# Exact $G_2$-structures on unimodular Lie algebras

## TL;DR

This paper investigates seven-dimensional unimodular Lie algebras with exact $G_2$-structures, especially those with vanishing third Betti number, and proves non-existence results for certain classes, impacting the search for compact examples.

## Contribution

It classifies and analyzes exact $G_2$-structures on unimodular Lie algebras, proving non-existence of such structures on (2,3)-trivial strongly unimodular Lie algebras.

## Key findings

- No exact $G_2$-structures on (2,3)-trivial strongly unimodular Lie algebras.
- Examples of solvable unimodular Lie algebras with exact $G_2$-structures when $b_2
eq 0$.
- Absence of compact manifolds with exact $G_2$-structures arising from these Lie groups.

## Abstract

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 $G$.