Hierarchical hyperbolicity of graphs of multicurves

We show that many graphs naturally associated to a connected, compact, orientable surface are hierarchically hyperbolic spaces in the sense of Behrstock, Hagen and Sisto. They also automatically have the coarse median property defined by Bowditch. Consequences for such graphs include a distance formula analogous to Masur and Minsky's distance formula for the mapping class group, an upper bound on the maximal dimension of quasiflats, and the existence of a quadratic isoperimetric inequality. The hierarchically hyperbolic structure also gives rise to a simple criterion for when such graphs are Gromov hyperbolic.