I am new to the study of lie groups, nilmanifolds etc. but the following question can be described very basically I think.
Take a complex Heisenberg group $G$
$ \left( \begin{array}{ccc} 1 & \mathbb{C} & \mathbb{C} \\ 0 & 1 & \mathbb{C} \\ 0 & 0 & 1 \end{array} \right)$
and the action on $G$ by matrix multiplication of the discrete subgroup $\Gamma$ of matrices like above with entries in $\mathbb{Z}[i]$ (the gaussian integers). Its convenient to denote the matrix
$ \left( \begin{array}{ccc} 1 & x & y \\ 0 & 1 & z \\ 0 & 0 & 1 \end{array} \right)$
by $(x,y,z)$ and multiplication by $(a,b,c)\cdot(x,y,z) = (a+x,b+y+az, c+z)$.
Now my question is to understand the geometry of the quotient space $G/\Gamma$. The equivalence classes or orbits of this action are elements of the form $(z,\omega +(n+im)\gamma, \gamma)$, where $z,\omega, \gamma \in \mathbb{C}$ and $n,m\in \mathbb{N}$ and the real and imaginary parts of the three coordinates are between 0 and 1.
Question: Describe somehow the fundamental domain of this action (if that is even possible geometrically, anything otherwise is also helpful).