It is known that a group $G$ acts geometrically on $\mathbb{H}^2$ if and only if its boundary $\partial G$ is homeomorphic to $S^1$ and $G$ is word-hyperbolic.

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.

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.