An acylindricity theorem for the mapping class group (Q634457)

The paper under review deals with the action of the mapping class group on the curve complex of a surface. It gives a computable upper bound for the number of mapping classes that move a long geodesic path a small distance. This bound is given in terms of the intersection of the curves at the end of the path and the topological type of the surface, as we well as in terms of a bound on the small distance that the path is displaced. Similar bounds that do not depend on intersection numbers have been obtained by \textit{M. Bestvina} and \textit{K. Fujiwara} in [``Bounded cohomology of subgroups of mapping class groups'', Geom. Topol. 6, 69--89 (2002; Zbl 1021.57001)] and \textit{B. H. Bowditch} in [``Tight geodesics in the curve complex'', Invent. Math. 171, No. 2, 281--300 (2008; Zbl 1185.57011)], but they are not computable. Let \(\Sigma\) be an orientable, connected, and compact surface of negative Euler characteristic. Assume moreover that it is not exceptional, namely that its Teichmüller space has dimension at least 4. The curve graph of \(\Sigma\) is defined to be the 1-skeleton of the curve complex of \(\Sigma\), and it is quasi-isometric to the whole complex for \(\Sigma\) non exceptional. The main theorem of the paper asserts that there is a computable function \(F\) depending on two natural parameters and a constant \(C\) such that for each natural \(r\) the following holds true: if two curves \(\alpha\) and \(\beta \) are at distance \( d(\alpha,\beta)\geq (c+2) r+9\) in the curve graph of \(\Sigma\), then the set of mapping classes that displace \(\alpha\) and \(\beta\) at most \(r\) has cardinality at most \(F(\iota(\alpha,\beta),r)\), where \(\iota(\alpha,\beta)\) denotes the intersection number between \(\alpha\) and \(\beta\).