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\(\newcommand{\Cat}{{\rm Cat}} \) \(\newcommand{\A}{\mathcal A} \) \(\newcommand{\freestar}{ \framebox[7pt]{$\star$} }\)

4. Morse and other boundaries

    1. Morse boundaries


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      Problem 4.05.

      [Charney] Take notions from the boundary of hyperbolic groups and extend them to Morse boundaries.


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          Problem 4.1.

          [Charney] Does a ‘quasi-symmetry’ of the Morse boundary come from a quasi-isometry? (so long as Morse boundary is big enough)


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              Problem 4.15.

              What do connectivity properties of the Morse boundary tell about $G$. Specifically mapping class group, right-angled Artin groups, right-angled Coxeter groups.


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                  Problem 4.2.

                  [Charney] If $G$ acts geometrically on two CAT(0) cube complexes $X_0, X_1$, are the Morse boundaries of $X_0, X_1$ homeomorphic? (What about the Croke–Kleiner group?)


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                      Problem 4.25.

                      [Cordes] Let $G$ be a finitely presented group with Morse boundary with at least 3 points. Must $G$ be acylindrically hyperbolic?
                        1. Remark. If $G$ is only assumed to be finitely generated, there are counterexamples (Olshanskii–Osin–Sapir).


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                              Problem 4.3.

                              [Fernos] Does the Morse boundary have a quasi-isometry invariant compactification?


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                                  Problem 4.35.

                                  [Cordes] Let $H\subset G$ be almost malnormal and such that the limit set of $H$ in the Morse boundary of $G$ is the Morse boundary of $H$. Is $H$ hyperbolically embedded?

                                    • Hierarchically hyperbolic groups


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                                      Problem 4.4.

                                      [Groves] Is there a convergence group characterization of hierarchically hyperbolic groups? What about acylindrical hyperbolic groups?


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                                          Problem 4.45.

                                          [Cordes] Does the hierarchical hyperbolic boundary depend on the hierarchically hyperbolic structure of a (non-hyperbolic) hierarchically hyperbolic group?


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                                              Problem 4.5.

                                              [Hagen] Is the hierarchically hyperbolic boundary of Teichmüller space (with the Teichmüller metric) homeomorphic to PML?
                                                1. Remark. There is a surjective map from PML to the HH boundary. To answer the question, it would suffice to show that this map is cell-like.


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                                                      Problem 4.55.

                                                      [Durham] The boundary of a hierarchically hyperbolic group is metrizable. Is there a natural metric?

                                                        • More questions


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                                                          Problem 4.6.

                                                          [Sisto] Characterize acylindrical hyperbolic groups in terms of the Poisson boundary or the Martin boundary.


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                                                              Problem 4.65.

                                                              [Haissinsky] Does every group that admits some convergence action on a compact space admit a universal convergence action?
                                                                1. Remark. “Universal" means that every convergence action is a quotient of this one.


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                                                                      Problem 4.7.

                                                                      What can be said about existence of Cannon–Thurston maps for Morse boundaries? hierarchically hyperbolic boundaries? other boundaries?


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                                                                          Problem 4.75.

                                                                          [Xie] For various boundaries, what is the weakest condition on a map between spaces that ensures that there is a map between the boundaries?


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                                                                              Problem 4.8.

                                                                              [Manning] What are useful examples of boundaries of group pairs beyond the Bowditch boundary?

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