Graduation Semester and Year

2015

Language

English

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematics

Department

Mathematics

First Advisor

Ruth Gornet

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.

Disciplines

Mathematics | Physical Sciences and Mathematics

Comments

Degree granted by The University of Texas at Arlington

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Included in

Mathematics Commons

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