# Posets of cosets and contractibility

For this question let $G$ be a group, perhaps infinite, and let $H_i$ for $i\in I$ be a (finite) family of subgroups closed under taking intersections. I am interested in the coset poset $\mathcal{C}(G,\{H_i\})$ which is defined as the set of cosets $g H_i$ with ordering by inclusion. Note that $g H_i\subseteq g'H_j$ implies that $H_i\subseteq H_j$ and $g^{-1} g'\in H_j$.

The family of subgroups $H_i$ defines a diagram of groups with maps given by inclusion inside $G$. Taking classifying spaces $B(H_i)$ gives a diagram of simplicial sets and I am interested in the colimit of this, denote it $B(G,H_i)$.

Question 1 Where is there a reference for:

The realisation $N(\mathcal{C}(G,H_i))$ is contractible if and only if $B(G,H_i)$ is a classifying space for $G$.

I've got a sketch proof which takes the coset poset and applies the Borel construction using the category $\mathcal{E}G$ with object set $G$ and singleton homsets:

$\mathcal{C}(G,H_i)\times\mathcal{E}(G)/G$

This quotient category can be seen to be equivalent to a category with objects the family $H_i$ and homsets $Hom(H_i,H_j)\cong H_j$. The nerve of this is then seen to be both equivalent to the colimit $B(G,H_i)$ and the Borel construction applied to $N(\mathcal{C}(G,H_i))$.

Question 2 What are some nice examples? I can think of right-angled Artin groups, with subgroups indexed by the simplices of the flag complex; I guess that the coset complex is CAT(0). Also if one of the subgroups is $G$ itself then the result holds.

Of course a valid answer to this whole question could be that I've got everything wrong.

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## 1 Answer

Here are some references that should be of use.

(i) H. Abels and S. Holz, Higher generation by subgroups , J. Alg, 160, (1993), 311– 341.

(ii) S. Holz, 1985, Endliche Identifizier zwishen Relationen , Ph.D. thesis, Univerist\"{a}t Bielefeld.

(iii) A. Bak, R. Brown, G. Minian and T.Porter, Global Actions, Groupoid Atlases and Applications, Journal of Homotopy and Related Structures, 1(1), 2006, pp.101 - 167.

Abels and Holz consisder a slightly different construction. They take the nerve of the covering given by all the cosets but discuss some of the other equivalent notions including yours. Several of the things you find, they prove in that paper (and some were in Holz's thesis.) They discuss some of the Artin group examples and give references to Soulé and others (I won't list their references here.) The paper on global actions includes some pretty (I think!) examples of the interpretation of these nerves in very elementary examples. (e.g. $S_4$, with its usual presentation by transpositions, with subgroups generated by pairs of those. The nerve, if I remember rightly, has realisation $S^2$, the 2-sphere.)

Abels and Holz's paper is very neat indeed and deserves to be more widely known. Hope this helps.

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Thanks for the reference, their covering is just what I want. You're right that this does deserve to be more widely known. I'll try to find the time to have a proper look at the paper on global actions, I think that I would learn a lot. –  James Griffin Jun 23 '11 at 9:52
@James These papers: authors me plus... (with R. Brown, and M. Bullejos), Crossed complexes, free crossed resolutions and graph products of groups, in Recent Advances in Group Theory and Low-Dimensional Topology , Research and Exposition in Mathematics vol 27, Helderman Verlag, 2003, pp. 11-26 (with R. Brown, E. Moore and C.D. Wensley), Crossed complexes, and free crossed resolutions for amalgamated sums and HNN-extensions of groups, Georgian Math. J., Special issue for the 70th birthday of H. Inissaridze, 9 (2002) 623 - 644, may be useful as well without involving directly the nerve. –  Tim Porter Jun 23 '11 at 10:32
I have been ferreting around since and have realised that there is more on this if you replace the nerve by the dual Vietoris complex. Then you have a wide range of methods developed for Volodin spaces and Volodin's K-theory. Look back at the papers in the 1970s on higher algebraic K-theory. –  Tim Porter Jan 24 '12 at 10:02
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