Graphs and Metric 2-step Nilpotent Lie Algebras

TL;DR

This paper explores the geometric properties of 2-step nilpotent Lie groups constructed from graphs, classifies their singularities, and identifies conditions for dense closed geodesics on associated compact nilmanifolds.

Contribution

It provides a classification of singularity properties of graph-derived Lie algebras and characterizes graphs leading to Heisenberg-like structures and dense geodesic sets.

Findings

Classification of singularity properties based on graph structure

Description of graphs producing Heisenberg-like Lie algebras

Conditions for dense closed geodesics on compact nilmanifolds

Abstract

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 $\Gamma \subseteq N$ for which the quotient $\Gamma \backslash N$, a compact nilmanifold, has a dense set of smoothly closed geodesics.