On the moduli space of quadruples of points in the boundary of complex hyperbolic space

We consider the space $\mathcal{M}$ of ordered quadruples of distinct points in the boundary of complex hyperbolic $n$-space, $\mathrm{ch}n,$ up to its holomorphic isometry group $\mathrm{P}\mathrm{U}(n,1)\mathrm{.}$ One of the important problems in complex hyperbolic geometry is to construct and describe a moduli space for $\mathcal{M}$. For $n=2$, this problem was considered by Falbel, Parker, and Platis. The main purpose of this paper is to construct a moduli space for $\mathcal{M}$ for any dimension $n\ge 1$. The major innovation in our paper is the use of the Gram matrix instead of a standard position of points.