It's not so obvious to me what this means, and kind of shifts the question over to filtered-complex land. I.e., given a s.e.s. of filtered complexes, in what way does the l.e.s. respect the filtrations? Reference?

This follows from carefully tracking the universal coefficient theorem for group cohomology. See, e.g., groupprops.subwiki.org/wiki/…

I think the question should be left open... at least I'd like to think about it a bit more, because I don't know the answer. I hope my first comment didn't prevent more people from thinking about it!

@PaulBroussous -- I now have doubts about the idea of reducing to the affine Weyl group. I don't think that's going to work, since there could be elements of G that send (x,y) to (x',y') without preserving the chosen apartment. Something else will be needed, and I don't know whether the result is true.

I think @PaulBroussous just about has a counterexample. Let $x = x'$. Then the subset of the affine Weyl group fixing $x$ is just a finite Weyl group. And that definitely does not act transitively on the set of pts in the apartment at a fixed distance from $x$, when that fixed distance is chosen appropriately. E.g., $S_3$ does not act transitively on the set of 12 vertices at distance $\sqrt{7}$, in type $A_2$.

@JohnBaez -- I guess you've seen the formulae for 3-cocycles on dihedral groups in the math-phys literature? It doesn't directly help, since you assume $n \geq 4$... but it gives some idea of what might be possible.

Maybe one can compute $b_2$ for the following non-division algebra example. Can't remember where I saw it, but it's a standard kind of thing. Take $F = Q(\sqrt{2})$ and $L = Q(\sqrt[4]{2})$. Take the unitary group $\Gamma = SU_3(A)$ , for the trivial Hermitian form and nontrivial element of $Gal(L/F)$, and $A$ a suitable ring of integers. Then $\Gamma$ will embed into $SL_3(R)$ (at one place of $F$) as a cocompact lattice... not sure about torsion.

@DesideriusSeverus, I learned all this stuff from Gross's paper On the Satake Isomorphism. math.harvard.edu/~gross/preprints/sat.pdf

Looks like the Cauchy-Schwarz Master Class to me... is it?

Well.. I should have included ostraka too (pottery shards). See ams.org/publicoutreach/feature-column/fc-2011-11 for an exposition. There's a little 2-dim diagram, accompanying a discussion of the icosahedron, from (perhaps) the 2nd centrury BCE.

One of the earliest (if not the earliest?) fragments we have of Euclid's Elements is 2nd century CE (maybe 1st), around 400 years after Euclid. See math.ubc.ca/~cass/Euclid/papyrus for a description. There's a little diagram. It doesn't represent anything 3-dimensional.

Is there any expectation that every absolutely convergent series of this form is an (exponential) period? Or a trick to show this?

You can always look at Bump's book for GL_2 things like this. Or Deligne's "Formes Modulaires" article in the Antwerp volume II (Modular Forms of One Variable II, 1973). But for a more general picture, I learned these things from Gross's paper "On the Satake Isomorphism". See math.harvard.edu/~gross/preprints/sat.pdf, especially Sections 1,2,3,5,8.

How about a formal definition of the local or global Weil group? Have you got that already?