[Paper Review] The Moduli Space of Points in the Boundary of Quaternionic Hyperbolic Space

Goashun Gou, Yue-Ping Jiang|Osaka City University (Osaka City University)|Dec 26, 2017

Algebraic and Geometric Analysis13 references3 citations

TL;DR

This paper constructs and describes the moduli space of ordered m-tuples (m ≥ 4) of distinct points in the boundary of quaternionic hyperbolic space up to isometry, using Moore’s determinant to define invariants—specifically quaternionic cross-ratios and Cartan’s angular invariants—thereby establishing a real algebraic variety of dimension $2m^2 - 6m + 5 - inom{m-2}{n-1} - inom{m-2}{n} - \cdots - inom{m-2}{m-n-1}$ when $m > n+1$. The approach provides a complete parameterization via normalized special-Gram quaternionic matrices.

ABSTRACT

Let $\mathcal{F}_1(n,m)$ be the space of ordered m-tuples of pairwise distinct points in $\partial \mathbf{H}_{\mathbb{H}}^n$ up to its isometry group $PSp(n,1)$. It is a real $2m^2-6m+5-\sum^{m-n-1}_{i=1}{m-2 \choose n-1+i}$ dimensional algebraic variety when $m>n+1$. In this paper, we construct and describe the moduli space of $\mathcal{F}_1(n,m)$, in terms of the Cartan's angle and cross-ratio invariants, by applying the Moore's determinant.

Motivation & Objective

• To construct and describe the moduli space of $\mathcal{F}_1(n,m)$, the space of ordered $m$-tuples of distinct points in $\partial\mathbf{H}_\mathbb{H}^n$ modulo $PSp(n,1)$-action.
• To extend the theory of moduli spaces from complex to quaternionic hyperbolic geometry, particularly for $m > n+1$.
• To establish a parameterization of the moduli space using invariants derived from Moore’s determinant and special-Gram quaternionic matrices.
• To analyze the deformation space of discrete, faithful, totally loxodromic representations into $PSp(2,1)$ using geometric invariants.
• To provide a complete algebraic description of the moduli space as a real algebraic variety with explicit dimension formula.

Proposed method

• Define normalized special-Gram quaternionic matrices as a canonical representative for each $PSp(n,1)$-orbit of $m$-tuples of distinct boundary points.
• Use Moore’s determinant to define invariants for quaternionic matrices, enabling the construction of cross-ratio and Cartan’s angular invariants.
• Parameterize the moduli space via the quaternionic cross-ratio $\mathbb{X}(p_1,p_2,p_3,p_4) = \langle\mathbf{p}_3,\mathbf{p}_1\rangle_1 \langle\mathbf{p}_3,\mathbf{p}_2\rangle_1^{-1} \langle\mathbf{p}_4,\mathbf{p}_2\rangle_1 \langle\mathbf{p}_4,\mathbf{p}_1\rangle_1^{-1}$ for quadruples of isotropic points.
• Define the quaternionic Cartan’s angular invariant $\mathbb{A}_H(p_1,p_2,p_3) = \arccos\left( \frac{\Re(-\langle\mathbf{p}_1,\mathbf{p}_2,\mathbf{p}_3\rangle_1)}{|\langle\mathbf{p}_1,\mathbf{p}_2,\mathbf{p}_3\rangle_1|} \right)$ for triples, where $\langle\mathbf{p}_1,\mathbf{p}_2,\mathbf{p}_3\rangle_1 = \langle\mathbf{p}_1,\mathbf{p}_2\rangle_1 \langle\mathbf{p}_2,\mathbf{p}_3\rangle_1 \langle\mathbf{p}_3,\mathbf{p}_1\rangle_1$.
• Prove that each $PSp(n,1)$-congruence class corresponds uniquely to a normalized special-Gram matrix, enabling a direct study of the moduli space via matrix invariants.

Experimental results

Research questions

• RQ1 What is the dimension and algebraic structure of the moduli space $\mathcal{F}_1(n,m)$ of $m$-tuples of distinct points in $\partial\mathbf{H}_\mathbb{H}^n$ modulo $PSp(n,1)$-action?
• RQ2 How can invariants such as the quaternionic cross-ratio and Cartan’s angular invariant be used to parameterize the moduli space in quaternionic hyperbolic geometry?
• RQ3 Can Moore’s determinant be effectively used to define invariants for quaternionic matrices in order to describe the moduli space of point configurations?
• RQ4 What is the relationship between the geometry of point configurations and the deformation space of discrete, faithful, totally loxodromic representations into $PSp(2,1)$?
• RQ5 How do the invariants derived from special-Gram matrices capture the full moduli space structure, and what is the role of normalization in achieving uniqueness?

Key findings

• The moduli space $\mathcal{F}_1(n,m)$ is a real algebraic variety of dimension $2m^2 - 6m + 5 - \sum_{i=1}^{m-n-1} \binom{m-2}{n-1+i}$ when $m > n+1$.
• Each $PSp(n,1)$-congruence class of an $m$-tuple of distinct boundary points corresponds uniquely to a normalized special-Gram quaternionic matrix.
• The quaternionic cross-ratio $\mathbb{X}(p_1,p_2,p_3,p_4)$ provides a key invariant for parameterizing configurations of four isotropic points.
• The quaternionic Cartan’s angular invariant $\mathbb{A}_H(p_1,p_2,p_3)$ is well-defined and real-valued for triples of distinct points with $\Re(-\langle\mathbf{p}_1,\mathbf{p}_2,\mathbf{p}_3\rangle_1) > 0$.
• The moduli space of discrete, faithful, totally loxodromic representations into $PSp(2,1)$ is parameterized by fixed point data $(p_i^+, p_i^-)$, and parameters $(r_i, \beta_i, \theta_i)$, with $r_i > 0$, $r_i \neq 1$, and $0 \leq \beta_i, \theta_i \leq \pi$ for $\mathbb{H}$.
• The conjugation class of each loxodromic element in $PSp(2,1)$ is uniquely determined by the triple $(r, \beta, \theta)$, and the full representation class is determined by the full set of such data for the generators.