3. Rigidity questions

Cannon and related conjectures
Conjecture 3.1.
[Cannon] If $G$ is hyperbolic and $\partial G\simeq S^2$, then $G$ is virtually a Kleinian group.
Remark. There are many equivalent formulations. For example, is such a $G$ cubulated? This problem can also be stated in terms of PI spaces.


Conjecture 3.2.
[KapovichKleiner] If $G$ is hyperbolic and $\partial G\simeq\mathscr{S}$ is a Sierpinski curve, then $G$ is virtually a Kleinian group.
Remark. This conjecture is implied by Cannon’s conjecture. (Is the converse true?) Like Cannon’s conjecture, it has many formulations: Is $G$ cubulated? Is $\partial G$ quasisymmetry equivalent to a subset of the round sphere?


Problem 3.3.
[Walsh] Let $(G,P)$ relative hyperbolic with Bowditch boundary $\mathscr S$ or $S^2$. Is $G$ virtually a Kleinian group?
Remark. By Groves–Manning, the Cannon conjecture implies this.


Quasiisometric rigidity
Problem 3.4.
[Lafont] Let $M$ compact $K<0$ and $\partial M\neq\varnothing$ totally geodesic. Is every quasiisometry of $\widetilde M$ bounded distance from an isometry? What about when $M$ is one of the Gromov–Thurston examples? 
Problem 3.5.
[Lafont] Is a random hyperbolic group QI rigid? (i.e. any quasiisometry is bounded distance from an isometry, i.e. QI$(G)=G$).
Remark. For a random hyperbolic group $\partial G$ is a Menger sponge.

Cite this as: AimPL: Boundaries of groups, available at http://aimpl.org/groupbdy.