For many examples of word-hyperbolic groups which I have seen in the context of low-dimensional topology, the ideal boundary is either homeomorphic to a n-sphere for some n or a Cantor set. So, I was wondering if this is generally true or there are some examples of hyperbolic groups whose boundaries are neither a sphere of some dimension nor a Cantor set. If there exist such examples, is there any known topological classification of boundaries of hyperbolic groups?
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$\begingroup$ The class of discrete hyperbolic groups is stable under free products, this provides low-cost examples with a different boundary. For more refined ones, see this survey by I. Kapovich and Benakli: math.uiuc.edu/~kapovich/PAPERS/bry1.pdf $\endgroup$– YCorCommented Jul 4, 2013 at 21:33
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There are plenty of other possibilities. Here are a few examples:
The boundary of the fundamental group of an acylindrical hyperbolic 3-manifold with totally geodesic boundary is homeomorphic to a Sierpinski carpet. (This appears in the Kapovich and Kleiner paper mentioned below, but must be standard---the point is the topological fact that any planar continuum with no local cut points is homeomorphic to a Sierpinski carpet. Indeed, they prove a converse result, modulo the Cannon Conjecture.)
The boundary of a random group is homeomorphic to a Menger sponge (Dahmani--Guirardel--Przytycki).
Bowditch proved that cyclic splittings of $\Gamma$ correspond to local cut points on the boundary. In particular, the boundary of a graph of free groups with cyclic edge groups can be decomposed along cut pairs into Cantor sets.
The classification of 1-dimensional Polish spaces implies that this is a complete list of boundaries of 2-dimensional hyperbolic groups, in the sense that if the boundary is connected with no local cut point (ie the group has no splitting over a virtually cyclic subgroup) then it must be a Sierpinski carpet or Menger sponge (Kapovich--Kleiner).
I imagine that a classification in higher dimensions is completely out of reach, though I'm no expert. (Misha Kapovich is active on MO and can provide an authoritative answer.)
I'll add full references tomorrow.
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$\begingroup$ Thanks a lot, Henry. This is very interesting. I wonder if there is any partial result in classification of boundaries of 3-dimensional hyperbolic groups. $\endgroup$ Commented Jul 5, 2013 at 2:36
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$\begingroup$ @harry: what do you mean by "3-dimensional hyperbolic group"? there are many inequivalent notions of dimension. $\endgroup$– YCorCommented Jul 5, 2013 at 6:42
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$\begingroup$ @Yves, in this case, the natural definition of dimension for a hyperbolic group $\Gamma$ is the topological dimension of the boundary +1. As you say, there are other relevant definitions. It's unknown which hyperbolic groups have boundary of conformal dimension 1, for instance. $\endgroup$– HJRWCommented Jul 5, 2013 at 9:02
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$\begingroup$ @YvesCornulier: Yes, I just meant the topological dimension of the boundary + 1. $\endgroup$ Commented Jul 6, 2013 at 2:37
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$\begingroup$ @HJRW I think you mean Sierpinski carpet as opposed to Sierpinski gasket. (The latter term is a synonym for the Sierpinski triangle.) $\endgroup$– Jim BelkCommented Jul 22, 2018 at 13:22