Exact G2-structures on unimodular Lie algebras

We consider seven-dimensional unimodular Lie algebras $mathfrak{g}$ admitting exact G$_2$-structures, focusing our attention on those with vanishing third Betti number $b_3(mathfrak{g})$. We discuss some examples, both in the case when $b_2(rg) eq0$ and in the case when the Lie algebra $rg$ is (2,3)-trivial, i.e., when both $b_2(mathfrak{g})$ and $b_3(mathfrak{g})$ vanish. These examples are solvable, as $b_3(mathfrak{g})=0$, but they are not strongly unimodular, a necessary condition for the existence of lattices on the simply connected Lie group corresponding to $mathfrak{g}$. More generally, we prove that any seven-dimensional (2,3)-trivial strongly unimodular Lie algebra does not admit any exact G$_2$-structure. From this, it follows that there are no compact examples of the form $(GammaackslashG,arphi)$, where $G$ is a seven-dimensional simply connected Lie group with (2,3)-trivial Lie algebra, $GammasubsetG$ is a co-compact discrete subgroup, and $arphi$ is an exact $G_2$-structure on $GammaackslashG$ induced by a left-invariant one on $G$.