5 problems

McMullen's Cannon–Thurston map conjecture. For any hyperbolic $3$-manifold $N$ with finitely generated fundamental group, there exists a continuous, $\pi_1(N)$-equivariant map

Let $\rho_i\to\rho$ be a converging sequence of non-elementary representations, with corresponding Cannon–Thurston maps. Suppose that $\Gamma$ is a surface group with cusps and tha…

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…