Arithmetic quotients of Bruhat-Tits buildings for groups over local fields of positive characteristic

Question

I have been led to believe that there is a result giving a description of the quotient of a Bruhat-Tits building $\Delta(G,k)$, for a semisimple algebraic group $G$ over a non-archimedean local field of positive characteristic $k$, by a non-uniform arithmetic lattice $\Gamma$. I believe it says that such a quotient is a union of a finite simplicial complex with finitely many cusps which are a product of $\mathbb{R}$ with a spherical building. I have had some difficulty in locating a reference for this and would be grateful if anyone could point me in the right direction.

Answer 1

I guess it is more convenient to collect some of the relevant literature references in an answer. The disclaimer is that I do not know of any results for general non-uniform arithmetic lattices $\Gamma\leq G(k)$; the results I know of always concern lattices of the form $G(\mathbb{F}_q[C])$ where $C$ is a smooth affine curve over the finite field $\mathbb{F}_q$ with a single point $P$ at infinity and the non-archimedean field $k$ is the completion of $\mathbb{F}_q(C)$ at the valuation corresponding to $P$.

The general observation as already mentioned in the comments is that the boundary of the Bruhat-Tits building $\Delta(G,k)$ is the spherical building for $G$ and $k$ (forgetting that $k$ also has a discrete valuation). The boundary of the quotient $\Delta(G,k)/\Gamma$ is the quotient of the boundary modulo the induced $\Gamma$-action. This quotient, in particular, is connected whenever the rank of $G$ is $\geq 2$.

Most of the computations of the quotient $\Delta(G,k)/\Gamma$ that have been done are for the case $G=GL_2$ or $G=SL_2$. As mentioned, the structure of cusps is determined in

• J.-P. Serre. Trees. Springer, 1980, in particular Sections II.1 and II.2.

• U. Stuhler. Homological properties of certain arithmetic groups in the function field case. Invent. Math. 57 (1980), 263-281. (This is the case of several valuations, with the action of $SL_2$ on a product of trees.)

In case you are interested in the structure of the finite subcomplex, this is computed for rational function fields in

• R. Köhl, B. Mühlherr and K. Struyve: Quotients of trees by arithmetic subgroups of $PGL_2$ over a rational function field. J. Group Theor. 18 (2015), 61-74.

For elliptic curves the quotient of the tree was computed in

• S. Takahashi. The fundamental domain of the tree of $GL(2)$ over the function field of an elliptic curve. Duke Math. J. (1993) 85-97.

For higher rank groups, there is a computation of the fundamental domain of the action of $GL_n(k[T])$ on the tree associated to the degree valuation on $k((T))$. For $G$ split, this is in

• C. Soulé. Chevalley groups over polynomial rings. In: Homological group theory. London Math. Soc. Lecture Notes 36, Cambridge University Press, 1979, 359-367,

and for $G$ non-split isotropic, this is done in

• B. Margaux. The structure of the group $G(k[t])$: variations on a theme of Soulé. Algebra Number Theory 3 (2009), 393-409.

These computations imply that the action of $G(k[t])$ on the boundary of the building $\Delta(G,k((t)))$ have fundamental domain a single simplex. In particular, the description as finitely many copies of buildings at infinity is not correct.

One of the major problems of going to higher rank is that for the "cusps" of $GL_n$, the structure of the whole quotient for smaller rank groups $GL_i$, $i\leq n$ is relevant. I have done a computation for $GL_3$ over an elliptic curve which may help illuminate this problem, see Section 3 of this paper on the arXiv. Apologies for the advertisement and the fact that no proofs are in that paper (this is sort of in the process of being generalized....) My point in bringing this up is that the structure of cusp/homotopy type at infinity for the quotient $\Delta(GL_3,k(E))/GL_3(k[E])$ is given by a graph describing decomposable rank three vector bundles on the elliptic curve. This graph contains subgraphs having the form of stars over $\mathbb{P}^1(k)$; these come from Takahashi's computation of the fundamental domain for $GL_2(k[E])$. In a way, to compute the structure of the cusp for $GL_3$, you have to know the whole quotient for $GL_2$. This makes higher-rank computations very infeasible. The computation for $GL_3(k[E])$ also shows that the cusps do not have the shape of buildings, they are more complicated things governed by vector bundle classification on the underlying curve.

Sorry for an oversized answer, I hope the references given are helpful.