Cannon–Thurston map nonexistence conjecture for hyperbolic normal subgroups of HHGs

Let $G$ be a hierarchically hyperbolic group (HHG), and let $H<G$ be an infinite hyperbolic normal subgroup of infinite index. A Cannon–Thurston map for $(H,G)$ is a continuous extension of the inclusion map from the boundary of $H$ to the boundary of $G$.

Cannon–Thurston map nonexistence conjecture. Such a map does not exist unless either $G$ is hyperbolic or $H$ is hierarchically quasiconvex in some HHG structure on $G$.

This conjecture proposes a general obstruction to extending boundary maps for infinite-index hyperbolic normal subgroups of nonhyperbolic HHGs, except when the subgroup is hierarchically quasiconvex in an appropriate HHG structure. The source gives motivating examples and known positive cases for hierarchically quasiconvex subgroups, but does not establish the conjecture in general.