Historically the most familiar fundamental domain for P n /GL n (Z) has been that of Minkowski. This paper develops a new fundamental domain more suited to applications in number theory. It is shown that these domains can be determined explicitly for given n and this is done for n = 3,4, 5,6. A reduction algorithm for an arbitrary element of P n is also determined.

We investigate parameters for the symmetric space H = G/K, G = GL{n, R), K = O(n), in the sense of positive definite quadratic forms. This leads to a description for the fundamental domain H/T where Γ is an arithmetic subgroup of G. We also see interesting relations with the Siegel sets. This enables us to explicitly describe Satake compactifcations of H/T. We will also consider the behavior at the "bottom" of the fundamental domains.

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Absrract.We report on a detailed study of the fundamental domain for the special linear group SL(3,Z) of 3 X 3 integral matrices with determinant one. Graphs of points coming from the action of Hecke operators are considered.