On the stable Cannon Conjecture

The Cannon Conjecture for a torsionfree hyperbolic group $G$ with boundary homeomorphic to $S^2$ says that $G$ is the fundamental group of an aspherical closed $3$-manifold $M$.  It is known that then $M$ is a hyperbolic $3$-manifold.  We prove the stable version that for any closed manifold $N$ of dimension greater or equal to $2$  there exists a closed manifold $M$ together with a simple homotopy equivalence $M \to N \times BG$. If $N$ is aspherical and $\pi_1(N)$ satisfies the Farrell-Jones Conjecture, then $M$ is unique up to homeomorphism.
This is joint work with Ferry and Weinberger.