Hierarchically hyperbolic groups (HHGs) expand on work of Masur and Minsky on mapping class groups of compact surfaces to provide a geometric model for certain acylindrically hyperbolic groups Gromov hyperbolic groups, and subgroups of right-angled Artin groups.
This geometric frame work has been especially useful for verifying finer properties of acylindrically hyperbolic groups such as the Tits alternative and uniform exponential growth.
I will describe why quotients obtained from subgroups normally generated by finitely many independently distributed random walks preserve hierarchical hyperbolicity.
These ideas also let us understand the geometry of random quotients of hyperbolic, and relatively hyperbolic groups. This is joint work in progress with Abbott, Berlyne, Mangioni, and Rasmussen.