Spherical CR uniformization of the magic 3-manifold

Abstract

We show the 3-manifold at infinity of the complex hyperbolic triangle group ${\Delta }_{3,\infty ,\infty ;\infty }$ is the three-cusped "magic" 3-manifold $6_1^3$. We also show the 3-manifold at infinity of the complex hyperbolic triangle group ${\Delta }_{3,4,\infty ;\infty }$ is the two-cusped 3-manifold $m295$ in the Snappy Census, which is a 3-manifold obtained by Dehn filling on one cusp of $6_1^3$. In particular, hyperbolic 3-manifolds $6_1^3$ and $m295$ admit spherical CR uniformizations. These results support our conjecture that the 3-manifold at infinity of the complex hyperbolic triangle group ${\Delta }_{3,n,m;\infty }$ is the one-cusped hyperbolic 3-manifold from the "magic" $6_1^3$ via Dehn fillings with filling slopes $(n-2)$ and $(m-2)$ on the first two cusps of it.

64 pages, 34 figures. Comments are welcome!

References in corpus (2)

• New non-arithmetic complex hyperbolic lattices
• A 1-parameter family of spherical CR uniformizations of the figure eight knot complement

Cited by in corpus (1)

• Slim curves, limit sets and spherical CR uniformisations