Many relationships exist between the geometry of a left-invariant Riemannian metric on a Lie group G and the Lie group structure of G . Thus it is natural to ask whether there exists a ´best´´ metric on a given Lie group and, if so, how does its geometry relate to the Lie group structure? We define a left-invariant Riemannian metric g on a Lie group G to be maximally symmetric if the isometry group of any other left-invariant metric on G is contained in that of g (or, more precisely, that of Phi^*g for some automorphism Phi of G). We prove the following: If a solvable Lie group admits a left- invariant Einstein metric g of negative Ricci curvature, then g is maximally symmetric. The proof is a blend of Lie group structure theory and geometric invariant theory, building on the deep work of J. Heber, J. Lauret and Y. Nikolayevsky concerning Einstein solvmanifolds.

The longstanding Alekseevskii conjecture asks whether every homogeneous Einstein manifold M of negative Ricci curvature is diffeomorphic to R^n; a slightly strengthened version asks whether every such M can be realized as a simply-connected solvable Lie group with a left-invariant Riemannian metric. The theorem lends philosophical support to the Alekseevskii conjecture, since any counterexample to the conjecture would not be maximally symmetric. In fact M. Jablonski and P. Petersen proved that any counterexample with semisimple isometry group would essentially be minimally symmetric!

This work is joint with Michael Jablonski.