  The general aim of this paper is to study which are the solvable Lie groups
admitting an Einstein left invariant metric. The space N of all nilpotent Lie
brackets on R^n parametrizes a set of (n+1)-dimensional rank-one solvmanifolds,
containing the set of all those which are Einstein in that dimension. The
moment map for the natural GL(n)-action on N evaluated in a point of N encodes
geometric information on the corresponding solvmanifold, allowing us to use
strong and well-known results from geometric invariant theory. For instance,
the functional on N whose critical points are precisely the Einstein
solvmanifolds is the square norm of this moment map. We also use a
GL(n)-invariant stratification for the space N following essentially a
construction given by F. Kirwan and show that there is a strong interplay
between the strata and the Einstein condition on the solvmanifolds. As
applications, we obtain several examples of graded (even 2-step) nilpotent Lie
algebras which are not the nilradicals of any standard Einstein solvmanifold,
as well as a classification in the 7-dimensional 6-step case and an existence
result for certain 2-step algebras associated to graphs.
