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We give a simple combinatorial criterion, in terms of an action on ahyperbolic simplicial complex, for a group to be hierarchically hyperbolic. Weapply this to show that quotients of mapping class groups by large powers ofDehn twists are hierarchically hyperbolic (and even relatively hyperbolic inthe genus 2 case). Under residual finiteness assumptions, we construct manynon-elementary hyperbolic quotients of mapping class groups. Using thesequotients, we reduce questions of Reid and Bridson-Reid-Wilton about finitequotients of mapping class groups to residual finiteness of specific hyperbolicgroups.

A combinatorial take on hierarchical hyperbolicity and applications to quotients of mapping class groups