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

TL;DR

This paper constructs a moduli space for quadruples of distinct boundary points in complex hyperbolic space of any dimension, using Gram matrices to generalize previous work limited to two dimensions.

Contribution

It introduces a novel approach employing Gram matrices to describe the moduli space of quadruples in complex hyperbolic space for all dimensions, extending prior two-dimensional results.

Findings

Developed a new method using Gram matrices for moduli space construction.

Extended the moduli space description from 2D to higher dimensions.

Provided a framework for analyzing boundary point configurations in complex hyperbolic geometry.

Abstract

We consider the space $\mathcal M$ of ordered quadruples of distinct points in the boundary of complex hyperbolic $n$-space, $\ch{n},$ up to its holomorphic isometry group ${\rm PU}(n,1).$ 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 $n \geq 1$. The major innovation in our paper is the use of the Gram matrix instead of a standard position of points.