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3. Rigidity questions

    1. 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.
        1. 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.

              [Kapovich-Kleiner] If $G$ is hyperbolic and $\partial G\simeq\mathscr{S}$ is a Sierpinski curve, then $G$ is virtually a Kleinian group.
                1. 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 $\partial G$ quasi-symmetry 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?
                        1. Remark. By Groves–Manning, the Cannon conjecture implies this.
                            • Quasi-isometric rigidity

                              Problem 3.4.

                              [Lafont] Let $M$ compact $K<0$ and $\partial M\neq\varnothing$ totally geodesic. Is every quasi-isometry of $\widetilde M$ bounded distance from an isometry? What about when $M$ is one of the Gromov–Thurston examples?
                                  The first question is true in the case of constant curvature, based on work of Kapovich, Kleiner, Leeb, and Schwartz (unpublished).
                                • Problem 3.5.

                                  [Lafont] Is a random hyperbolic group QI rigid? (i.e. any quasi-isometry is bounded distance from an isometry, i.e. QI$(G)=G$).
                                    1. 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.