##
**The asymptotic dimension of a curve graph is finite.**
*(English)*
Zbl 1135.57010

The curve graph of a compact orientable surface has the isotopy classes of essential non-parallel non-peripheral simply closed curves as vertices in which two vertices are adjacent if the corresponding curves can be realised simultaneously by pairwise disjoint curves. The curve graph equals the 1-skeleton of the curve complex defined in [W. J. Harvey, Riemann surfaces and related topics: Proc. 1978 Stony Brook Conf., Ann. Math. Stud. 97, 245–251 (1981; Zbl 0461.30036)]. In the first part of their paper the authors study the asymptotic dimension of a \(\delta\)-hyperbolic graph satisfying B. H. Bowditch’s boundedness property B [Invent. Math. 171, No. 2, 281–300 (2008; Zbl 1185.57011)]. They then combine their findings with the following two powerful results on the curve graph: By [H. A. Masur and Y. N. Minsky, Invent. Math. 138, No. 1, 103–149 (1999; Zbl 0941.32012)] the curve graph of a compact orientable surface of genus \(g\) with \(p\) punctures satisfying \(3g-3+p > 1\), is hyperbolic, while by Bowditch [loc. cit.] the set of tight geodesics on such a curve graph satisfies the above-mentioned property B.

Altogether the authors obtain that the asymptotic dimension of the curve graph of a compact orientable surface of genus \(g\) with \(p\) punctures satisfying \(3g-3+p > 1\), is finite. The authors then turn to G. Yu’s property A [Invent. Math. 139, No. 1, 201–240 (2000; Zbl 0956.19004)] and J.-L. Tu’s and A. N. Dranishnikov’s property \(A_1\) [Bull. Soc. Math. Fr. 129, No. 1, 115–139 (2001; Zbl 1036.58021) and Geom. Dedicata 119, 1–15 (2006; Zbl 1113.20036)]. These are equivalent for discrete metric spaces with bounded geometry [N. Higson and J. Roe, J. Reine Angew. Math. 519, 143–153 (2000; Zbl 0964.55015)]. It turns out that the curve graph of a compact orientable surface of genus \(g\) with \(p\) punctures satisfying \(3g-3+p > 1\), has property \(A_1\). Finally, the authors show that Artin groups of finite type \(A_n\) or \(C_n\) of rank at least three have asymptotic dimension at most \(n\). Moreover, Artin groups of affine type \(\widetilde A_{n-1}\) or \(\widetilde C_{n-1}\) of rank at least three have asymptotic dimension \(n-1\). Here, the asymptotic dimension of a finitely generated group is defined to be the asymptotic dimension of the Cayley graph with respect to some finite set of generators; since two such Cayley graphs for the same group are quasi-isometric, this indeed defines an invariant of the group that does not depend on a specific finite set of generators. This result about the asymptotic dimensions of Artin groups depends on the authors’ work on the asymptotic dimensions of modular groups/mapping class groups of compact orientable surfaces and the fact that the considered Artin groups are central extensions of certain finite index subgroups of the mapping class group of the \((n+2)\)-punctured sphere [R. Charney and J. Crisp, Math. Res. Lett. 12, No. 2-3, 321–333 (2005; Zbl 1077.20055)].

Altogether the authors obtain that the asymptotic dimension of the curve graph of a compact orientable surface of genus \(g\) with \(p\) punctures satisfying \(3g-3+p > 1\), is finite. The authors then turn to G. Yu’s property A [Invent. Math. 139, No. 1, 201–240 (2000; Zbl 0956.19004)] and J.-L. Tu’s and A. N. Dranishnikov’s property \(A_1\) [Bull. Soc. Math. Fr. 129, No. 1, 115–139 (2001; Zbl 1036.58021) and Geom. Dedicata 119, 1–15 (2006; Zbl 1113.20036)]. These are equivalent for discrete metric spaces with bounded geometry [N. Higson and J. Roe, J. Reine Angew. Math. 519, 143–153 (2000; Zbl 0964.55015)]. It turns out that the curve graph of a compact orientable surface of genus \(g\) with \(p\) punctures satisfying \(3g-3+p > 1\), has property \(A_1\). Finally, the authors show that Artin groups of finite type \(A_n\) or \(C_n\) of rank at least three have asymptotic dimension at most \(n\). Moreover, Artin groups of affine type \(\widetilde A_{n-1}\) or \(\widetilde C_{n-1}\) of rank at least three have asymptotic dimension \(n-1\). Here, the asymptotic dimension of a finitely generated group is defined to be the asymptotic dimension of the Cayley graph with respect to some finite set of generators; since two such Cayley graphs for the same group are quasi-isometric, this indeed defines an invariant of the group that does not depend on a specific finite set of generators. This result about the asymptotic dimensions of Artin groups depends on the authors’ work on the asymptotic dimensions of modular groups/mapping class groups of compact orientable surfaces and the fact that the considered Artin groups are central extensions of certain finite index subgroups of the mapping class group of the \((n+2)\)-punctured sphere [R. Charney and J. Crisp, Math. Res. Lett. 12, No. 2-3, 321–333 (2005; Zbl 1077.20055)].

Reviewer: Ralf Gramlich (Darmstadt)