6 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 $S$ be any surface of finite type, and let $g_1,\dots,g_l\in \mathcal{MCG}^\pm(S)$. For a subgroup $H$ of a group $G$, write $\langle\langle H\rangle\rangle$ for its normal clo…

Let $G$ act properly, cocompactly, and by isometries on a $\cat$ cube complex. Cubical HHG conjecture. The group $G$ is a hierarchically hyperbolic group (HHG). This would characte…

Let $M$ be the $3$-manifold under consideration, and let $\pi_1M$ denote its fundamental group. The group $\pi_1M$ acts freely and cocompactly on a CAT(0) cube complex if the actio…

Let $M$ be a non-geometric graph manifold, and let $G=\pi_1M$ be its fundamental group. A group is virtually compact special if it has a finite-index subgroup that is the fundament…