  We introduce two new families of solutions to the vacuum Einstein equations
with negative cosmological constant in 5 dimensions. These solutions are static
black holes whose horizons are modelled on the 3-geometries nilgeometry and
solvegeometry. Thus the horizons (and the exterior spacetimes) can be foliated
by compact 3-manifolds that are neither spherical, toroidal, hyperbolic, nor
product manifolds, and therefore are of a topological type not previously
encountered in black hole solutions. As an application, we use the
solvegeometry solutions to construct Bianchi VI$_{-1}$ braneworld cosmologies.
