The higher-rank quasi-isometry classification conjecture for function-field arithmetic groups

Let $K$ be a global function field, let $S$ be a nonempty set of valuations of $K$, and let $\mathbf{G}$ be a connected, absolutely simple, $K$-isotropic algebraic $K$-group of adjoint type. For each $v\in S$, write $K_v$ for the completion of $K$ at $v$, set $G_S=\prod_{v\in S}\mathbf{G}(K_v)$, and let $\mathbf{G}(\mathcal{O}_S)$ be embedded diagonally as a discrete subgroup of $G_S$. The group $\mathcal{QI}(\mathbf{G}(\mathcal{O}_S))$ denotes its quasi-isometry group, and $\operatorname{Comm}_{\operatorname{Aut}(G_S)}(\mathbf{G}(\mathcal{O}_S))$ denotes the group of topological automorphisms of $G_S$ that commensurate $\mathbf{G}(\mathcal{O}_S)$. The higher-rank quasi-isometry classification conjecture. If

$$\sum_{v\in S}\operatorname{rank}_{K_v}(\mathbf{G})>1,$$

then there is an isomorphism

$$\mathcal{QI}(\mathbf{G}(\mathcal{O}_S))\cong \operatorname{Comm}_{\operatorname{Aut}(G_S)}(\mathbf{G}(\mathcal{O}_S)).$$

This is presented as the expected quasi-isometric classification for non-cocompact, irreducible arithmetic groups over global function fields. The preceding results establish the analogous number-field theorem, while the function-field statement was open in general at the time of the paper; the paper notes that mixed-rank cases were already proved.