Complex Hyperbolic Elliptics Preserving Lagrangian Planes

Abstract

We prove that a regular elliptic isometry $f$ of complex hyperbolic space $\mathbf{H}_{\mathbb{C}}^2$ preserves a Lagrangian plane through its fixed point as a non-involution if and only if $f$ is real elliptic. In this case, the isometry $f$ actually preserves a continuous one-parameter family of Lagrangian planes through the fixed point. The boundaries of these planes form a torus $\mathbb{T}^2_f \subset \partial \mathbf{H}_{\mathbb{C}}^2$, called the fixed torus of $f$. For torsion $f$, we show that all Ford domains of $\langle f \rangle$ with respect to the extended Cygan metric and centred on $\mathbb{T}^2_f$ admit the same explicit cellular structure. As an application, we classify all discrete and faithful complex hyperbolic $(n,\infty,\infty)$-triangle groups for $n = 3, 4, 5$.

Figures (4)

• Figure 1: The standard torus $\mathbb{T}^2$ is foliated by the circles $\{C_q \mid q \in \mathbb{T}^2\}$, which can be explicitly parameterised as $\mathbb{T}^2 = \bigsqcup_{\phi \in [0, 2\pi)} C_{q(\phi)}$ with $q(\phi)\coloneqq [-(3+2\sqrt{2}),(2+\sqrt{2})e^{i \phi},1]^t$. The left side of this figure shows the leaves $C_{q(\phi)}$ for $\phi \in \{k \pi / 16 \mid k = 0, 1, \ldots, 31\}$. Now, let $T$ be the Heisenberg translation sending $q_0 = [-1-2i, \sqrt{2}, 1]^t \in \mathbb{T}^2$ to $[0,0] \in \mathcal{N}$. The right side of this figure displays the images of the leaves $C_{q(\phi)}$ under $H_2 \cdot T$. In particular, the leaf $C_{q(0)}$ containing $q_0$ is mapped to the one-point compactification of an affine line.
• Figure 2: We list nine different selections of $(\theta_1, \theta_2, \theta_3)$. In the corresponding figures, parameterised by $(\psi_1, \psi_2) \in (0, 2\pi)^2$ (with $\psi_1$ on the horizontal axis and $\psi_2$ on the vertical axis), the intersection $I(\theta_1) \cap I(\theta_2)$ is a disk bounded by the provided circle. Although the locus of $Q(X,Y)=0$ varies significantly across configurations, where $X=\cot(\psi_1/2)$ and $Y=\cot(\psi_2/2)$, it always forms a crossing intersection with the disk $W(X,Y) \le 0$. Integer grids are also provided in each figure.
• Figure 3: We present three distinct choices for the pair $(\theta_1, \theta_2)$. For each, the intersection $I(\theta_1) \cap I(\theta_2)$ forms a disk, bounded by the displayed circle and visualized in coordinates parameterised by $(\psi_1, \psi_2) \in (0, 2\pi)^2$ (with $\psi_1$ horizontal and $\psi_2$ vertical). For each pair, we provode the loci of $Q_{\theta_1,\theta_2,\theta_3}(X,Y)=0$ for $\theta_3 \in \{k\pi/16 \mid k=0, 1, ..., 31\}$. Singular loci are plotted as solid curves, while generic loci are dotted. Integer grids are also provided in each figure.
• Figure 4: The functions $\rho_{j',j,k}(t)$ are plotted for $n=3,4,5$ and $n=6$. In each figure, the horizontal axis represents $t \in (0,\pi)$ and we display all functions $\rho_{j',j,k}(t)$ satisfying $1 \le k < \lfloor 2/\sin(\pi/n) \rfloor$, except for the cases $k = 1$ and $j'+j=n$, which may overlap. Dotted lines indicate $\rho = 0$ (horizontal) and $t = \tan(\pi/n)$ (vertical). An inset in the lower right corner shows a zoomed-in view of a very small region centred at the point $(\tan(\pi/n),0)$.

Theorems & Definitions (116)

• Theorem A
• Theorem B
• Theorem C
• Proposition 1.1
• Remark 1.2
• Theorem D
• Remark 1.3
• Remark 1.4
• Conjecture 1.5: Conjecture 5.1 in Schwartz2002
• Theorem E