Bux–Wortman conjecture on distortion dimension of S-arithmetic subgroups

Let $\mathbf G$ be a reductive group defined over a global field $\mathbb K$, let $S$ be a finite set of valuations, and let

$$\mathbf G_S=\prod_{v\in S}\mathbf G(\mathbb K_v)$$

be endowed with a left-invariant metric. Let $X=S\times B$ be the product of a symmetric space and Euclidean buildings, quasi-isometric to $\mathbf G_S$, and let $r=\sum_{s\in S}\operatorname{rank}_{\mathbb K_v}\mathbf G$ be its Euclidean rank. Bux–Wortman conjecture. The $S$-arithmetic subgroup $\mathbf G(\mathcal O_S)$ has distortion dimension $r-1$ as a subgroup of the reductive group $\mathbf G_S$ with respect to these metrics.

The conjecture is formulated using coarse manifolds and measures the dimensions in which the lattice is homotopically undistorted in the ambient group. It provides a geometric reduction-theoretic description of the metric distortion of $S$-arithmetic subgroups.