Purely coclosed $\mathrm{G}_2$-structures on nilmanifolds -- II

TL;DR

This paper completes the classification of seven-dimensional nilpotent Lie groups with purely coclosed G2-structures, focusing on indecomposable groups with higher step nilpotency, advancing understanding of special geometric structures on nilmanifolds.

Contribution

It extends previous classification results by addressing indecomposable 5- and 6-step nilpotent Lie groups with G2-structures.

Findings

Classified indecomposable 5-step nilpotent Lie groups with G2-structures.

Classified indecomposable 6-step nilpotent Lie groups with G2-structures.

Completed the overall classification of such structures on nilmanifolds.

Abstract

This paper completes the classification of seven-dimensional nilpotent Lie groups endowed with a left-invariant purely coclosed $\text{G}_2$-structure, initiated by the first-named author and collaborators. In this previous work, the authors provided the classification of decomposable seven-dimensional nilpotent Lie groups and of the indecomposable ones up to step $4$ of nilpotency. Here, we address the case of indecomposable $5$- and $6$-step nilpotent Lie groups.