  In this paper we construct negatively curved Einstein spaces describing
gravitational waves having a solvegeometry wave-front (i.e., the wave-fronts
are solvable Lie groups equipped with a left-invariant metric). Using the
Einstein solvmanifolds (i.e., solvable Lie groups considered as manifolds)
constructed in a previous paper as a starting point, we show that there also
exist solvegeometry gravitational waves. Some geometric aspects are discussed
and examples of spacetimes having additional symmetries are given, for example,
spacetimes generalising the Kaigorodov solution. The solvegeometry
gravitational waves are also examples of spacetimes which are indistinguishable
by considering the scalar curvature invariants alone.
