Date of Award

1995

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

First Advisor

Leonard F. Richardson

Abstract

This dissertation arose from efforts to investigate an example which appeared in (G) of a phenomenon which has been considered to be rare: namely, the existence of two discrete cocompact subgroups $\Gamma\sb1$ and $\Gamma\sb2$ in a Lie group G such that $\Gamma\sb1/G$ and $\Gamma\sb2/G$ have the same (unitary) spectrum but $\Gamma\sb1$ is not isomorphic to $\Gamma\sb2.$ This phenomenon may be called representation equivalence of $\Gamma\sb1$ and $\Gamma\sb2$ with $\Gamma\sb1$ non-isomorphic to $\Gamma\sb2.$. In (G) the first known example of this phenomenon in the class of solvable Lie groups was given. In this example G was a specific three-step nilpotent Lie group and two discrete cocompact subgroups $\Gamma\sb1$ and $\Gamma\sb2$ of G such that $\Gamma\sb1$ is representation equivalent to $\Gamma\sb2$ with non-isomorphic to $\Gamma\sb2$ were presented. In the present dissertation we have been able to generalize this example and prove the following result: Let G be any three-step nilpotent Lie algebra with rational structure constants and all coadjoint orbits flat. Then there exist discrete cocompact subgroups $\Gamma\sb1$ and $\Gamma\sb2$ in G such that $\Gamma\sb1$ is representation equivalent to $\Gamma\sb2$ but $\Gamma\sb1$ is not isomorphic to $\Gamma\sb2.$. This theorem contains the example in (G) as a special case and demonstrates that this phenomenon occurs surprisingly often. Chapter 2 contains the proof of this new result. The author investigated the role of flatness of orbits in this phenomenon of non-isomorphic representation equivalence by considering the lowest dimensional example of a nilpotent Lie group with non-flat coadjoint orbits. The author has been able to show that for a large category of discrete cocompact subgroups of this group this phenomenon of non-isomorphic representation equivalence cannot occur. Chapter 3 contains the proof of this result. The author has also proven some short but apparently new structural results for three-step nilpotent Lie groups with one-dimensional center. Chapter 4 contains these. Chapter 1 contains the background material for the work undertaken in Chapters 2, 3 and 4.

Pages

49

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