Non-existence of Cannon-Thurston maps for hierarchically hyperbolic groups

We prove that Cannon–Thurston maps do not exist for hyperbolic normal subgroups of hierarchically hyperbolic groups, both for Morse and hierarchically hyperbolic boundaries. This result includes a large class of normal subgroups of the mapping class group of a surface of genus $g\ge 2$. As a corollary, we also prove a non-existence result for subgroups of most free by cyclic groups. Moreover, we prove that the visual boundary of a CAT(0) group with isolated flats is homeomorphic to the relatively hierarchically hyperbolic boundary of the space it acts on, thus recovering a previously known non-existence result for CAT(0) groups with isolated flats.