Inhomogeneous deformations of Einstein solvmanifolds

For each non-flat, unimodular Ricci soliton solvmanifold $({\mathsf{S}}_0,g_0)$, we construct a one-parameter family of complete, expanding, gradient Ricci solitons that admit a cohomogeneity one isometric action by ${\mathsf{S}}_0$. The orbits of this action are hypersurfaces homothetic to $({\mathsf{S}}_0,g_0)$. These metrics are asymptotic at one end to an Einstein solvmanifold. In the one-parameter family, exactly one metric is Einstein, and exactly one has orbits that are isometric to $({\mathsf{S}}_0,g_0)$.