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 $n = 3, 4, 5$.