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In every dimension $n\ge 3$ we introduce a class of orthogonalgraph-manifolds and prove that the fundamental group of any orthogonalgraph-manifold quasi-isometrically embeds into a product of $n$ trees. As aconsequence, we obtain that asymptotic and linearly-controlled asymptoticdimensions of such group are equal to $n$.

Quasi-isometric embedding of the fundamental group of an orthogonal graph-manifold into a product of metric trees