The Moduli Space of Points in the Boundary of Quaternionic Hyperbolic Space

Abstract

Let $\mathcal{F}_1(n,m)$ be the ${\rm PSp}(n,1)$-configuration space of ordered $m$-tuple of pairwise distinct points in the boundary of quaternionic hyperbolic n-space $\partial\mathbf{H}_\mathbb{H}^n$ , i.e., the $m$-tuple of pairwise distinct points in $\partial\mathbf{H}_\mathbb{H}^n$ up to the diagonal action of ${\rm PSp}(n,1)$. In terms of Cartan's angular invariant and cross-ratio invariants, the moduli space of $\mathcal{F}_1(n,m)$ is described by using Moore's determinant. We show that the moduli space of $\mathcal{F}_1(n,m)$ is a real $2m^2-6m+5-\sum^{m-n-1}_{i=1}{m-2 \choose n-1+i}$ dimensional subset of a algebraic variety with the same real dimension when $m>n+1$.