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.