  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.
