2015/01/07 by Ray, Allie
#05C99 #17B30 #22E25 #Combinatorics (math.CO) #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1501.01605
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.