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? 



Topology of boundaries of hyperbolic groups
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](http://www.ams.org/mathscinet/search/publdoc.html?amp=&loc=refcit&refcit=1834498&vfpref=html&r=6&mx-pid=2755002)).
* Bowditch [proved](http://www.ams.org/mathscinet/search/publdoc.html?r=1&pg1=CNO&s1=1638764&loc=fromrevtext) 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](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=AUCN&pg6=PC&pg7=ALLF&pg8=ET&review_format=html&s4=kapovich&s5=kleiner&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=4&mx-pid=1834498)).


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.


