| We generalize a construction by Grenier of exact fundamental domains for the action of GLn( Z ) on GLn( R )/O(n) which are well-suited to spectral theory and harmonic analysis. In the generalization, GLn( R )/O(n) with its standard Iwasawa coordinates, is replaced by a manifold X equipped with a coordinate system &phis;. Further, GLn( Z ) is replaced with a group Gamma with an action on X such that the action satisfies certain axioms with respect to &phis;. We show that if G is one of the groups GLn( R ), Gamma = GLn( Z ) the full group of integer points in the standard representation, then the action of r on the symmetric space X = G/ K satisfies the axioms with respect to the standard Iwasawa coordinates. We make brief comments about how to extend this argument to the cases of G = GLn( C ), SLn( R ), SLn( C ), or SO3( C ) and Gamma = G( o ). We conclude the first chapter by making the conjecture that the axioms are satisfied whenever G is the R points of a Chevalley group and Gamma = G( Z ) the group of integer points of G. The second chapter is devoted to proving the following result special to one case of the Grenier domain. Let Gamma = SL2( Z [i]), F the Picard fundamental domain for Gamma, T the standard fundamental domain for Gammainfinity containing F . We prove that the tube T=FGinfinity has a tiling by Gamma-translates of F . This means that there is a covering of T by Gamma-translates of F such that the Gamma-translates in the covering stay within T . We provide a counter-example to the most natural generalization of this fact in the case of Gamma = SL3( Z ), and raise the question as to what the correct generalization should be. As is indicated at several points in the dissertation, particular motivation for all of these investigations is provided by the heat kernel analysis project of Jorgenson-Lang. |
