I don't know much about Kac-Moody algebras/groups, but perhaps $PGL_2(Z)$ arises from a BN-pair through something like the following: there's a Kac-Moody Lie algebra (Frenkel-Feingold, Math. Ann. 1983) with Weyl group $PGL_2(Z)$ as desired. Then, I dunno... maybe Tits associates a Kac-Moody group (via a few constructions), and someone else (perhaps) describes "twin buildings" and BN-pairs. I'd start sniffing in those directions, if an expert doesn't weigh in. Good luck! Hoping for a better answer soon.

I seem to recall Noah Snyder thinking about such a question group-theoretically. Paging @Noah Snyder? (Is that how you do it?)

A comment for now. Here's a construction that doesn't work. Choose a quadratic form q over Q (the rationals) of signature (4,4) over the reals. But require q to be anisotropic at some nonempty set of finite places. Take $\Gamma$ an arithmetic subgroup of $Spin(q)$. Then $\Gamma \backslash Spin(4,4) / K$ might do it. But sadly quadratic forms in 8 variables over $Q_p$ are never anisotropic, so this fails. Similar failure for $G_2$ instead of $Spin(4,n)$. Sad!

See Lorscheid, Theorem 3.14 of his 2nd blueprints paper at arxiv.org/pdf/1201.1324.pdf. He describes $G(F_{1^2})$ in his framework, at least. For $n > 2$... why not email Lorscheid?

There should be a special tag for questions titled "A silly/naive/stupid question about [insert fancy technical topic]" which are then followed by a perfectly intelligent and well-formed question.

You might be interested in the article of Norbert Schappacher here: www-irma.u-strasbg.fr/~schappa/NSch/Publications_files/…. Starting at p.16, it describes some relevant details about the document history.

Some say the integral will end in pi, some say it's not so nice. From what I've tasted of periods, I hold with those who favor pi. But only if n is twice An integer or else I think, The integral won't be so nice, but still pretty great, and zetas sould suffice.

I think the condition is that $BwB \subset P w_0 P$ if and only if $w = w_1 w_0 w_2$ for some $w_1, w_2 \in W_I$. Here $W_I$ is the Weyl group of a standard Levi of $P$. See mathoverflow.net/questions/118974/… for a related question. This should give the answer to your explicit question, with a bit of work with minimal-length representatives, I think.

@TitoPiezasIII, thanks! I don't know much of anything about these continued fractions and recurrences (but now I know a bit more). I'm enjoying reading the Q&A from the sidelines.

@TitoPiezasIII: and have you cooked up numbers $a,b,c,d,\ldots$ satisfying $$\zeta(5) = \frac{a} {b - \frac{1^{10}} {c - \frac{2^{10}} {d - \frac{3^{10}} }}}?$$

I applaud this question with one hand.

A step towards an inductive proof: if $G$ is a unipotent $k$-group, and $c \in Z^2(G(k), Z / p Z)$ is a 2-cocycle (for trivial action), does there exist a polynomial 2-cocycle $\gamma \in Z^2(G, G_a)$ such that $\gamma$ gives $c$ upon taking $k$-points? I want to attack this one inductively with inflation-restriction with respect to $G$ and $[G,G]$. But no time for more than a sketchy comment.

I took the liberty of editing the opening for clarity and conciseness (since I know the person who posed the question).

I've made edits based on the two comments above.