# The stable Cannon Conjecture for torsion-free Farrell-Jones groups

# The stable Cannon Conjecture for torsion-free Farrell-Jones groups

The torsion-free case of a well-known (open!!) conjecture of Cannon says: Let G be a torsion free hyperbolic group. Suppose that its boundary is homeomorphic to S^2. Then G is the fundamental group of a closed hyperbolic 3-manifold. By the class of Farrell-Jones groups (FJ), we will mean the class of groups which satisfy both the K-theoretic and the L-theoretic Farrell-Jones Conjectures with coefficients in additive categories with finite wreath products. This class contains all hyperbolic groups, CAT(0)-groups, fundamental groups of $3$-manifolds, lattices in almost connected Lie groups, and virtually solvable groups, and is stable under direct and free products. We will prove the following stable analog of this conjecture: Let G be a torsion-free hyperbolic group. Suppose that its boundary is homeomorphic to S^{n-1}. Let N be any aspherical closed manifold of dimension d such that d + n > 5 and \pi_1(N) belongs to FJ. Then there is an aspherical closed manifold M with fundamental group G x pi_1 N. This is joint work with Wolfgang Lück and Shmuel Weinberger.