Minimal Height fundamental Domains for Arithmetic Groups

We generalize a construction by D. Grenier of exact
fundamental domains for the action of $\mathrm{GL}(n,\mathbb{Z})$
on $\mathrm{GL}(n,\mathbb{R})/K$ which are well-suited to spectral
theory and harmonic analysis. In the generalization,
$\mathrm{GL}(n,\mathbb{R})/K$, with its standard Iwasawa
coordinates, is replaced by a manifold $X$ equipped with a
coordinate system $\phi$. Further, $\mathrm{GL}(n,\mathbb{Z})$ is
replaced with a group $\Gamma$ with an action on $X$ such that the
action satisfies certain axioms with respect to $\phi$. We show
that if $G$ is one of the groups $\mathrm{GL}(n,\mathbb{C})$,
$\Gamma=\mathrm{GL}(n,\mathbb{Z}[\mathbf{i}])$ the full group of
integer points in the standard representation, then the action of
$\Gamma$ 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 case of
$\mathrm{SO}(3,\mathbb{C})$ and
$\Gamma=\mathrm{SO}(3,\mathbb{Z}(\mathbf{i}))$. We make the
conjecture that the axioms are satisfied whenever $G$ is the
$\mathbb{R}$ points of a Chevalley group and
$\Gamma=G(\mathbb{Z})$ the group of integer points of $G$. We
make some comments concerning one possible motivation for this work, from
the theory of theta relations associated to arithmetic quotients.