  Among closed G2-structures there are two very distinguished classes:
Laplacian solitons and Extremally Ricci-pinched G2-structures. We study the
existence problem and explore possible interplays between these concepts in the
context of left-invariant G2-structures on solvable Lie groups. Also, some
Ricci pinching properties of G2-structures on solvmanifolds are obtained, in
terms of the extremal values and points of the Ricci pinching functional
F=scal/|Ric|. Many natural open problems have been included.
