Two weeks ago a paper was posted (by Benjamin Barrett) on the arXiv (arXiv:2004.11650) proving the following theorem about Gromov boundaries of word hyperbolic groups:
Let \(G\) be a one-ended hyperbolic group. Then \(\partial G\) is locally simply-connected if and only if for every point \(\xi\in \partial G\) the space \(\partial G \setminus \xi\) is simply-connected.
This is reminiscent of a theorem of Bestvina-Mess (Section 3 in DOI:10.2307/2939264) about one-ended hyperbolic groups: if for every point \(\xi\in \partial G\) the space \(\partial G \setminus \xi\) is connected, then \(\partial G\) is locally path-connected.
Note that Swarup (https://eudml.org/doc/231415) later proved that the assumption in the theorem of Bestvina-Mess is always satisfied for one-ended hyperbolic groups, i.e., their Gromov boundaries are always locally path-connected.
Since Barrett’s result is somehow a \(\pi_1\)-version of the result of Bestvina-Mess (which can be considered a \(\pi_0\)-result), I’m wondering whether (or under which additional conditions) one can prove corresponding \(\pi_k\)-results?
I’m calling these results “local-to-global principles” since they relate a local topological property (locally \(\pi_k\)-connected) to a global property (\(\pi_k\)-connectedness). Well … the global property is in these cases only required to hold after removal of single points, so maybe one should invent a slightly different name for this than “global”. I’m open for suggestions … (just put them as comments to this post).
Let me comment on the requirement that the hyperbolic group should be one-ended: it is known that this condition is equivalent to the fact that the boundary \(\partial G\) is connected. So it is not unreasonable to assume this condition when proving results like the above ones. Example of one-ended hyperbolic groups are the following:
- fundamental groups of negatively curved, closed Riemannian manifolds: in this case the boundary is a sphere
- almost surely a random group is a one-ended hyperbolic group: the boundaries of these groups are Menger curves (as always, enjoy such statements with caution since properties of random groups depend heavily on the random process used to produce the groups)
UPDATE (19.05.2020): I was talking to Benjamin Barrett a bit about his result. As I thought, the following statements are true:
- The ends of an arbitrary hyperbolic group are in one-to-one corresponds to the connected components of its boundary at infinity.
- His result holds for any hyperbolic group in the sense that it holds for each of its ends separately.
Assuming that the \(\pi_k\)-version of his result also holds true, a corollary of this would be that spheres are the only manifolds that can occur as boundaries at infinity of one-ended hyperbolic groups. As Benjamin told me, this conclusion is actually already known.