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

1. End invariants and topological questions

    1. Semi-stability conjecture


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      Problem 1.1.

      Let $G$ be a finitely-presented one-ended group, and let $K_G$ be a presentation 2-complex. Does $\text{pro}-\pi_1(E(\widetilde K_g), r)$ depend on the ray $r$?
          This is known in many cases, e.g. hyperbolic groups. It’s open for CAT(0) groups. Some possible cases to consider: Does it hold if $G$ is hyperbolic relative to groups satisfying the conjecture? Does it hold if $G$ is CAT(0) and relatively hyperbolic?
        1. 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 pro-fundamental group is well-defined. 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?


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                  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?
                    1. Remark. May want to assume that the boundary of the coned off graph is connected.

                        • Existence and uniqueness of Z-set compactifications


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                          Problem 1.3.

                          Does every group of type F admit a $Z$-set compactification? $EZ$-set compactification?


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                              Problem 1.4.

                              If $K$ is finite $K(\pi,1)$, is $\widetilde K$ inward tame?
                                1. Remark. If $\widetilde K$ admits a $Z$-set compactification, then $\widetilde K$ is inward tame.


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                                      Problem 1.5.

                                      [Bestvina] Are any two $Z$-boundaries $Z_1,Z_2$ of a group cell-like equivalent?
                                          This question is open for CAT(0) groups, although it’s been solved for the Croke–Kleiner examples (Guilbault-Mooney).

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