On the Relationship Between Representation Equivalence and Isomorphism of Fundamental Groups of Three-Step Nilmanifolds.

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.