What surfaces(manifolds) can be the boundary of hyperbolic groups?

Question

Question: What surfaces can be the Gromov boundary of a hyperbolic group? (You could also ask the same question except for higher dimensional manifolds.)

I know that spheres appear as the boundary of some hyperbolic group (fundamental groups of closed hyperbolic three manifolds), but that is about it. Intuitively I feel like surfaces of genus $g>0$ should not appear as boundaries (it "feels" like they don't have enough nice symmetries/self similarity, although their homeomorphisms groups are huge...).

Bowditch gave topological(dynamical) characterization of hyperbolic groups in terms of uniform convergence actions, which I am not familiar with, but looks like it could help show that some boundaries do or do not exist. The naive idea I was thinking, if surfaces with genus could be boundaries, would be choose some homeomorphisms of the surface with "north" and "south" poles and hopefully the group generated by those would be nice enough that the group would have that surface as its boundary. Maybe a Poincare-Hopf type theorem could rule this out.

At the end of the summary section of Sphere boundaries of hyperbolic groups by Benjamin Beeker and Nir Lazarovich there is a descriptions of a hyperbolic group with an interesting boundary, and when it was explained to me months ago I seem to remember it was surface-ish but I don't remember the details(I don't think it was a surface though).

Answer 1

In this survey by I. Kapovich and Benakli (arxiv link, Theorem 4.4), you can find the result that a the only topological manifolds occurring as boundaries of hyperbolic groups, up to homeomorphism, are spheres. This holds in any dimension $n\ge 0$.

Answer 2

As YCor points out the answer is in these notes, but I do want to add some detail.

First, the "self similarity" of the boundary statement I was thinking about basically comes from the north-south dynamic. So, if you act on the boundary by a loxodromic there is an attracting point and repelling, and the fact that you can get the boundary, without repelling point, into any neighborhood of the attracting fixed point is the self-similarity I was for some reason failing to pin point.

The proof in the above survey that the only manifold boundaries are spheres basically follows the above idea.