## LSU Historical Dissertations and Theses

1990

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics

Leonard Richardson

#### Abstract

The compact quotients of three-dimensional solvable non-nilpotent Lie groups by discrete subgroups fall into two categories; those which are quotients of $S\sb{R}$ (semidirect product, R acts on R$\sp2$ by rotation), and those which are quotients of $S\sb{H}$ = $R\ \propto\ R\sp2$ (semidirect product, $R$ acts on $R\sp2$ via translation along a hyperbolic orbit). Continuous functions in the primary summands of $L\sp2$ of each type of solvmanifold are examined. It is shown that there exists a Fejer theorem on solvmanifolds $M\sb{R}$ which are quotients of $S\sb{R}$; if $P\sb{i}$: $L\sp2(M\sb{R})\mapsto$ $L\sp2(M\sb{R})$ is orthogonal projection onto the ith primary summand of such a quotient, there exists a sequence of operators $S\sb{n}$ such that each $S\sb{n}$ is a finite linear combination of the $P\sb{i}$, and such that for each continuous function $f\in L\sp2(M\sb{R}),\ S\sb{n}(f)\to f$ uniformly as $n\to\infty$. It is shown that no such theorem holds for quotients of $S\sb{H}$, for which it is known that orthogonal projections do not preserve continuity of functions. It is shown that if $f\in L\sp2(M\sb{H})$ is continuous and $P\sb{i}(f)$ is essentially bounded, then $P\sb{i}(f)$ must be continuous. Together with a result of L. Richardson, this result implies that there are continuous $f\in L\sp2(M\sb{H})$ for which $P\sb{i}(f)$ is essentially unbounded. It is shown in Section 2 that continuous functions in primary summands of $L\sp2(M\sb{H})$ must vanish on $M\sb{H}$ for all compact quotients of $M\sb{H}$ by discrete subgroups. There are five compact quotients of $S\sb{R}$; one quotient manifold is homeomorphic to the three-dimensional torus, and its continuous primary summand functions need not vanish. For three of the other quotients, it is shown that all continuous primary summand functions need not vanish. For the remaining quotient it is known that functions in certain subspaces of the primary summands must vanish, but it is not known whether all continuous primary summand functions must vanish in this case.

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