Quasiisometric embedding of the fundamental group of an orthogonal graph-manifold into a product of metric trees (Q2849204)

\textit{D. Hume} and \textit{A. Sisto} [Proc. Am. Math. Soc. 141, No. 10, 3337--3340 (2013; Zbl 1326.20044)] have shown that the universal cover of a graph manifold quasi-isometrically embeds into a product of three metric trees. In the current work, the author introduces a class of orthogonally glued higher-dimensional graph-manifolds in every dimension \(n\geq 3\) and then shows that fundamental groups of such ``orthogonal graph-manifolds'' embed quasi-isometrically into a product of \(n\) metric trees. It follows that the asymptotic dimension of such a fundamental group is \(n\) and the linearly controlled asymptotic dimension of such a group is \(n\). The proof is constructive.