Symmetries of higher dimensional black holes

We prove that if a stationary, real analytic, asymptotically flat vacuum black hole spacetime of dimension n >= 4 contains a non-degenerate horizon with compact cross-sections that are transverse to the stationarity generating Killing vector field then, for each connected component of the black hole's horizon, there is a Killing field which is tangent to the generators of the horizon. For the case of rotating black holes, the stationarity generating Killing field is not tangent to the horizon generators and therefore the isometry group of the spacetime is at least two dimensional. Our proof relies on significant extensions of our earlier work on the symmetries of spacetimes containing a compact Cauchy horizon, allowing now for non-closed generators of the horizon.