Discrete subgroups acting transitively on vertices of a Bruhat–Tits building

Abstract

We describe all the discrete subgroups of $Ad\left({\mathbb{G}}_0\right)\left(F\right)⋊Aut\left(F\right)$ that act transitively on the set of vertices of $\mathfrak{B}=\mathfrak{B}(F,{\mathbb{G}}_0)$, the Bruhat–Tits building of a pair $(F,{\mathbb{G}}_0)$ of a characteristic $0$ nonarchimedean local field, and a simply connected, absolutely almost simple $F$-group if $\mathfrak{B}$ is of dimension at least $4$. In fact, we classify all such maximal subgroups. We show that there are exactly eleven families of such subgroups and explicitly construct them. Moreover, we show that four of these families act simply transitively on the vertices. In particular, we show that there is no such action if either the dimension of the building is larger than $7$, if $F$ is not isomorphic to ${\mathbb{Q}}_p$ for some prime $p$, or if the building is associated to ${\mathbb{SL}}_{n,D_0}$, where $D_0$ is a noncommutative division algebra. Along the way we also give a new proof of the Siegel–Klingen theorem on the rationality of certain Dedekind zeta functions and $L$-functions.