  We give a simple combinatorial criterion, in terms of an action on a
hyperbolic simplicial complex, for a group to be hierarchically hyperbolic. We
apply this to show that quotients of mapping class groups by large powers of
Dehn twists are hierarchically hyperbolic (and even relatively hyperbolic in
the genus 2 case). Under residual finiteness assumptions, we construct many
non-elementary hyperbolic quotients of mapping class groups. Using these
quotients, we reduce questions of Reid and Bridson-Reid-Wilton about finite
quotients of mapping class groups to residual finiteness of specific hyperbolic
groups.
