TITLE: Counterexamples to analogue of Cannon conjecture in higher dimensions
QUESTION [8 upvotes]: It is known that a group $G$ acts geometrically on $\mathbb{H}^2$ if and only if $G$ is word-hyperbolic and its boundary $\partial G$ is homeomorphic to $S^1$.
The analogous statement for $\mathbb{H}^3$ and $S^2$ is open and is a conjecture of Cannon. 
I read somewhere that this fails in higher dimensions, but I can't find an explicit counterexample. Could somebody provide one (in dimension 4)?
Thanks.

REPLY [10 votes]: There are various compact manifolds of negative curvature which are not homnotopy-equivalent to closed hyperbolic manifolds: Locally symmetric ones (complex hyperbolic, etc) as well as Gromov-Thurston and Mostow-Siu examples. Their $\pi_1$'s are Gromov-hyperbolic, boundary is a topological sphere. The examples exist in all dimensions $\ge 4$ (Gromov-Thurston).