1. End invariants and topological questions

Semistability conjecture
Problem 1.1.
Let $G$ be a finitelypresented oneended group, and let $K_G$ be a presentation 2complex. Does $\text{pro}\pi_1(E(\widetilde K_g), r)$ depend on the ray $r$?
Remark. Equivalent formulation: Are any two proper rays in $\widetilde K_G$ properly homotopic?

Remark. One motivation for this question is to show that the profundamental group is welldefined. An alternate approach to this problem would be to show that, given a group $G$, there is a way to choose a ray in $\widetilde K_G$ canonically. Is this possible?


Problem 1.2.
[Hagen] Suppose $(G,P)$ is relatively hyperbolic and consider rays in $G$ for which there is a $K$ such that no coset of a parabolic is visited for longer than $K$. Are any two such rays properly homotopic?
Remark. May want to assume that the boundary of the coned off graph is connected.


Existence and uniqueness of Zset compactifications
Problem 1.3.
Does every group of type F admit a $Z$set compactification? $EZ$set compactification? 
Problem 1.4.
If $K$ is finite $K(\pi,1)$, is $\widetilde K$ inward tame?
Remark. If $\widetilde K$ admits a $Z$set compactification, then $\widetilde K$ is inward tame.


Problem 1.5.
[Bestvina] Are any two $Z$boundaries $Z_1,Z_2$ of a group celllike equivalent?
Cite this as: AimPL: Boundaries of groups, available at http://aimpl.org/groupbdy.