We show that many graphs naturally associated to a connected, compact,
oriented 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.
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