[1/20] Complexes of groups are concrete objects living at the intersection of many abstract fields of mathematics: algebraic topology, algebraic+differential geometry, and even higher categories.

But their most natural home is in geometric group theory...

[2/20] To build one, start with a simplicial complex X (eg the bow-tie) that exhibits some global symmetries. If the vertex marked △ is at the origin, then reflections about the horizontal and vertical coordinate axes give an action of the klein four group G=(Z/2 x Z/2) on X

[3/20] Call the up-down reflection sigma and the left-right reflection tau. The vertex labels correspond to the orbits: everything fixes vertex △; the two ⃞ vertices and the two ○ vertices are fixed by sigma and interchanged by tau, etc.

[4/20] Quite conveniently, there is a simplicial subcomplex which meets each orbit exactly once. These are called fundamental domains. These don't exist for every group G acting on every simplicial complex X, but in our example there are four of them. Here's one:

[5/20] If an adversary just showed you such a fundamental domain and asked you to recover X and G-action, you would have no hope: there are too many other X's and G's with the same fundamental domain. We need extra data for the recovery challenge, and the race is on.

[6/20] Maybe if you knew how many △ vertices and how many ⃞ vertices, how many ⃞ -----○ edges etc., you could at least get the graph comprising the 1-skeleton of X, right?

Wrong. Things can go horribly wrong if the extra data you have is purely numerical.

[7/20] But knowing the simplex counts is a good start! The number of ○ vertices in X is in fact the subgroup index [G:G'] where G' is the *stabilizer*, i.e., the subgroup of G that fixes this vertex ○. The key idea is to endow each simplex with knowledge of its stabilizer.

[8/20] Under (mild) restrictions on the G-action, we can ensure that the stabilizer of a big simplex (eg, an edge) includes into the stabilizer of its faces (namely, its boundary vertices). Here's the picture:

[9/20] Sheafy types will call this a (pre)cosheaf, but never mind them. We just have a stabilizing subgroup of G on each simplex of the fundamental domain, and an injective group homomorphism from [big simplex's group] to [face simplex's group] for each face.

[10/20] The thing that makes life complicated is that not every group action admits a nice fundamental domain! If you imagine the group G = Z/3 rotating this (subdivided) 2-simplex X, there is no subcomplex that meets each orbit exactly once!

[11/20] Of course, morally we might know what the correct "quotient" simplicial complex Y = X/G should be in this case, even if it is not a subcomplex... here it is.

[12/20] There is an "orbit map" from X down to Y = X/G so that the inverse image of each Y-simplex is its G-orbit in X. Don't believe me? Boy, do I have a picture for you...

[13/20] Okay, so the absence of a fundamental domain is surmountable. A more technical annoyance occurs when the acting group G is not abelian. Consider three simplices x > y > z in the quotient Y, and let G_x, G_y, G_z be their stabilizers.

[14/20] Here's the trouble: there are three injective group homomorphisms: one from G_x to G_y, one G_y to G_z and a third G_x to G_z. The composite of the first two need not equal the third when G is nonabelian. This makes the cosheaf crowd cry.

[15/20] Fortunately, the direct map d from G_x to G_z *is* related to the composite c from G_x to G_y to G_z. There is a group element h in G_z so that for each p in G_x, we have d(p) = h c(x) h-inverse.

This is a non-issue when G is abelian because conjugation is trivial...

[16/20] We now have everything we need. A complex of groups for the G-action on X with quotient Y is an assignment of (1) a G-subgroup to each simplex x of Y, (2) an injective homomorphism to each face relation x > y, and (3) a conjugating h for each x > y > z.

There's more...

[17/20] There are technical conditions that must be satisfied by the injective maps and conjugators. How best to state them?

Treat Y = X/G as a poset of simplices ordered by the co-face relation, and let 2Grp be the (2,1) category of groups, homomorphisms, and conjugation.

[18/20] Definition: A complex of groups for the G-action on X is a pseudofunctor from the poset X/G to the (2,1) category 2Grp.

No geometric group theory text defines CoG's like this; instead, the associativity on 2-morphisms is spelled out and called "the cocycle condition"

[19/20] But now we (hopefully) know what it is, and why it is necessary! The process of recovering X and G from the associated complex of groups is a fundamental part of geometric group theory, often called *the basic construction*.

[20/20] This higher-categorical perspective on CoG's is described in a paper https://arxiv.org/abs/1807.09396  with Lisa Carbone and Yusra Naqvi, soon to appear in SIAGA ( https://tinyurl.com/y5bglrk4 )