Nilpotent Lie Algebras And Nilmanifolds Constructed From Graphs

Abstract

The interaction between graph theory and differential geometry has been studied previously, but S. Dani and M. Mainkar brought a new approach to this study by associating a two-step nilpotent Lie algebra (and thereby a two-step nilmanifold) with a simple graph. We prsesent a new construction that associates a two-step nilpotent Lie algebra to an arbitrary (not necessarily simple) directed edge-labeled graph. We then use properties of a Schreier graph to determine necessary and sufficient conditions for this Lie algebra to extend to a three-step nilpotent Lie algebra.After considering the curvature of the two-step nilmanifolds associated with the graphs, we show that if we start with pairs of non-isomorphic Schreier graphs coming from Gassmann-Sunada triples, the pair of associated two-step nilpotent Lie algebras are always isometric. In contrast, we use a well-known pair of Schreier graphs to show that the associated three-step nilpotent extensions need not be isometric.