Abért–Virág conjecture on Hausdorff dimension of linear groups in W_p

Let $R$ be a commutative pro-$p$ ring and let $G\le \mathrm{GL}_n(R)$ be a linear group over $R$. The group $W_p$ is the group of $p$-adic automorphisms, namely a Sylow pro-$p$ subgroup of the automorphism group of the $p$-adic tree, equipped with its level-stabilizer metric and Hausdorff dimension. Abért–Virág conjecture. For any embedding of $G$ into $W_p$, the image of $G$ has zero Hausdorff dimension. This conjecture is the main result of the paper in greater generality: the corresponding zero-dimensionality statement is proved for embeddings of linear groups over any integral domain into the automorphism group of a bounded rooted tree.