3 problems

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 is a continuous extension of th…

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 $G$.

Let $H<G$ be a hyperbolic subgroup of a hyperbolic group $G$. Suppose the Cannon–Thurston map … exists, and suppose that $H$ is not quasiconvex in $G$. Non-quasiconvex subgroup con…