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.



Counterexamples to analogue of Cannon conjecture in higher dimensions
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). 


