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Journal of Lie Theory 26 (2016), No. 2, 479--495
Copyright Heldermann Verlag 2016

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

Allie Ray
Mathematics Department, University of Texas at Arlington, Box 19408, Arlington, TX 76019-0408, U.S.A.

We associate a two-step nilpotent Lie algebra to an arbitrary Schreier graph. We then use properties of the Schreier graph to determine necessary and sufficient conditions for this Lie algebra to extend to a three-step nilpotent Lie algebra. As an application, if we start with pairs of non-isomorphic Schreier graphs coming from Gassmann-Sunada triples, we prove that 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.

Keywords: Metric Nilpotent Lie Algebras, Schreier Graphs, Gassmann-Sunada Triples.

MSC: 05C99, 17B30, 22E25

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