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 $b_2(\mathfrak{g})\neq0$, and in the case when the Lie algebra $\mathfrak{g}$ 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 $(\Gamma\backslash G,\varphi)$, where $G$ is a seven-dimensional simply connected Lie group with (2,3)-trivial Lie algebra, $\Gamma\subset G$ is a co-compact discrete subgroup, and $\varphi$ is an exact $G_2$-structure on $\Gamma\backslash G$ induced by a left-invariant one on $G$.

中文翻译：

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单模李代数上的精确 $$\hbox {G}_{2}$$-结构

我们考虑七维单模李代数 $\mathfrak{g}$ 承认精确的 $G_2$-结构，将我们的注意力集中在那些第三个 Betti 数 $b_3(\mathfrak{g})$ 消失的结构上。我们讨论一些例子，包括 $b_2(\mathfrak{g})\neq0$ 的情况，以及李代数 $\mathfrak{g}$ 是 (2,3)-平凡的情况，即，当$b_2(\mathfrak{g})$ 和 $b_3(\mathfrak{g})$ 都消失了。这些例子是可解的，如 $b_3(\mathfrak{g})=0$，但它们不是强单模，这是对应于 $\mathfrak{g}$ 的单连通李群上格存在的必要条件。更一般地，我们证明任何七维 (2,3)-平凡的强单模李代数不承认任何精确的 $G_2$-结构。由此可知，没有 $(\Gamma\backslash G,\varphi)$ 形式的紧凑示例，