\mathrm{ab}{)}^2=1)$, (so $a$ denotes, say, the 3-cycle $(123)$ and $b$, a transposition $(12)$).

  * Take $H_1=⟨a⟩=\{1,(123),(132)\}$, yielding two cosets $H_1$ and $H_{1}b$.

  * Similarly take $H_2=⟨b⟩=\{1,(12)\}$ giving cosets $H_2$, $H_{2}a$ and $H_2{a}^2$.

The covering of $S_3$ is then $ℌ=\{H_1,H_{1}b,H_2,H_{2}a,H_2{a}^2\}$, and the nerve is a complete bipartite graph? on $2+3$-vertices.

$n$-generating families

Definition

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

Rephrasing and extending comments made earlier, we have

Proposition

There are isomorphisms:

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

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

Examples

1. Keeping in the case where $\mathscr{H}=\{H_1,H_2\}$, so two subgroups of $G$ then we have that $\bigsqcup _{\cap }\mathscr{H}\to G$ is an isomorphism if and only if $N(ℌ)$ is a tree. This gives one of the basic types of a graph of groups?. In more generality, if $\mathscr{H}=\{H_1,\dots ,H_n\}$ then there is a complex of groups associated with $N(ℌ)$.

2. A more complex family of examples of the above situation occurs in algebraic K-theory. This has the general linear group, $G{\ell }_n(R)$, of invertible $n\times n$ 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 $\{1,\dots ,n\}$. 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 $\mathrm{PO}(n)$ for the set of partial orders of $\{1,\dots ,n\}$.

Definition

We say an $n\times n$ matrix, $A=(a_{\mathrm{ij}})$ 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 $T_n^{\sigma }(R)$ be the subgroup of $G{\ell }_n(R)$ formed by the $\sigma$-triangular matrices and then look at all such subgroups for all $n$, considering the stable general linear group $G\ell (R)$ as the colimit of the nested sequence of all the $G{\ell }_n(R)$, take $G=G\ell (R)$. Considering the family, $\mathscr{H}$, of all the $T_n^{\sigma }(R)$, form the corresponding nerve, $N(ℌ)$. 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.