Coclosed G2-structures inducing nilsolitons

Abstract We show obstructions to the existence of a coclosed G 2 {\mathrm{G}_{2}} -structure on a Lie algebra ???? {\mathfrak{g}} of dimension seven with non-trivial center. In particular, we prove that if there exists a Lie algebra epimorphism from ???? {\mathfrak{g}} to a six-dimensional Lie algebra ???? {\mathfrak{h}} , with the kernel contained in the center of ???? {\mathfrak{g}} , then any coclosed G 2 {\mathrm{G}_{2}} -structure on ???? {\mathfrak{g}} induces a closed and stable three form on ???? {\mathfrak{h}} that defines an almost complex structure on ???? {\mathfrak{h}} . As a consequence, we obtain a classification of the 2-step nilpotent Lie algebras which carry coclosed G 2 {\mathrm{G}_{2}} -structures. We also prove that each one of these Lie algebras has a coclosed G 2 {\mathrm{G}_{2}} -structure inducing a nilsoliton metric, but this is not true for 3-step nilpotent Lie algebras with coclosed G 2 {\mathrm{G}_{2}} -structures. The existence of contact metric structures is also studied.