Overflying Nilpotent Horizons

We study solutions of Einstein equations with negative cosmological constant in five dimensions that describe black holes whose event horizons are homogeneous, anisotropic spaces. We focus on the case where the constant-time slices of the horizon are the Nil geometry, the Thurston geometry associated to the Heisenberg group. For such spaces, we analyze the symmetries both in the asymptotic region and in the near horizon region. We compute the associated conserved charges, which turn out to be finite and admit a sensible physical interpretation. We analyze the thermodynamics of the Nil black hole, and we present a stationary spinning generalization of it in the slowly rotating approximation.