G_2-structures on Einstein solvmanifolds

We study the $G_2$ analogue of the Goldberg conjecture on non-compact solvmanifolds, i.e. on simply conneted solvable Lie groups no having a lattice. In contrast to the almost-K\"ahler case we prove that a 7-dimensional solvmanifold cannot admit any left-invariant calibrated $G_2$-structure $\varphi$ such that the induced metric $g_{\varphi}$ is Einstein, unless $g_{\varphi}$ is flat. Moreover, we show that a 7-dimensional (non-flat) Einstein solvmanifold $(S,g)$ of rank one cannot admit any left-invariant cocalibrated $G_2$-structure $\varphi$ such that the induced metric $g_{\varphi} = g$.