TITLE: $G_\mathbb{Z}$-homotopy type of rational Tits building $\Delta_{G, \mathbb{Q}}$
QUESTION [9 upvotes]: Take $G$ to be a standard semisimple algebraic $\mathbb{Q}$-group, e.g. $Sp_{2g}$ or $SO(h)$ for $h$ a nondegenerate quadratic form over $\mathbb{Q}$. The arithmetic group $\Gamma=G_{\mathbb{Z}}$ has an obvious simplicial action on the associated rational Tits building $\Delta_{G, \mathbb{Q}}$. It is well known (a theorem of Borel) that $G_{\mathbb{Z}}$ acts with finitely many orbits on the vertices of $\Delta_{G, \mathbb{Q}}$, i.e. finiteness of the double cosets $G_\mathbb{Z} \backslash G_\mathbb{Q} /P_\mathbb{Q}$ for $P_\mathbb{Q}$ a rational parabolic. Hence the naive quotient $\Gamma \backslash \Delta$ is some finite complex.
Question: are the combinatorics of the naive quotient $\Gamma \backslash \Delta$ known?
My true motivation: I would like to know more about $\Gamma$-equivariant maps with target $\Delta$. Tits-Solomon theorem says $\Delta$ has the homotopy-type of a countable wedge of spheres $S^d$ (where $d=d(G)$ is known). 
Question: can somebody give me a reference or explanation describing the $\Gamma$-homotopy type of $\Delta$? Is the equivariant homotopy type of the associated adele groups known, i.e. $G_\mathbb{A}(\infty)$-homotopy type of $\Delta_{G, \mathbb{A}}$?

REPLY [8 votes]: $\newcommand\SL{\text{SL}}\newcommand\O{\mathcal{O}}\newcommand\cl{\text{cl}}$For $\text{SL}_n$ over a number field $K$, the rational Tits building is the classical Tits building $T_n(K)$, namely the poset of proper nonzero $K$-subspaces of $K^n$. In fact, starting here we can assume simply that $\O_K$ is a Dedekind domain with fraction field $K$.
For each $d=1,\ldots,n-1$, the orbits of $d$-dimensional subspaces under $\SL_n\O_K$ can be identified with the ideal class group $\widetilde{K}_0(\O_K)$: for each $d$-dimensional $K$-subspace $V$, the intersection $V\cap \O_K^n$ is a rank-$d$ projective $\O_K$-module, and $V$ and $W$ lie in the same $\SL_n\O_K$-orbit iff $V\cap \O_K^n$ and $W\cap \O_K^n$ are isomorphic.
More than this, using the fact that $\O_K$ is a Dedekind domain, you can show that one chain of subspaces is in the same orbit as another iff the ideal classes match up (i.e. iff the $i$th subspaces lie in the same orbit, for each $i$).
This shows that the quotient $T_n(K)/\SL_n\O_K$ is the $(n-1)$-fold join of the discrete set $\widetilde{K}_0(\O_K)$ with itself. This is an $\ell^{n-1}$-fold wedge of $(n-2)$-spheres, where $\ell=\lvert \widetilde{K}_0(\O_K)\rvert-1$, and indeed a spherical building of type $A_1\times \cdots \times A_1$: a chamber is determined by a sequence $(c_1,\ldots,c_{n-1})$ of ideal classes $c_i\in\widetilde{K}_0(\O_K)$. If $\O_K$ is a PID (i.e. $\lvert\widetilde{K}_0(\O_K)\rvert=1$), this tells you that the quotient $T_n(K)/\SL_n\O_k$ is contractible.
All of this so far is elementary, but the following theorem is not. [Edited to add reference to paper.] It is joint with Benson Farb and Andrew Putman and appears in our just-posted paper "Integrality in the Steinberg module and the top-dimensional cohomology of $\textrm{GL}_n\O_K$".
Theorem: The projection map $T_n(K)\twoheadrightarrow T_n(K)/\SL_n\O_K$ is surjective on homology.
An even stronger theorem is true: for every apartment in the quotient $T_n(K)/\SL_n\O_K$, there is an apartment in $T_n(K)$ whose fundamental class hits its homology class, on the nose. (This appears as Proposition 5.5 of our paper. Section 5 is self-contained, so if you are only interested in this result it suffices to read Sections 5.1 and 5.2 on pages 21-28. Still, I encourage you to check out the introduction while you're there!)
To get a feeling for the strength of this result, it's a fun game to pick your favorite Dedekind domain with nontrivial class group, pick $2n-2$ ideal classes to determine an apartment in the quotient, and then try to find an apartment in $T_n(K)$ that hits it. (Warning: you probably won't be able to do this, except for $n=2$ or $n=3$! --- at least we weren't able to for a while. But the theorem we prove says that you always can.)