4. Morse and other boundaries

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 ‘quasisymmetry’ of the Morse boundary come from a quasiisometry? (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, rightangled Artin groups, rightangled 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?
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 quasiisometry 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 (nonhyperbolic) hierarchically hyperbolic group? 
Problem 4.5.
[Hagen] Is the hierarchically hyperbolic boundary of Teichmüller space (with the Teichmüller metric) homeomorphic to PML?
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 celllike.


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?
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.