Einstein metrics: Homogeneous solvmanifolds, generalised Heisenberg groups and Black Holes

In this paper we construct Einstein spaces with negative Ricci curvature in various dimensions. These spaces -- which can be thought of as generalised AdS spacetimes -- can be classified in terms of the geometry of the horospheres in Poincare-like coordinates, and can be both homogeneous and static. By using simple building blocks, which in general are homogeneous Einstein solvmanifolds, we give a general algorithm for constructing Einstein metrics where the horospheres are any product of generalised Heisenberg geometries, nilgeometries, solvegeometries, or Ricci-flat manifolds. Furthermore, we show that all of these spaces can give rise to black holes with the horizon geometry corresponding to the geometry of the horospheres, by explicitly deriving their metrics.