Complex Hyperbolic Elliptics Preserving Lagrangian Planes

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 ${\mathbb{T}}_f^2\subset \mathrm{\partial }{\mathbf{H}}_{\mathbb{C}}^2$, called the fixed torus of $f$. For torsion $f$, we show that all Ford domains of $⟨f⟩$ with respect to the extended Cygan metric and centred on ${\mathbb{T}}_f^2$ admit the same explicit cellular structure. As an application, we classify all discrete and faithful complex hyperbolic $(n,\mathrm{\infty },\mathrm{\infty })$-triangle groups for $n=3,4,5$.