2023/10/09 by Jiming Ma, Ma, Jiming, Baohua Xie +1
Mathematics · Computer Science · #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2310.05408
It is well-known that complex hyperbolic triangle groups Δ(3,3,4) generated by three complex reflections I1,I2,I3 in PU(2,1) has 1-dimensional moduli space. Deforming the representations from the classical ℝ-Fuchsian one to Δ(3,3,4; ∞), that is, when I3I2I1I2 is accidental parabolic, the 3-manifolds at infinity change from a Seifert 3-manifold to the figure-eight knot complement. When I3I2I1I2 is loxodromic, there is an open set Ω⊂ ∂\mathbf H2\mathbb C=\mathbb S3 associated to I3I2I1I2, which is a subset of the discontinuous region. We show the quotient space Ω/ Δ(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 Δ(3,3,4; ∞) is the figure-eight knot complement. In particular, this confirms a conjecture of Falbel-Guilloux-Will.