Hierarchical hyperbolicity of graphs of multicurves

TL;DR

This paper demonstrates that various graphs associated with surfaces are hierarchically hyperbolic spaces, enabling new geometric insights such as distance formulas, bounds on quasiflats, and criteria for hyperbolicity.

Contribution

It establishes that many surface-related graphs are hierarchically hyperbolic, providing a unified framework with new geometric and combinatorial properties.

Findings

Graphs have the coarse median property.

Distance formulas analogous to Masur-Minsky's.

Quadratic isoperimetric inequality.

Abstract

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.