Only two examples of extremally Ricci pinched G2-structures can be found in the literature and they are both homogeneous. We study in this paper the existence and structure of such very special closed G2-structures on Lie groups. Strong structural conditions on the Lie algebra are proved to hold. As an application, we ob-tain three new examples of extremally Ricci pinched G2-structures and that they are all necessarily steady Laplacian solitons. The deformation and rigidity of such structures are also studied.
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