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All known examples of homogeneous Einstein spaces with negative scalar curvature (non compact) are isometric to standard Einstein solvmanifolds . we prove that any nilpotent Lie algebra having a codimension-one abelian ideal is the nilradical of a rank –one Einstein solvmanifold . In other words this nilpotent Lie algebra admits a rank-one solvable extension which can be endowed with an Einstein left invariant Riemannian metric . also a curve of pairwise non-isometric 8-dimensional rank-one Einstein solvmanifold is given .
 

In this thesis, we review the proof to standardness of Einstein solvamanifolds which is based on some results from Geometric Invariant Theory and stratification of topological spaces. Standardness is a very simple and yet powerful algebraic condition on the lie algebra of a solvmanifold which yields to remarkable existence and uniqueness and obstruction results