Appears in collection : 2022 - T2 - WS1 - Mapping class groups and Out(Fn)

We show that the mapping class group of a closed surface of higher genus admits a proper action on a nonpositively curved cube complex (which is however not simply connected). We use this information together with a construction of Ji and McPhearson to study its asymptotic geometry. As an application, we show that the covering dimension of the Gromov boundary of the curve graph is at most $4g − 6$ (which slightly improves a result of Gabai and is conjectured to be sharp).