  We consider extremal black hole solutions to the vacuum Einstein equations in
dimensions greater than five. We prove that the near-horizon geometry of any
such black hole must possess an SO(2,1) symmetry in a special case where one
has an enhanced rotational symmetry group. We construct examples of vacuum
near-horizon geometries using the extremal Myers-Perry black holes and boosted
Myers-Perry strings. The latter lead to near-horizon geometries of black ring
topology, which in odd spacetime dimensions have the correct number rotational
symmetries to describe an asymptotically flat black object. We argue that a
subset of these correspond to the near-horizon limit of asymptotically flat
extremal black rings. Using this identification we provide a conjecture for the
exact ``phase diagram'' of extremal vacuum black rings with a connected horizon
in odd spacetime dimensions greater than five.
