"Simplicial complex" product of groups?

Question

Let $X=(V,E)$ be a graph, and to each vertex $v \in V$, associate a group $G_v$. The graph product of the groups $G_v$ (as defined e.g. here) is $F/R$; the quotient of the free product of the $G_v$ by the by the normal subgroup generated by commutators $[G_{u_1},G_{u_2}]$, where $\{u_1,u_2\} \in E$.

Let $K$ be a simplicial complex on vertex set $V$, and let $G_v$ be a collection of groups. One can also associate a graph product of groups $G^K$ to $K$ by taking the graph which is the $1$-skeleton of $K$. $G^K$ is not then dependent on the higher simplices of $K$. In particular, it doesn't depend on missing faces of dimension greater than $1$.

I wish to find a generalization of the construction of the graph product of groups to take into account the higher simplices (or missing faces) of $K$, but which agrees with the construction of the graph product of groups in the case that $K$ is flag.

One can attempt to make this generalization by taking the colimit of the appropriate diagram in $\mathbf{Grp}$. That is, take the colimit of the diagram $\mathcal{D}: Cat(K) \rightarrow \mathbf{Grp}$, a functor from the face category of $K$ to $\mathbf{Grp}$, which associates to each simplex $\{v_{i_1},...,v_{i_j} \} = \sigma \in K$ the direct product $\prod_{j} G_{v_{i_j}}$, and takes inclusions of simplices to inclusions of groups

This doesn't work, because this still only depends on the $1$-skeleton of $K$. The issue appears to be an absense of "higher commutativity" in $\mathbf{Grp}$.

In their book Metric Spaces of Non-Positive curvature, Bridson and Haefliger define complexes of groups, and the fundamental group of such a complex. We can use $K$ to construct a simple complex of groups (associating to every simplex the product of the vertex groups) and then take the fundamental group. That construction seems like it could get me somewhere, but it does not seem to be possible to use/adapt this into generalization of the graph product of groups - despite the fact that the resulting fundamental group does seem to depend on the higher simplices of $K$.

Intuitively (to me), the fundamental group seems to be an invariant which depends on "loops" in $K$, rather than missing faces.

Viewing groups as discrete topological groups and taking the homotopy colimit seems like a potential way to go, but the construction of homotopy colimits in $\mathbf{TGrp}$ seems very involved.

Does a construction exist which makes such a generalization possible?

Answer 1

I don't know if this will help, but in https://arxiv.org/pdf/math/0101220.pdf, we used graph products of spaces (which I think had been introduced by Danny Cohen but I'm not sure) This leads to a graph tensor product of crossed resolutions of the groups. This will give a resolution of the graph product. I'm not sure of the categorical interpretation of that graph product of spaces.

A point may be in your case to replace the groups, not by (discrete) topological groups, but either by groupoids (using each group as a groupoid with a single object) or by their nerves, which are classifying spaces of the groups. This corresponds to our situation but without the higher commutativity that comes from using crossed complexes. It also enables one to think of other more general settings. Remember that the category of groups can be thought of as a 2-category, by embedding it in the category of groupoids and that involves conjugation, so may be useful for your question. There are also the $cat^n$-groups of Loday modelling of homotopy $n$-types, where a sense to higher commutativity is certainly in evidence, as these are also crossed $n$-cubes of groups. (I can provide more details if you need.)

BTW The higher commutativity that you want and which you say is not there in $\mathbf{Grp}$ is there in simplicial group(oid)s, so expand your groups into hjigher dimensions so as to encode more of the higher commutativity, that comes from multiple commutators. (This relates to a lot of problems in rewriting theory, but that would lead me too far away from your question!)

Another setting that may give you some ideas is that of an idea explored by Abels and Holz (H. Abels and S. Holz. Higher generation by subgroups. J. Alg, 160:311– 341, 1993.) There the higher simplices are interpreted as coming from a covering of a group by subgroups (and one knows homological information on those subgroups). This is related to Haefliger's ideas but does not go in the same direction. In all this ideas of orbifolds and stacks are probably lurking but again I won't explore those here (as I am not an expert in those areas.)

I hope some of these ideas help.

Edit: I just had another idea on the connection between some of the above thoughts. That is to think of the commutator subgroups within $F$ as being crossed modules thus not forming the quotient $F/R$ as such but leaving $[G_u,G_v]$ as if in a higher dimension to $F$. The resulting object will take the presentation that you give as the important thing, rather than the group that it presents. One should be able to give a meaning for the triple commutator and hence get a 2-crossed module or something similar. (Again if you need references I can give them but here would not be a good place.)