Figure-eight knot is always over there

Abstract

It is well-known that complex hyperbolic triangle groups $\Delta (3,3,4)$ generated by three complex reflections $I_1,I_2,I_3$ in $\mbox{PU(2,1)}$ has 1-dimensional moduli space. Deforming the representations from the classical $\mathbb{R}$-Fuchsian one to $\Delta (3,3,4;\infty )$, that is, when $I_3{I}_2{I}_1{I}_2$ is accidental parabolic, the 3-manifolds at infinity change from a Seifert 3-manifold to the figure-eight knot complement. When $I_3{I}_2{I}_1{I}_2$ is loxodromic, there is an open set $\Omega \subset \partial {\mathbf{H}}_{\mathbb{C}}^2={\mathbb{S}}^3$ associated to $I_3{I}_2{I}_1{I}_2$, which is a subset of the discontinuous region. We show the quotient space $\Omega /\Delta (3,3,4)$ is always the figure-eight knot complement in the deformation process. This gives the topological/geometrical explain that the 3-manifold at infinity of $\Delta (3,3,4;\infty )$ is the figure-eight knot complement. In particular, this confirms a conjecture of Falbel-Guilloux-Will.