Asymptotic dimensions of the arc graphs and disk graphs

We give quadratic upper bounds for the asymptotic dimensions of the arc graphs and disk graphs.20F65, 20F69, 57K20; 57M60 IntroductionThe asymptotic dimension, denoted by dim asym X, of a metric space X was introduced by Gromov 1993, page 29 as a large-scale analogue of the covering dimension.The curve graph, Ꮿ.S /, for a surface S D S g;b was introduced by Harvey 1981, page 246 as a sort of Bruhat-Tits building for Teichmüller space.It has since been generalised in many ways.Bell and Fujiwara 2008, Corollary 1 first proved that the asymptotic dimension of Ꮿ.S / is finite.More recently, Bestvina and Bromberg 2019, Corollary 1.1 proved that dim asym Ꮿ.S g / Ä 4g 4 (when g > 1) and that dim asym Ꮿ.S g;b / Ä 4g 3 C b D 0 .S g;b / when g > 0 and b > 0 (or g D 0 and b > 2).Here we combine the machineries of Bestvina et al. 2015;Masur and Schleimer 2013 to produce a quasi-isometric embedding of the arc graph Ꮽ.S; / into a finite product of quasitrees of curve complexes.From this we deduce the following: Corollary 3.11 Suppose that S D S g;b has nonempty boundary.Suppose that @S is a nonempty union of components.Finally, suppose that 0 .S/ 1.ThenSisto (private communication, 2022) suggests that the machineries of Behrstock et al. 2017, Theorem 5.2; Vokes 2022, Theorem A.2 can be combined to obtain a similar result.We also obtain the following result for the disk graph Ᏸ.M; S/ of a compression body: Corollary 4.18 Suppose M is a nontrivial spotless compression body with upper boundary S D S g;b .Suppose that 0 .S/ 1.Then dim asym Ᏸ.M; S/ Ä 1 2 .4gC b/.4g C b 3/ 2: Hamenstädt 2019, Theorem 3.6 has obtained a similar result when M is a handlebody; see Remark 4.20.