As we know the figure-8 ($4\_1$) complement can be obtained by quotienting $\mathbb{H}^3$ with an arithmetic Kleinian group, which has index 12 inside $PSL(2,\mathcal{O}\_3)$. The resulting complete hyperbolic manifold has a cusp corresponding to the torus boundary.


On the other hand, the "conformal boundary" (boundary at infinity) of $\mathbb{H}^3$ is the sphere at infinity $S^2\_{\infty}$. For a Kleinian group $\Gamma$, the hyperbolic manifold $\mathbb{H}^3/\Gamma$ inherits a conformal boundary $\Lambda/\Gamma$, where $\Lambda$ is the domain of discontinuity ($S^2\_{\infty}$ with the limit set of $\Gamma$ excluded).


Then **how is the *empty* "conformal boundary" of $4\_1$ complement related to its cusp?** (I know that the conformal structure of the torus boundary is called "cusp shape"); i.e., **should both of them correspond to the $z=0$ "end" in Poincare half-space model?** (since the torus boundary results from the truncated vertices on $S^2\_{\infty}$ of 2 ideal tetrahedra under developing map upon gluing).


**Update:** the limit set of $\Gamma$ here is the entire $S^2\_{\infty}$ (Corollary 11.8 in [Bonahon](https://books.google.com/books?id=F5qIAwAAQBAJ&pg=PA302&lpg=PA302&dq=figure%20eight%20knot%20kleinian%20group%20limit%20set&source=bl&ots=jTdmMINns4&sig=Bu4s-zwx0PmfV4qxhDl7JoT68ic&hl=en&sa=X&ved=0ahUKEwiOvYe57v_aAhVpxlQKHXy6AEEQ6AEIczAJ#v=onepage&q=figure%20eight%20knot%20kleinian%20group%20limit%20set&f=false)), so domain of discontinuity as well as the conformal boundary is $\emptyset$.


**Confusion:** according to section 5 of [Thurston](https://pdfs.semanticscholar.org/721c/f2a9c02111f01da0d6f24775aa3fd40f19a8.pdf), the *entire* boundary of a Kleinian manifold $(\mathbb{H}^3\cup\Lambda)/\Gamma$ is $\Lambda/\Gamma$, which is empty for $4\_1$ complement. Then what is the role of the torus boundary (cusp)?


P.S.: As a physicist, I may not have stated the definition of "conformal boundary" rigorously, but in my mind it is the boundary in the sense of AdS/CFT correspondence (in Euclidean signature, $AdS\_3$ is $\mathbb{H}^3$).



Conformal boundary and cusp of figure-8 complement
The group of parabolic isometries fixing a point at infinity is isomorphic to ${\mathbb C}$. (Because it acts simply transitively on a horosphere $H$.) The discrete group $\Gamma$ intersects this stabilizer of the parabolic fixed point (i.e., the stabilizer of $H$) in a lattice $\Lambda\subset{\mathbb C}$. The quotient $H/\Lambda={\mathbb C}/\Lambda$ is a torus with a conformal structure.


