**Speaker:** Jiming Ma (Fudan University)

**Abstract:** It is well known that the complex hyperbolic triangle group $\Delta(3,3,4)$ generated by three complex reflections $I_1,I_2,I_3$ in $\mathbf{PU}(2,1)$ has a 1-dimensional moduli space. Deforming the representations from the classical real 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 explanation that the 3-manifold at infinity of $\Delta(3,3,4; \infty)$ is the figure-eight knot complement. In particular, this confirms the conjecture of Falbel-Guilloux-Will. This is joint work with Baohua Xie.

**Time:** 4pm, Friday, October 18, 2024

**Location:** Room 1610 at SIMIS, Block A, No. 657 Songhu Road, Yangpu District, Shanghai