Embedding universal covers of graph manifolds in products of trees. (Q2845408)

The authors prove that the universal cover of any graph manifold quasi-isometrically embeds into the product of three metric trees, compare the result of \textit{S. Buyalo} and \textit{V. Schroeder} [Geom. Dedicata 113, 75-93 (2005; Zbl 1090.54029)] where it was shown that \(\mathbb H^3\) can be quasi-isometrically embedded into a product of three infinite valence trees. As a corollary, the authors prove that if \(\widetilde M\) is the universal cover of a closed graph manifold, then the Assouad-Nagata dimension of \(\widetilde M\) is three. \textit{A. Smirnov} [St. Petersbg. Math. J. 22, No. 2, 307-319 (2011); translation from Algebra Anal. 22, No. 2, 185-203 (2010; Zbl 1227.57025)] showed that the Assouad-Nagata dimension of such an object was finite and conjecture it to be 3. Thus, the authors' theorem implies Smirnov's conjecture. The asymptotic dimension provides the lower bound of \(3\) in the proof of the conjecture. A graph manifold is said to be non-geometric if its decomposition into Seifert fibered pieces is non-trivial. The paper concludes with an open question, which asks whether every non-geometric graph manifold has a fundamental group with asymptotic dimension \(3\). The authors' proof of the main theorem constructs the three trees and shows the map to their product is a quasi-isometric embedding.