
## 4. Morse and other boundaries

1. ### Morse boundaries

#### Problem 4.05.

[Charney] Take notions from the boundary of hyperbolic groups and extend them to Morse boundaries.
• #### 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)
• #### 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.
• #### 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?)
• #### 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).
• #### Problem 4.3.

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

#### Problem 4.4.

[Groves] Is there a convergence group characterization of hierarchically hyperbolic groups? What about acylindrical hyperbolic groups?
• #### Problem 4.45.

[Cordes] Does the hierarchical hyperbolic boundary depend on the hierarchically hyperbolic structure of a (non-hyperbolic) hierarchically hyperbolic group?
• #### 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.
• #### Problem 4.55.

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

#### Problem 4.6.

[Sisto] Characterize acylindrical hyperbolic groups in terms of the Poisson boundary or the Martin boundary.
• #### 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.
• #### Problem 4.7.

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