Graphs and Metric 2-step Nilpotent Lie Algebras

Dani and Mainkar introduced a method for constructing a 2-step nilpotent Lie algebra ${\mathfrak{n}}_G$ from a simple directed graph $G$ in 2005. There is a natural inner product on ${\mathfrak{n}}_G$ arising from the construction. We study geometric properties of the associated simply connected 2-step nilpotent Lie group $N$ with Lie algebra ${\mathfrak{n}}_G$. We classify singularity properties of the Lie algebra ${\mathfrak{n}}_G$ in terms of the graph $G$. A comprehensive description is given of graphs $G$ which give rise to Heisenberg-like Lie algebras. Conditions are given on the graph $G$ and on a lattice $\mathrm{\Gamma }\subseteq N$ for which the quotient $\mathrm{\Gamma }\mathrm{\backslash }N$, a compact nilmanifold, has a dense set of smoothly closed geodesics.