
## 5. Miscellaneous

1. #### Problem 5.05.

[Lafont] Let $G$ hyperbolic, acting geometrically on $X_0,X_1$. Is there a quasi-isometry $X_0\rightarrow X_1$ with smallest multiplicative constant?
The answer is Yes for surfaces.
1. Remark. The quasi-isometry need not be $G$-equivariant in the statement. As an interesting case, are there examples with quasi-isometry constant 1 but where there doesn’t exist a map with quasi-isometry constant 1?
• ### Group analogues to manifold constructions

#### Problem 5.1.

Is there a group theoretic notion of drilling? (If $\partial G$ is connected, then the new Bowditch boundary should have no cut points.) If $G$ is Kleinian, what happens to $\partial G$ after drilling?
• #### Problem 5.15.

[Lafont] What is the group theoretic analogue of taking a ramified cover along a codimension-2 subgroup?
• ### CAT(0) groups

#### Problem 5.2.

[Bell] When does a Coxeter group have a unique CAT(0) boundary up to homeomorphism?
• #### Problem 5.25.

[Ruane] Is there a CAT(0) group with 2 visual boundaries so that one is locally connected and the other is not? Is there an algebraic condition on a CAT(0) group that characterizes when the visual boundary is locally connected?
• ### Characterization questions

#### Problem 5.3.

What spaces can be the Gromov boundary of a hyperbolic group? How about the Bowditch or Morse boundary?
• #### Problem 5.35.

Which metric spaces are quasi-symmetric to a Gromov boundary and a Bowditch boundary?
• #### Problem 5.4.

[Manning] If $(G,P)$ is relatively hyperbolic and $G\rightarrow\overline{G}$ is a long Dehn filling with $\bar G$ hyperbolic and the cusped space of $\overline{(G,P)}$ quasi-isometric to the Cayley graph of $\overline{G}$, does $(G,P)$ have to be either Kleinian or Fuchsian?
• #### Problem 5.45.

[Walsh] Let $K$ be a Kleinian group with Sierpinski boundary. Is there a quasi-symmetry invariant that detects whether or not $K$ has a rank-2 cusp?
• #### Problem 5.5.

Is there an analogue of the Geoghegan–Ontaneda result about Cech cohomology of the boundaries of CAT(0) groups for other boundaries? (c.f. Bestvina–Mess)
• ### Manifold questions

#### Problem 5.55.

[Lafont] Let $M$ be a locally CAT($-1$) closed manifold with $\partial\widetilde M=S^{n-1}$. Does $M$ admit a Riemannian metric with $K<0$?
1. Remark. Davis–Januszkiewicz give examples of CAT(0) $M^n$ that don’t admit a Riemannian metric with $K\le0$ for $n\ge5$ (but these examples don’t have sphere boundary – in fact this is how you know that they don’t have Riemannian metric of negative curvature). Davis–Januszkiewicz–Lafont construct examples for $n=4$ (do they have boundary a sphere?).
• #### Problem 5.6.

[Lafont] Does there exist a hyperbolic group $G$ whose boundary is not a sphere, but is
1. (i) locally contractible
2. (ii) a homology $n$-manifold
3. (iii) satisfies the disjoint disk property
1. Remark. Non-manifold finite dimensional examples of spaces satisfying (i) – (iii) have been constructed by Bryant-Ferry-Mio-Weinberger, but it is unknown whether these can occur as the boundary of a group. If such examples exist, then the boundary $B$ would be an “exotic" homology manifold, i.e. one whose Quinn index is distinct from 1. One could then exploit the dynamics of the group action to show some soft form of homogeneity for $B$. (Whether such spaces are homogenous is unknown.)
• Remark. If one drops condition (iii), then (non-sphere) examples satisfying (i) and (ii) can be found in Davis–Januszkiewicz’s JDG paper. In their examples, the boundary $B$ contains points $p$ with contractable neighborhoods $U$ such that $U\setminus\{p\}$ is not simply connected. Taking two loops in $U\setminus\{p\}$ and finding bounding 2-disks in $U$ gives a pair of 2-dimensional disks intersecting at $p$, which cannot be perturbed to be disjoint.

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