# Fundamental domain of $\operatorname{GL}(n,\mathbb R)$ acted on by $\operatorname{GL}(n, \mathbb Z)$

What is a simple description of a fundamental domain of $\operatorname{GL}(n,\mathbb R)$ acted on by $\operatorname{GL}(n,\mathbb Z)$?

$\operatorname{GL}(n,\mathbb R)$ is the group of all real invertible matrices with matrix multiplication, $\operatorname{GL}(n,\mathbb Z)$ the group of all matrices with integer entries, whose inverses also have integer entries, with matrix multiplication. $h \in \operatorname{GL}(n, \mathbb Z) \subset \operatorname{GL}(n,\mathbb R)$ acts on $g \in \operatorname{GL}(n,\mathbb R)$ by letting $h\cdot g := hg$

Remarks:

A fundamental domain $F$ is a subset of $\operatorname{GL}(n,\mathbb R)$ such that for any $x$ in $\operatorname{GL}(n,\mathbb R)$ there is exactly one $h$ in $\operatorname{GL}(n, \mathbb Z)$ such that $hx \in F$. I'm looking for an as clean as possible description of some $F$ in terms of the matrix entries. Clearly $\operatorname{GL}(n,\mathbb R)$ can be replaced with any set of (possibly not invertible) matrices with $n$ rows (possibly with few or more than n columns). The case with one column and $n=2$ is not too different from finding a fundamental domain of the upper half plane with respect to MÃ¶bius transformations. Also, $\operatorname{GL}$ could have been replaced with $\operatorname{SL}$.

This appears as a very basic question to me, and if it turns out I'm ignorant of some useful tools or theorems I will accept pointers to such. In fact this would be even better than a direct answer to the specific question (since I have many related seemingly basic questions), as long as it helps significantly in answering the specific question.

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You may wish to consult Dave Witte Morris's pre-book on Arithmetic groups, especially chapter 7 where he constructs a fundamental domain for $\operatorname{SL}(n,\mathbb Z)$ via Siegel sets (generalizing the classical case $n=2$ rather directly). In our research group we had a seminar on an earlier version about 10 years ago and it was already quite good at that point. I'm sure the quality hasn't decreased... – t.b. Jul 17 '12 at 19:04
@all: concerning the (lattices-in-lie-groups) tag, I was unsure how to tag this, so I decided to create this one (even if it's not a perfect fit here). Feel free to remove it or replace it if you can think of a better one -- I was even less happy with (arithmetic-groups) due to the possible confusion with $\mathbb{Z}/(n)$, say. – t.b. Jul 17 '12 at 19:15