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 $I_3I_2I_1I_2$ is accidental parabolic, the 3-manifolds at infinity change from a Seifert 3-manifold to the figure-eight knot complement. When $I_3I_2I_1I_2$ is loxodromic, there is an open set $\Omega \subset \partial\mathbf H^{2}_{\mathbb C}=\mathbb S^3$ associated to $I_3I_2I_1I_2$, 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.