Asymptotic dimensions of the arc graphs and disk graphs (Q6614583)

The asymptotic dimension \(\dim_{\mathsf{asym}} X\) of a metric space \(X\) was introduced by M. Gromov as a large-scale analogue of the covering dimension. The curve graph, \(\mathcal{S}\), for a surface \(S=S_{g,b}\) was introduced by \textit{W. J. Harvey}, in [Riemann surfaces and related topics: Proc. 1978 Stony Brook Conf., Ann. Math. Stud. 97, 245--251 (1981; Zbl 0461.30036)] as a sort of Bruhat-Tits building for Teichmüller space. \textit{M. Bestvina} and \textit{K. Bromberg}, in [Geom. Topol. 23, No. 5, 2227--2276 (2019; Zbl 1515.20209)], proved that \(\dim_{\mathsf{asym}} \mathcal{C}(S_{g,b}) \leq 4g-3+b\) when \(g >0\) and \(b >0\) or \(g=0\) and \(b>2\).\N\NIn the paper under review, the authors, combining \textit{M. Bestvina} et al. [Publ. Math., Inst. Hautes Étud. Sci. 122, 1--64 (2015; Zbl 1372.20029)] and \textit{H. Masur} and \textit{S. Schleimer} [J. Am. Math. Soc. 26, No. 1, 1--62 (2013; Zbl 1272.57015)], produce a quasi-isometric embedding of the arc graph \(\mathcal{A}(S, \Delta)\) into a finite product of quasitrees of curve complexes. From this they deduce that if \(S=S_{g,b}\) has non-empty boundary, \(\Delta \subset \partial S\) is a non-empty union of components and \(4g-3+b \geq 1\), then \(\dim_{\mathsf{asym}} \mathcal{A}(S, \Delta) \leq \frac{1}{2}(4g+b)(4g-3+b)\). They also obtain a similar result the disk graph \(\mathcal{D}(M,S)\) of a compression body.