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higher generation by subgroups

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 GG and a family, \mathcal{H}, of subgroups of GG, consider the nerve of the covering of the set of elements of GG by the cosets HgH g, for HH\in \mathcal{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()N(\mathfrak{H}) as in the source.

  • The group GG is generated by the union of the subgroups HH in \mathcal{H} if and only if N()N(\mathfrak{H}) is connected.

  • The group GG is the coproduct of the subgroups, amalgamated along their intersections, if and only if N()N(\mathfrak{H}) is simply connected.

The point of the article is to explore the consequences of the higher connectivity of N()N(\mathfrak{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()N(\mathfrak{H}) influence the properties of GG.

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, GG, and a family, ={H iiI}\mathcal{H} = \{H_i\mid i\in I\} of subgroups of GG. Each subgroup, HH, determines a family of right cosets, H gH_g, which cover the set, GG. Of course, these partition GG, so there are no non-trivial intersections between them. If we use all the right cosets, H igH_ig, for all the H iH_i in \mathcal{H}, then, of course, we expect to get non-trivial intersections.

Let = iIH i\G={H igH i},\mathfrak{H} = \coprod_{i\in I}H_i\backslash G= \{ H_i g\mid H_i\in \mathcal{H}\}, where the gg is more as an indicator of right cosets than strictly speaking an index. This is the family of all right cosets of subgroups in \mathcal{H}. This covers GG and we write N()N(\mathfrak{H}) for the corresponding simplicial complex, which is the nerve of this covering.

More formally:

Definition

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

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

  2. If ={H 1,H 2}\mathcal{H} = \{H_1,H_2\}, (and H 1H_1 and H 2H_2 are not equal!), then any right H 1H_1 coset, H 1gH_1g, will intersect some of the right H 2H_2-cosets, for instance, H 1gH 2gH_1g\cap H_2g always contains gg. The nerve, N()N(\mathfrak{H}), is a bipartite graph, considered as a simplicial complex. (The number of edges will depend on the sizes of H 1H_1, H 2H_2 and H 1H 2H_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(a,b:a 3=b 2=(ab) 2=1)S_3 \equiv ( a,b : a^3=b^2=(ab)^2=1), (so aa denotes, say, the 3-cycle (123)(1 2 3) and bb, a transposition (12)(1 2)).
    • Take H 1=a={1,(123),(132)}H_1 = \langle a \rangle = \{1, (1 2 3), (1 3 2)\}, yielding two cosets H 1H_1 and H 1bH_1b.

    • Similarly take H 2=b={1,(12)}H_2 = \langle b\rangle = \{1, (1 2)\} giving cosets H 2H_2, H 2aH_2a and H 2a 2H_2a^2.

The covering of S 3S_3 is then ={H 1,H 1b,H 2,H 2a,H 2a 2}\mathfrak{H} = \{H_1,H_1b,H_2,H_2a,H_2a^2\}, and the nerve is a complete bipartite graph? on 2+32+3-vertices.

nn-generating families

Definition

A family, \mathcal{H}, of subgroups of GG is called nn-generating if the nerve, N()N(\mathfrak{H}), of the corresponding coset covering is (n-1)-connected, i.e., π iN()=0\pi_i N(\mathfrak{H}) = 0 for i<ni\lt n.

Rephrasing and extending comments made earlier, we have

Proposition

There are isomorphisms:

(a) π 0N()G/H j;\pi_0N(\mathfrak{H}) \cong G/\langle\bigcup H_j\rangle;

(b) π 1N()Ker(G).\pi_1N(\mathfrak{H}) \cong Ker(\underset{\cap}{\sqcup} \mathcal{H}\to G).

Examples
  1. Keeping in the case where ={H 1,H 2}\mathcal{H} = \{H_1,H_2\}, so two subgroups of GG then we have that G\underset{\cap}{\sqcup} \mathcal{H}\to G is an isomorphism if and only if N()N( \mathfrak{H}) is a tree. This gives one of the basic types of a graph of groups?. In more generality, if ={H 1,,H n}\mathcal{H} = \{H_1,\ldots ,H_n\} then there is a complex of groups associated with N()N( \mathfrak{H}).

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

    • Let RR be an associative ring, and now let σ\sigma be a partial order on {1,,n}\{1,\ldots,n\}. If ii is less that jj in the partial order σ\sigma, it is convenient to write i<σji\stackrel{\sigma}{\lt} 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 σ\sigma.) We will write PO(n)PO(n) for the set of partial orders of {1,,n}\{1,\ldots,n\}.

    • We say an n×nn\times n matrix, A=(a ij)A = (a_{ij}) is σ\sigma-triangular if, when iσji\stackrel{\sigma}{\leq} j is false, a ij=0a_{ij}=0, and all diagonal entries, a iia_{ii} are 11.

    • We let T n σ(R)T_n^\sigma(R) be the subgroup of G n(R)G\ell_n(R) formed by the σ\sigma-triangular matrices and then look at all such subgroups for all nn, considering the stable general linear group G(R)G\ell(R) as the colimit of the nested sequence of all the G n(R)G\ell_n(R), take G=G(R)G = G\ell(R). Considering the family, \mathcal{H}, of all the T n σ(R)T_n^\sigma(R), form the corresponding nerve, N()N(\mathfrak{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 \mathfrak{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 April 29, 2014 08:08:19 by Tim Porter (2.26.39.195)