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[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...
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
![[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](https://pbs.twimg.com/media/Eh8kP_0X0AAVPWv.jpg "[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& #39;t exist for every group G acting on every simplicial complex X, but in our example there are four of them. Here& #39;s one:
![[4/20] Quite conveniently, there is a simplicial subcomplex which meets each orbit exactly once. These are called fundamental domains. These don& #39;t exist for every group G acting on every simplicial complex X, but in our example there are four of them. Here& #39;s one:](https://pbs.twimg.com/media/Eh8oUd0XcAA-5M8.png "[4/20] Quite conveniently, there is a simplicial subcomplex which meets each orbit exactly once. These are called fundamental domains. These don& #39;t exist for every group G acting on every simplicial complex X, but in our example there are four of them. Here& #39;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& #39;s and G& #39;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.
![[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.](https://pbs.twimg.com/media/Eh8qQhYXsAABzVj.jpg "[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.")
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& #39;] where G& #39; 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& #39;s the picture:
![[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& #39;s the picture:](https://pbs.twimg.com/media/Eh8r6izXgAAylTM.jpg "[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& #39;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& #39;s group] to [face simplex& #39;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!
![[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!](https://pbs.twimg.com/media/Eh8t8uGWAAM_2gd.png "[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.
![[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.](https://pbs.twimg.com/media/Eh8uoTtXcAE5EHO.png "[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& #39;t believe me? Boy, do I have a picture for you...
![[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& #39;t believe me? Boy, do I have a picture for you...](https://pbs.twimg.com/media/Eh8vao1X0AAV7ml.jpg "[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& #39;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& #39;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...
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& #39;s more...
There& #39;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.
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& #39;s like this; instead, the associativity on 2-morphisms is spelled out and called "the cocycle condition"
No geometric group theory text defines CoG& #39;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& #39;s is described in a paper https://arxiv.org/abs/1807.09396 ">https://arxiv.org/abs/1807.... with Lisa Carbone and Yusra Naqvi, soon to appear in SIAGA ( https://tinyurl.com/y5bglrk4 )">https://tinyurl.com/y5bglrk4&...
