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We associate a two-step nilpotent Lie algebra to an arbitrary Schreier graph.We then use properties of the Schreier graph to determine necessary andsufficient conditions for this Lie algebra to extend to a three-step nilpotentLie algebra. As an application, if we start with pairs of non-isomorphicSchreier graphs coming from Gassmann-Sunada triples, we prove that the pair ofassociated two-step nilpotent Lie algebras are always isometric. In contrast,we use a well-known pair of Schreier graphs to show that the associatedthree-step nilpotent extensions need not be isometric.

Two-Step and Three-Step Nilpotent Lie Algebras Constructed from Schreier Graphs